# Gelfand–Phillips type properties of locally convex spaces

Saak Gabriyelyan<EMAIL_ADDRESS>Department of Mathematics, Ben-Gurion
University of the Negev, Beer-Sheva, P.O. 653, Israel

###### Abstract

Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of the
Gelfand–Phillips property and the (coarse) Gelfand–Phillips property of order
$p$ of Banach spaces, we introduce and study different types of the
Gelfand–Phillips property of order $(p,q)$ (the $GP_{(p,q)}$ property) and the
coarse Gelfand–Phillips property of order $p$ in the realm of all locally
convex spaces. We compare these classes and show that they are stable under
taking direct product, direct sums and closed subspaces. It is shown that any
locally convex space is a quotient space of a locally convex space with the
$GP_{(p,q)}$ property. Characterizations of locally convex spaces with the
introduced Gelfand–Phillips type properties are given.

###### keywords:

Gelfand–Phillips property of order $(p,q)$ , coarse Gelfand–Phillips property
of order $p$ , $p$-Gelfand–Phillips sequentially compact property of order
$(q^{\prime},q)$

###### MSC:

[2010] 46A3 , 46E10

##  1 Introduction

All locally convex spaces are assumed to be Hausdorff and over the field
$\mathbb{F}$ of real or complex numbers. We denote by $E^{\prime}$ the
topological dual of a locally convex space (lcs for short) $E$. The
topological dual space $E^{\prime}$ of $E$ endowed with the weak∗ topology
$\sigma(E^{\prime},E)$ and the strong topology $\beta(E^{\prime},E)$ is
denoted by $E^{\prime}_{w^{\ast}}$ and $E^{\prime}_{\beta}$, respectively. For
a bounded subset $B\subseteq E$ and a functional $\chi\in E^{\prime}$, we put

$\|\chi\|_{B}:=\sup\big{\\{}|\chi(x)|:x\in B\cup\\{0\\}\big{\\}}.$

If $E$ is a Banach space, we denote by $B_{E}$ the closed unit ball of $E$.

###### Definition 1.1

A bounded subset $B$ of a Banach space $E$ is called limited if each weak∗
null sequence $\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$ converges to zero
uniformly on $B$, that is $\lim_{n\to\infty}\|\chi_{n}\|_{B}=0$.

Let us recall the classical notion of Gelfand–Phillips spaces.

###### Definition 1.2

A Banach space $E$ is said to have the Gelfand–Phillips property or is a
Gelfand–Phillips space if every limited set in $E$ is relatively compact.

In [28] Gelfand proved that every separable Banach space is Gelfand–Phillips.
On the other hand, Phillips [35] showed that the non-separable Banach space
$\ell_{\infty}$ is not Gelfand–Phillips.

Following Bourgain and Diestel [6], a bounded linear operator $T$ from a
Banach space $L$ into $E$ is called limited if $T(B_{L})$ is a limited subset
of $E$. In [15], Drewnowski noticed the next characterization of Banach spaces
with the Gelfand–Phillips property.

######  Theorem 1.3 ([15])

For a Banach space $E$ the following assertions are equivalent:

  1. (i)

$E$ is Gelfand–Phillips;

  2. (ii)

every limited weakly null sequence in $E$ is norm null;

  3. (iii)

every limited operator with range in $E$ is compact.

This characterization of Gelfand–Phillips Banach spaces plays a crucial role
in many arguments for establishing the Gelfand–Phillips property in Banach
spaces. The Gelfand–Phillips property was intensively studied in particular in
[8, 15, 16, 42, 43, 44]. It follows from results of Schlumprecht [42, 43] that
the Gelfand–Phillips property is not a three space property (see also Theorem
6.8.h in [7]). In our recent paper [3], we give several new characterizations
of Gelfand–Phillips Banach spaces.

Another direction for studying the Gelfand–Phillips property is to
characterize Gelfand–Phillips spaces that belong to some important classes of
Banach spaces. In the next proposition, whose short proof is given in
Corollary 2.2 of [3], we provide some of the most important and general
results in this direction (for all relevant definitions used in what follows
see Section 2).

###### Theorem 1.4

A Banach space $E$ is Gelfand–Phillips if one of the following conditions
holds:

  1. (i)

([3, Cor. 2.2]) the closed unit ball $B_{E^{\prime}}$ of the dual space
$E^{\prime}$ endowed with the weak∗ topology is selectively sequentially
pseudocompact;

  2. (ii)

([28]) $E$ is separable;

  3. (iii)

([8, Prop. 2]) $E$ is separably weak∗-extensible;

  4. (iv)

(cf. [15, Th. 2.2] and [43, Prop. 2]) the space $E^{\prime}_{w^{\ast}}$ is
selectively sequentially pseudocompact at some $E$-norming set $S\subseteq
E^{\prime}$;

  5. (v)

([16, Th. 4.1]) $E=C(K)$ for some compact selectively sequentially
pseudocompact space $K$.

The notion of a limited set in locally convex spaces was introduced by
Lindström and Schlumprecht in [34] and independently by Banakh and Gabriyelyan
in [5]. Since limited sets in the sense of [34] are defined using
equicontinuity, to distinguish both notions we called them in [5] by
${\mathcal{E}}$-limited sets:

###### Definition 1.5

A subset $A$ of a locally convex space $E$ is called

  1. (i)

${\mathcal{E}}$-limited  if $\|\chi_{n}\|_{A}\to 0$ for every equicontinuous
weak∗ null sequence $\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$ ([34]);

  2. (ii)

limited  if $\|\chi_{n}\|_{A}\to 0$ for every weak∗ null sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$ ([5]).

It is clear that if $E$ is a $c_{0}$-barrelled space, then $A$ is limited if
and only if it is ${\mathcal{E}}$-limited, and for Banach spaces both these
notions of limited sets coincide.

Lindström and Schlumprecht [34] proposed the following extention of the
Gelfand–Phillips property to the class of all locally convex spaces.

######  Definition 1.6 ([34])

A locally convex space $E$ is called an ${\mathcal{E}}$-Gelfand–Phillips space
or it has the ${\mathcal{E}}$-Gelfand–Phillips property if every
${\mathcal{E}}$-limited set in $E$ is precompact.

Also in Definition 1.6 we use the notation the
“${\mathcal{E}}$-Gelfand–Phillips property” instead of the “Gelfand–Phillips
property” as in [34] to emphasize the equicontinuity of weak∗ null sequences
in $E^{\prime}$ (as we noticed above this extra-condition can be omitted for
$c_{0}$-barrelled spaces but not in general) and the fact that this notion can
be naturally extended from the case of Banach spaces to the general case not
in a unique way.

Fréchet spaces with the ${\mathcal{E}}$-Gelfand–Phillips property were
thoroughly studied by Alonso in [2]. The following operator characterization
of ${\mathcal{E}}$-Gelfand–Phillips spaces was obtained by Ruess in [39].

######  Theorem 1.7 ([39])

A locally convex space $E$ is an ${\mathcal{E}}$-Gelfand–Phillips space if and
only if for each locally convex (equivalently, Fréchet or Banach) space $L$, a
subset $H$ of the $\epsilon$-product $E\epsilon L$ of $E$ and $L$ is
precompact if and only if

  1. (i)

$\\{T(\chi):T\in H\\}$ is precompact in $L$ for all $\chi\in E^{\prime}$;

  2. (ii)

for any equicontinuous, weak∗ null sequence $\\{\chi_{n}\\}_{n\in\omega}$ in
$E^{\prime}$, it follows that $T(\chi_{n})\to 0$ in $L$ uniformly over all
$T\in H$.

A generalization of the Gelfand–Phillips property to the class of all locally
convex spaces can be done not in a unique way as in Definition 1.6. Being
motivated by the notion of the Josefson–Nissenzweig property introduced and
studied in [4] and as it is explained in [4], the following notion can be
considered as a version of the Josefson–Nissenzweig property which holds
everywhere. We shall say that a subset $A$ of a locally convex space $E$ is
barrel-bounded or barrel-precompact if $A$ is bounded or, respectively,
precompact in the strong topology $\beta(E,E^{\prime})$ on the space $E$.

###### Definition 1.8

A locally convex space $E$ is said to have the $b$-Gelfand–Phillips property
or else $E$ is a $b$-Gelfand–Phillips space if every limited barrel-bounded
subset of $E$ is barrel-precompact.

It is easy to see that a Banach space $E$ has the Gelfand–Phillips property if
and only if it is $b$-Gelfand–Phillips in the sense of Definition 1.8 (where
the prefix “$b$-” is added to emphasize the “barreled” nature of the notion);
and if $E$ is a barrelled space, then $E$ has the $b$-Gelfand–Phillips
property if and only if it is an ${\mathcal{E}}$-Gelfand–Phillips space or a
Gelfand–Phillips space in the sense of Lindström and Schlumprecht [34].

Let $E$ and $H$ be locally convex spaces. Denote by $\mathcal{L}(E,H)$ the
family of all operators (= continuous linear maps) from $E$ to $H$. Let
$p\in[1,\infty]$. Recall that a sequence $\\{x_{n}\\}_{n\in\omega}$ in $E$ is
called weakly $p$-summable if for every $\chi\in E^{\prime}$ it follows that
$(\langle\chi,x_{n}\rangle)\in\ell_{p}$ if $p\in[1,\infty)$ and
$(\langle\chi,x_{n}\rangle)\in c_{0}$ if $p=\infty$. The families
$\ell_{p}^{w}(E)$ and $c_{0}^{w}(E)$ of all weakly $p$-summable sequences in
$E$ are vector spaces which admit natural locally convex vector topologies
such that they are complete if so is $E$, for details see Section 19.4 in [32]
or Section 4 in [22].

Unifying the notion of unconditional convergent (u.c.) operator and the notion
of completely continuous operators (i.e., they transform weakly null sequences
into norm null), Castillo and Sánchez selected in [9] the class of
$p$-convergent operators. An operator $T:X\to Y$ between Banach spaces $X$ and
$Y$ is called $p$-convergent if it transforms weakly $p$-summable sequences
into norm null sequences. Using this notion they introduced and study Banach
spaces with the Dunford–Pettis property of order $p$ ($DPP_{p}$ for short) for
every $p\in[1,\infty]$. A Banach space $X$ is said to have the $DPP_{p}$ if
every weakly compact operator from $X$ into a Banach space $Y$ is
$p$-convergent.

The influential article of Castillo and Sánchez [9] inspired an intensive
study of $p$-versions of numerous geometrical properties of Banach spaces. In
particular, the following $p$-versions of limitedness were introduced by Karn
and Sinha [33] and Galindo and Miranda [27].

###### Definition 1.9

Let $p\in[1,\infty]$, and let $X$ be a Banach space. A bounded subset $A$ of
$X$ is called

  1. (i)

a $p$-limited set if

$\big{(}\sup_{a\in
A}|\langle\chi_{n},a\rangle|\big{)}\in\ell_{p}\;\;\Big{(}\mbox{or
}\big{(}\sup_{a\in A}|\langle\chi_{n},a\rangle|\big{)}\in c_{0}\;\mbox{ if
}p=\infty\Big{)}$

for every $(\chi_{n})\in\ell_{p}^{w}\big{(}X^{\ast},\sigma(X^{\ast},X)\big{)}$
(or $(\chi_{n})\in c_{0}^{w}\big{(}X^{\ast},\sigma(X^{\ast},X)\big{)}$ if
$p=\infty$) ([33]);

  2. (ii)

a coarse $p$-limited set if for every $T\in\mathcal{L}(E,\ell_{p})$ $($or
$T\in\mathcal{L}(E,c_{0})$ if $p=\infty$$)$, the set $T(A)$ is relatively
compact ([27]).

Now, it is natural to define the (coarse) $p$-version of the Gelfand–Phillips
property of Banach spaces as follows.

###### Definition 1.10

Let $p\in[1,\infty]$. A Banach space $X$ is said to have

  1. (i)

the $p$-Gelfand–Phillips property (the $GP_{p}$ property) if every limited
weakly $p$-summable sequence in $X$ is norm null ([18]);

  2. (ii)

the coarse $p$-Gelfand–Phillips property (the coarse $GP_{p}$ property) if
every coarse $p$-limited set in $E$ is relatively compact ([27]).

Taking into account Theorem 1.3 it is easy to see that a Banach space $X$ has
the $GP_{\infty}$ property if and only if it has the $GP$ property. Numerous
results concerning the Gelfand–Phillips type properties were obtained by
Deghhani at al. [12], Fourie and Zeekoei [17], Ghenciu [29, 30, 31], and
Galindo and Miranda [27].

The classes of limited, ${\mathcal{E}}$-limited, $p$-limited and coarse
$p$-limited sets can be defined in any locally convex space as follows.

######  Definition 1.11 ([24])

Let $1\leq p\leq q\leq\infty$. A non-empty subset $A$ of a locally convex
space $E$ is called

  1. (i)

a $(p,q)$-limited set (resp., $(p,q)$-${\mathcal{E}}$-limited set) if

$\Big{(}\|\chi_{n}\|_{A}\Big{)}\in\ell_{q}\;\mbox{ if $q<\infty$, }\;\mbox{ or
}\;\;\|\chi_{n}\|_{A}\to 0\;\mbox{ if $q=\infty$},$

for every (resp., equicontinuous) weak∗ $p$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$;

  2. (ii)

a coarse $p$-limited set if for every $T\in\mathcal{L}(E,\ell_{p})$ $($or
$T\in\mathcal{L}(E,c_{0})$ if $p=\infty$), the set $T(A)$ is relatively
compact.

We denote by $\mathsf{L}_{(p,q)}(E)$ and $\mathsf{EL}_{(p,q)}(E)$ the family
of all $(p,q)$-limited subsets and all $(p,q)$-${\mathcal{E}}$-limited subsets
of $E$, respectively. $(p,p)$-(${\mathcal{E}}$-)limited sets and
$(\infty,\infty)$-(${\mathcal{E}}$-)limited sets will be called simply
$p$-(${\mathcal{E}}$-)limited sets and (${\mathcal{E}}$-)limited sets,
respectively. The family of all coarse $p$-limited sets is denoted by
$\mathsf{CL}_{p}(E)$.

Taking into consideration that the classes of precompact, sequentially
precompact, relatively sequentially compact and relatively compact subsets of
a locally convex space are distinct in general, one can naturally define the
following types of the Gelfand–Phillips property generalizing the
corresponding notions from Definitions 1.2, 1.6, 1.8 and 1.10.

###### Definition 1.12

Let $1\leq p\leq q\leq\infty$. A locally convex space $E$ is said to have

  1. (i)

a $(p,q)$-Gelfand–Phillips property  (a precompact $(p,q)$-Gelfand–Phillips
property, a sequential $(p,q)$-Gelfand–Phillips property or a sequentially
precompact $(p,q)$-Gelfand–Phillips property) if every $(p,q)$-limited set in
$E$ is relatively compact (resp., precompact, relatively sequentially compact
or sequentially precompact); for short, the $GP_{(p,q)}$ property (resp., the
$prGP_{(p,q)}$ property, the $sGP_{(p,q)}$ property or the $spGP_{(p,q)}$
property);

  2. (ii)

a $(p,q)$-equicontinuous Gelfand–Phillips property  (a precompact
$(p,q)$-equicontinuous Gelfand–Phillips property, a sequential
$(p,q)$-equicontinuous Gelfand–Phillips property or a sequentially precompact
$(p,q)$-equicontinuous Gelfand–Phillips property) if every
$(p,q)$-${\mathcal{E}}$-limited set in $E$ is relatively compact (resp.,
precompact, relatively sequentially compact or sequentially precompact); for
short, the $EGP_{(p,q)}$ property (resp., the $prEGP_{(p,q)}$ property, the
$sEGP_{(p,q)}$ property or the $spEGP_{(p,q)}$ property);

  3. (iii)

a $b$-$(p,q)$-Gelfand–Phillips property  (a $b$-$(p,q)$-equicontinuous
Gelfand–Phillips property ) if every $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) set in $E$ is barrel-precompact; for short,
the $b$-$GP_{(p,q)}$ property or the $b$-$EGP_{(p,q)}$ property, respectively;

  4. (iv)

a coarse $p$-Gelfand–Phillips property  (a coarse precompact
$p$-Gelfand–Phillips property, a coarse sequential $p$-Gelfand–Phillips
property or a coarse sequentially precompact $p$-Gelfand–Phillips property) if
every coarse $p$-limited set in $E$ is relatively compact (resp., precompact,
relatively sequentially compact or sequentially precompact); for short, the
coarse $GP_{p}$ property (resp., the coarse $prGP_{p}$ property, the coarse
$sGP_{p}$ property or the coarse $spGP_{p}$ property).

In the case $q=p$ we shall write only one subscript $p$, and if $q=p=\infty$
the subscripts will be omitted (so that we say that $E$ has the $GP_{p}$
property or the $GP$ property etc., respectively).

The main purpose of the article is to study locally convex spaces with
Gelfand–Phillips type properties introduced in Definition 1.12. Now we
describe the content of the article.

In Section 2 we fix basic notions and prove some necessary results used in the
article. It is a well known result due to Odell and Stegall (see [37, p. 377])
that any limited set in a Banach space is weakly sequentially precompact. In
Theorem 2.6 we essentially generalize this result using the same idea. By the
celebrated Rosenthal $\ell_{1}$-theorem, any bounded sequence in a Banach
space $E$ has a weakly Cauchy subsequence if and only if $E$ has no an
isomorphic copy of $\ell_{1}$. The remarkable result of Ruess [40] shows that
an analogous statement holds true for much wider classes of locally convex
spaces. These facts motive to introduce the weak Cauchy subsequence property
of order $p$ ($wCSP_{p}$ for short) for any $p\in[1,\infty]$, see Definition
2.7. In Proposition 2.9 we prove that if $1<p<\infty$, then $\ell_{p}$ has the
$wCSP_{p}$ if and only if $p\geq 2$. We also show that the class of locally
convex spaces with the $wCSP_{p}$ is stable under taking countable products
and arbitrary direct sums, see Proposition 2.10.

In Section 3 we select natural relationships between Gelfand–Phillips type
properties introduced in Definition 1.12, see Lemmas 3.1, 3.2 and 3.5 and
Propositions 3.8 and 3.8. As a corollary we show in Proposition 3.6 that if
$E$ is an angelic, complete and barrelled space (for example, $E$ is a strict
$(LF)$-space), then all Gelfand–Phillips type properties from (i)-(iii) of
Definition 1.12 coincide. If the space $E$ has some of Schur type properties,
then $V^{\ast}$ type properties of Pełczyński’ imply corresponding
Gelfand–Phillips type properties, see Proposition 3.9. It is easy to see
(Lemma 3.1) that the Gelfand–Phillips type properties of order $(1,\infty)$
are the strongest ones in the sense that if $E$ has for example the
$spGP_{(1,\infty)}$ property, then it has the $spGP_{(p,q)}$ property for all
$1\leq p\leq q\leq\infty$. Therefore one can expect that spaces with the
$spGP_{(1,\infty)}$ property have some additional properties. This is indeed
so, in Proposition 3.22 we show that if $E$ is a sequentially complete
barrelled locally convex space whose bounded subsets are weakly sequentially
precompact, then $E$ has the $spGP_{(1,\infty)}$ property if and only if it
has the Schur property. In Proposition 3.11 we show that the classes of
locally convex spaces with Gelfand–Phillips type properties are stable under
taking direct products and direct sums, and Proposition 3.13 stands heredity
properties of Gelfand–Phillips type properties. In Theorem 3.14 we
characterize spaces $C_{p}(X)$ which have one of the equicontinuous $GP$
properties from (ii) of Definition 1.12. In Theorem 3.16 we show that any lcs
is a quotient space of an lcs with the $GP_{(p,q)}$ property; consequently,
the class of locally convex spaces with the $GP_{(p,q)}$ property is
sufficiently rich.

Characterizations of locally convex spaces with Gelfand–Phillips type
properties are given in Section 4. In Theorem 4.2 we obtain an operator
characterization of locally convex spaces with the sequential $GP_{(p,q)}$
property and the sequentially precompact $GP_{(p,q)}$ property; our results
generalize a characterization of Banach spaces with the Gelfand–Phillips
property of order $p$ obtained by Ghenciu in Theorem 15 of [29]. In Theorem
4.4 we characterize locally convex spaces with the sequentially precompact
$GP_{(p,q)}$ property in an important class of locally convex spaces which
includes all strict $(LF)$-spaces. In Theorem 4.5 we essentially generalize
the equivalence (i)$\Leftrightarrow$(ii) in Theorem 1.3. A natural direct
extension of the Gelfand–Phillips property for Banach spaces is the precompact
$GP_{(p,\infty)}$ property with $p\in[1,\infty]$ (so that the precompact
Gelfand–Phillips property and the precompact $GP_{(1,\infty)}$ property are
two extreme cases). Locally convex spaces with the precompact
$GP_{(p,\infty)}$ property are characterized in Theorem 4.7. Sufficient
conditions to have the precompact $GP_{(p,\infty)}$ property are given in
Theorem 4.8 and Corollary 4.9. Banach spaces with the coarse
$p$-Gelfand–Phillips property are characterized by Galindo and Miranda in
Proposition 13 of [27]. In Theorem 4.10 we generalize their result to all
locally convex spaces.

Recall that a Banach space $E$ has the Gelfand–Phillips property if and only
if every limited weakly null sequence in $E$ is norm null, see Theorem 1.3 (or
the more general Theorem 4.5 below). This characterization motivates to
introduce the following type of Gelfand–Phillips property which is studied in
Section 5.

###### Definition 1.13

Let $p,q,q^{\prime}\in[1,\infty]$, $q^{\prime}\leq q$. A locally convex space
$(E,\tau)$ is said to have the $p$-Gelfand–Phillips sequentially compact
property of order $(q^{\prime},q)$ ($p$-$GPscP_{(q^{\prime},q)}$ for short) if
every weakly $p$-summable sequence in $E$ which is also a
$(q^{\prime},q)$-limited set is $\tau$-null. If $q^{\prime}=q$,
$q^{\prime}=q=\infty$, $p=\infty$ or $p=q^{\prime}=q=\infty$, we shall say
simply that $E$ has the $p$-$GPscP_{q}$, the $p$-$GPscP$,
$GPscP_{(q^{\prime},q)}$ or the $GPscP$, respectively.

We characterize locally convex spaces $E$ with the
$p$-$GPscP_{(q^{\prime},q)}$ in Theorem 5.4. This theorem extends and
generalizes characterizations of Banach spaces with the Gelfand–Phillips
property obtained in Theorem 2.2 of [41] and Corollary 13(ii) of [30]. Spaces
with the $GPscP_{(q,\infty)}$ are characterized in Theorem 5.7.

In Section 6 we introduce strong versions of $V^{\ast}$ type properties of
Pełczyński, which usually are stronger than the corresponding Gelfand–Phillips
type properties. These new classes of locally convex spaces have nice
stability properties, see Propositions 6.5 and 6.6. Locally convex spaces with
the strong (resp., sequentially) precompact $V^{\ast}_{p}$ property are
characterized in Theorem 6.7.

##  2 Preliminaries results

We start with some necessary definitions and notations used in the article.
Set $\omega:=\\{0,1,2,\dots\\}$. The closed unit ball of the field
$\mathbb{F}$ is denoted by $\mathbb{D}$. All topological spaces are assumed to
be Tychonoff (= completely regular and $T_{1}$). The closure of a subset $A$
of a topological space $X$ is denoted by $\overline{A}$. The space $C(X)$ of
all continuous functions on $X$ endowed with the pointwise topology is denoted
by $C_{p}(X)$. A Tychonoff space $X$ is called Fréchet–Urysohn if for any
cluster point $a\in X$ of a subset $A\subseteq X$ there is a sequence
$\\{a_{n}\\}_{n\in\omega}\subseteq A$ which converges to $a$. A Tychonoff
space $X$ is called an angelic space if (1) every relatively countably compact
subset of $X$ is relatively compact, and (2) any compact subspace of $X$ is
Fréchet–Urysohn. Note that any subspace of an angelic space is angelic, and a
subset $A$ of an angelic space $X$ is compact if and only if it is countably
compact if and only if $A$ is sequentially compact, see Lemma 0.3 of [36].

Let $E$ be a locally convex space. The span of a subset $A$ of $E$ and its
closure are denoted by $E_{A}:=\mathsf{span}(A)$ and
$\overline{\mathsf{span}}(A)$, respectively. We denote by $\mathcal{N}_{0}(E)$
(resp., $\mathcal{N}_{0}^{c}(E)$) the family of all (resp., closed absolutely
convex) neighborhoods of zero of $E$. The family of all bounded subsets of $E$
is denoted by $\mathsf{Bo}(E)$. The value of $\chi\in E^{\prime}$ on $x\in E$
is denoted by $\langle\chi,x\rangle$ or $\chi(x)$. A sequence
$\\{x_{n}\\}_{n\in\omega}$ in $E$ is said to be Cauchy if for every
$U\in\mathcal{N}_{0}(E)$ there is $N\in\omega$ such that $x_{n}-x_{m}\in U$
for all $n,m\geq N$. If $E$ is a normed space, $B_{E}$ denotes the closed unit
ball of $E$. The family of all operators from $E$ to an lcs $L$ is denoted by
$\mathcal{L}(E,L)$.

For an lcs $E$, we denote by $E_{w}$ and $E_{\beta}$ the space $E$ endowed
with the weak topology $\sigma(E,E^{\prime})$ and with the strong topology
$\beta(E,E^{\prime})$, respectively. The topological dual space $E^{\prime}$
of $E$ endowed with weak∗ topology $\sigma(E^{\prime},E)$ or with the strong
topology $\beta(E^{\prime},E)$ is denoted by $E^{\prime}_{w^{\ast}}$ or
$E^{\prime}_{\beta}$, respectively. The polar of a subset $A$ of $E$ is
denoted by $A^{\circ}:=\\{\chi\in E^{\prime}:\|\chi\|_{A}\leq 1\\}.$ A subset
$B$ of $E^{\prime}$ is equicontinuous if $B\subseteq U^{\circ}$ for some
$U\in\mathcal{N}_{0}(E)$.

A subset $A$ of a locally convex space $E$ is called

  1. $\bullet$

precompact if for every $U\in\mathcal{N}_{0}(E)$ there is a finite set
$F\subseteq E$ such that $A\subseteq F+U$;

  2. $\bullet$

sequentially precompact if every sequence in $A$ has a Cauchy subsequence;

  3. $\bullet$

weakly $($sequentially$)$ compact if $A$ is (sequentially) compact in $E_{w}$;

  4. $\bullet$

relatively weakly compact if its weak closure
$\overline{A}^{\,\sigma(E,E^{\prime})}$ is compact in $E_{w}$;

  5. $\bullet$

relatively weakly sequentially compact if each sequence in $A$ has a
subsequence weakly converging to a point of $E$;

  6. $\bullet$

weakly sequentially precompact if each sequence in $A$ has a weakly Cauchy
subsequence.

Note that each sequentially precompact subset of $E$ is precompact, but the
converse is not true in general, see Lemma 2.2 of [22]. We shall use
repeatedly the next lemma, see Lemma 4.4 in [23].

###### Lemma 2.1

Let $\tau$ and $\mathcal{T}$ be two locally convex vector topologies on a
vector space $E$ such that $\tau\subseteq\mathcal{T}$. If
$S=\\{x_{n}\\}_{n\in\omega}$ is $\tau$-null and $\mathcal{T}$-precompact, then
$S$ is $\mathcal{T}$-null. Consequently, if $S$ is weakly $\mathcal{T}$-null
and $\mathcal{T}$-precompact, then $S$ is $\mathcal{T}$-null.

In what follows we shall actively use the following classical completeness
type properties and weak barrelledness conditions. A locally convex space $E$
is

  1. $\bullet$

quasi-complete if each closed bounded subset of $E$ is complete;

  2. $\bullet$

sequentially complete if each Cauchy sequence in $E$ converges;

  3. $\bullet$

locally complete if the closed absolutely convex hull of a null sequence in
$E$ is compact;

  4. $\bullet$

(quasi)barrelled if every $\sigma(E^{\prime},E)$-bounded (resp.,
$\beta(E^{\prime},E)$-bounded) subset of $E^{\prime}$ is equicontinuous;

  5. $\bullet$

$c_{0}$-(quasi)barrelled if every $\sigma(E^{\prime},E)$-null (resp.,
$\beta(E^{\prime},E)$-null) sequence is equicontinuous.

It is well known that $C_{p}(X)$ is quasibarrelled for every Tychonoff space
$X$.

Denote by $\bigoplus_{i\in I}E_{i}$ and $\prod_{i\in I}E_{i}$ the locally
convex direct sum and the topological product of a non-empty family
$\\{E_{i}\\}_{i\in I}$ of locally convex spaces, respectively. If
$0\not=\mathbf{x}=(x_{i})\in\bigoplus_{i\in I}E_{i}$, then the set
$\mathrm{supp}(\mathbf{x}):=\\{i\in I:x_{i}\not=0\\}$ is called the support of
$\mathbf{x}$. The support of a subset $A$, $\\{0\\}\subsetneq A$, of
$\bigoplus_{i\in I}E_{i}$ is the set $\mathrm{supp}(A):=\bigcup_{a\in
A}\mathrm{supp}(a)$. We shall also consider elements
$\mathbf{x}=(x_{i})\in\prod_{i\in I}E_{i}$ as functions on $I$ and write
$\mathbf{x}(i):=x_{i}$.

Denote by $\mathsf{ind}_{n\in\omega}E_{n}$ the inductive limit of a (reduced)
inductive sequence $\big{\\{}(E_{n},\tau_{n})\big{\\}}_{n\in\omega}$ of
locally convex spaces. If in addition
$\tau_{m}{\restriction}_{E_{n}}=\tau_{n}$ for all $n,m\in\omega$ with $n\leq
m$, the inductive limit $\mathsf{ind}_{n\in\omega}E_{n}$ is called strict and
is denoted by $\underrightarrow{\mbox{ s-$\mathsf{ind}$}}_{n}\,E_{n}$. In the
partial case when all spaces $E_{n}$ are Fréchet, the strict inductive limit
is called a strict $(LF)$-space.

Below we recall some of the basic classes of compact-type operators.

###### Definition 2.2

Let $E$ and $L$ be locally convex spaces. An operator $T\in\mathcal{L}(E,L)$
is called compact (resp., sequentially compact, precompact, sequentially
precompact, weakly compact, weakly sequentially compact, weakly sequentially
precompact, bounded) if there is $U\in\mathcal{N}_{0}(E)$ such that $T(U)$ a
relatively compact (relatively sequentially compact, precompact, sequentially
precompact, relatively weakly compact, relatively weakly sequentially compact,
weakly sequentially precompact or bounded) subset of $E$.

Let $p\in[1,\infty]$. Then $p^{\ast}$ is defined to be the unique element of
$[1,\infty]$ which satisfies $\tfrac{1}{p}+\tfrac{1}{p^{\ast}}=1$. For
$p\in[1,\infty)$, the space $\ell_{p^{\ast}}$ is the dual space of $\ell_{p}$.
We denote by $\\{e_{n}\\}_{n\in\omega}$ the canonical basis of $\ell_{p}$, if
$1\leq p<\infty$, or the canonical basis of $c_{0}$, if $p=\infty$. The
canonical basis of $\ell_{p^{\ast}}$ is denoted by
$\\{e_{n}^{\ast}\\}_{n\in\omega}$. In what follows we usually identify
$\ell_{1^{\ast}}$ with $c_{0}$. Denote by $\ell_{p}^{0}$ and $c_{0}^{0}$ the
linear span of $\\{e_{n}\\}_{n\in\omega}$ in $\ell_{p}$ or in $c_{0}$ endowed
with the induced norm topology, respectively. We shall use also the following
well known description of relatively compact subsets of $\ell_{p}$ and
$c_{0}$, see [11, p. 6].

###### Proposition 2.3

(i) A bounded subset $A$ of $\ell_{p}$, $p\in[1,\infty)$, is relatively
compact if and only if

$\lim_{m\to\infty}\sup\Big{\\{}\sum_{m\leq n}|x_{n}|^{p}:x=(x_{n})\in
A\Big{\\}}=0.$

(ii) A bounded subset $A$ of $c_{0}$ is relatively compact if and only if
$\lim_{n\to\infty}\sup\\{|x_{n}|:x=(x_{n})\in A\\}=0.$

Let $p\in[1,\infty]$. A sequence $\\{x_{n}\\}_{n\in\omega}$ in a locally
convex space $E$ is called

  1. $\bullet$

weakly $p$-convergent to $x\in E$ if $\\{x_{n}-x\\}_{n\in\omega}$ is weakly
$p$-summable;

  2. $\bullet$

weakly $p$-Cauchy if for each pair of strictly increasing sequences
$(k_{n}),(j_{n})\subseteq\omega$, the sequence
$(x_{k_{n}}-x_{j_{n}})_{n\in\omega}$ is weakly $p$-summable.

A sequence $\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$ is called weak∗
$p$-summable (resp., weak∗ $p$-convergent to $\chi\in E^{\prime}$ or weak∗
$p$-Cauchy) if it is weakly $p$-summable (resp., weakly $p$-convergent to
$\chi\in E^{\prime}$ or weakly $p$-Cauchy) in $E^{\prime}_{w^{\ast}}$.
Following [22], $E$ is called $p$-barrelled  (resp., $p$-quasibarrelled) if
every weakly $p$-summable sequence in $E^{\prime}_{w^{\ast}}$ (resp., in
$E^{\prime}_{\beta}$) is equicontinuous.

Generalizing the corresponding notions in the class of Banach spaces
introduced in [9] and [29], the following $p$-versions of weakly compact-type
properties are defined in [22]. Let $p\in[1,\infty]$. A subset $A$ of a
locally convex space $E$ is called

  1. $\bullet$

(relatively) weakly sequentially $p$-compact if every sequence in $A$ has a
weakly $p$-convergent subsequence with limit in $A$ (resp., in $E$);

  2. $\bullet$

weakly sequentially $p$-precompact if every sequence from $A$ has a weakly
$p$-Cauchy subsequence.

An operator $T:E\to L$ between locally convex spaces $E$ and $L$ is called
weakly sequentially  $p$-(pre)compact if there is $U\in\mathcal{N}_{0}(E)$
such that $T(U)$ is relatively weakly sequentially $p$-compact (resp., weakly
sequentially $p$-precompact) subset of $L$.

The following class of subsets of an lcs $E$ was introduced and studied in
[22], it generalizes the notion of $p$-$(V^{\ast})$ subsets of Banach spaces
defined in [10].

###### Definition 2.4

Let $p,q\in[1,\infty]$. A non-empty subset $A$ of a locally convex space $E$
is called a $(p,q)$-$(V^{\ast})$ set (resp., a $(p,q)$-$(EV^{\ast})$ set) if

$\Big{(}\sup_{a\in A}|\langle\chi_{n},a\rangle|\Big{)}\in\ell_{q}\;\mbox{ if
$q<\infty$, }\;\mbox{ or }\;\;\Big{(}\sup_{a\in
A}|\langle\chi_{n},a\rangle|\Big{)}\in c_{0}\;\mbox{ if $q=\infty$},$

for every (resp., equicontinuous) weakly $p$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}_{\beta}$.
$(p,\infty)$-$(V^{\ast})$ sets and $(1,\infty)$-$(V^{\ast})$ sets will be
called simply $p$-$(V^{\ast})$ sets and $(V^{\ast})$ sets, respectively.

Let us recall $V^{\ast}$ type properties of Pełczyński defined in [22].

###### Definition 2.5

Let $1\leq p\leq q\leq\infty$. A locally convex space $E$ is said to have

  1. $\bullet$

the property $V^{\ast}_{(p,q)}$ (a property $EV^{\ast}_{(p,q)}$) if every
$(p,q)$-$(V^{\ast})$ (resp., $(p,q)$-$(EV^{\ast})$) set in $E$ is relatively
weakly compact;

  2. $\bullet$

the property $sV^{\ast}_{(p,q)}$ (a property $sEV^{\ast}_{(p,q)}$) if every
$(p,q)$-$(V^{\ast})$ (resp., $(p,q)$-$(EV^{\ast})$) set in $E$ is relatively
weakly sequentially compact;

  3. $\bullet$

the weak property $sV_{(p,q)}^{\ast}$ or a property $wsV_{(p,q)}^{\ast}$
(resp., a weak property $sEV_{(p,q)}^{\ast}$ or a property
$wsEV_{(p,q)}^{\ast}$) if each $(p,q)$-$(V^{\ast})$-subset (resp.,
$(p,q)$-$(EV^{\ast})$-subset) of $E$ is weakly sequentially precompact.

In the case when $q=\infty$ we shall omit the subscript $q$ and say that $E$
has the property $V^{\ast}_{p}$ etc., and in the case when $q=\infty$ and
$p=1$ we shall say that $E$ has the property $V^{\ast}$ etc.

Let $p\in[1,\infty]$. The $p$-Schur property of Banach spaces was defined in
[13] and [19]. Generalizing this notion and following [22], an lcs $E$ is said
to have a $p$-Schur property if every weakly $p$-summable sequence is a null-
sequence. In particular, $E$ has the Schur property if and only if it is an
$\infty$-Schur space. Following [23], $E$ is called a weakly sequentially
$p$-angelic space if the family of all relatively weakly sequentially
$p$-compact sets in $E$ coincides with the family of all relatively weakly
compact subsets of $E$. The space $E$ is a weakly $p$-angelic space if it is a
weakly sequentially $p$-angelic space and each weakly compact subset of $E$ is
Fréchet–Urysohn.

Following [21], a sequence $A=\\{a_{n}\\}_{n\in\omega}$ in an lcs $E$ is said
to be equivalent to the standard unit basis $\\{e_{n}:n\in\omega\\}$ of
$\ell_{1}$ if there exists a linear topological isomorphism $R$ from
$\overline{\mathsf{span}}(A)$ onto a subspace of $\ell_{1}$ such that
$R(a_{n})=e_{n}$ for every $n\in\omega$ (we do not assume that the closure
$\overline{\mathsf{span}}(A)$ of the $\mathsf{span}(A)$ of $A$ is complete or
that $R$ is onto). We shall say also that $A$ is an $\ell_{1}$-sequence.
Following [26], a locally convex space $E$ is said to have the Rosenthal
property if every bounded sequence in $E$ has a subsequence which either (1)
is Cauchy in the weak topology, or (2) is equivalent to the unit basis of
$\ell_{1}$. The following remarkable extension of the celebrated Rosenthal
$\ell_{1}$-theorem was proved by Ruess [40]: each locally complete locally
convex space $E$ whose every separable bounded set is metrizable has the
Rosenthal property. Thus every strict $(LF)$-space has the Rosenthal property.

Recall that an lcs $X$ is called injective if for every subspace $H$ of a
locally convex space $E$, each operator $T:H\to X$ can be extended to an
operator ${\bar{T}}:E\to X$.

In [37, p. 377] Rosenthal pointed out a theorem of Odell and Stegall which
states that any $\infty$-$(V^{\ast})$ set of a Banach space is weakly
sequentially precompact. In what follows we use the following generalization
of this remarkable result which is of independent interest (our detailed proof
follows the Odell–Stegall idea, cf. also Theorem 5.21 of [24]).

###### Theorem 2.6

Let $2\leq p\leq q\leq\infty$, and let $E$ be a locally convex space with the
Rosenthal property. Then every $(p,q)$-$(V^{\ast})$ subset of $E$ is weakly
sequentially precompact. Consequently, each $(p,q)$-limited subset of $E$ is
weakly sequentially precompact.

###### Proof 1

Suppose for a contradiction that there is a $(p,q)$-$(V^{\ast})$ subset $A$ of
$E$ which is not weakly sequentially precompact. So there is a sequence
$S=\\{x_{n}\\}_{n\in\omega}$ in $A$ which does not have a weakly Cauchy
subsequence. By the Rosenthal property of $E$ and passing to a subsequence if
needed, we can assume that $S$ is an $\ell_{1}$-sequence. Set
$H:=\overline{\mathsf{span}}(S)$, and let $P:H\to\ell_{1}$ be a topological
isomorphism of $H$ onto a subspace of $\ell_{1}$ such that $P(x_{n})=e_{n}$
for every $n\in\omega$ (where $\\{e_{n}\\}_{n\in\omega}$ is the standard unit
basis of $\ell_{1}$). Let $J:\ell_{1}\to\ell_{p}$,
$I_{1}:\ell_{1}\to\ell_{2}$, and $I_{2}:\ell_{2}\to\ell_{p}$ be the natural
inclusions; so $J=I_{2}\circ I_{1}$. By the Grothendieck Theorem 1.13 of [14],
the operator $I_{1}$ is $1$-summing. By the Ideal Property 2.4 of [14], $J$ is
also $1$-summing, and hence, by the Inclusion Property 2.8 of [14], the
operator $J$ is $2$-summing. By the discussion after Corollary 2.16 of [14],
the operator $J$ has a factorization

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
15.29227pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{J:\;\ell_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.01381pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{R}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 39.29227pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
39.29227pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{L_{\infty}(\mu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
76.5237pt\raise 6.46666pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.17778pt\hbox{$\scriptstyle{J_{2}^{\infty}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 95.50113pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
95.50113pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{L_{2}(\mu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
127.03687pt\raise 6.07222pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.71112pt\hbox{$\scriptstyle{Q}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 148.90997pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
148.90997pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ell_{p}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where $\mu$ is a regular probability measure on some compact space $K$ and
$J_{2}^{\infty}:L_{\infty}(\mu)\to L_{2}(\mu)$ is the natural inclusion. By
Theorem 4.14 of [14], the Banach space $L_{\infty}(\mu)$ is injective.
Therefore, by Lemma 5.20 of [24], $L_{\infty}(\mu)$ is an injective locally
convex space. In particular, the operator $R\circ P:H\to L_{\infty}(\mu)$ can
be extended to an operator $T_{\infty}:E\to L_{\infty}(\mu)$. Set $T:=Q\circ
J_{2}^{\infty}\circ T_{\infty}$. Then $T$ is an operator from $E$ to
$\ell_{p}$ such that

$T(x_{n})=Q\circ J_{2}^{\infty}\circ R\circ P(x_{n})=J\circ
P(x_{n})=e_{n}\quad\mbox{ for every $n\in\omega$},$

where $\\{e_{n}\\}_{n\in\omega}$ is the standard unit basis of $\ell_{p}$.
Since $A$ and hence also $S$ are $(p,q)$-$(V^{\ast})$ sets, it follows that
the canonical basis $\\{e_{n}\\}_{n\in\omega}$ of $\ell_{p}$ is also a
$(p,q)$-$(V^{\ast})$ set. However this is impossible because the standard unit
basis $\\{e_{n}^{\ast}\\}_{n\in\omega}$ of the dual $\ell_{p^{\ast}}$ is
weakly $p$-summable (see Example 4.4 of [22]), however since $\sup\\{|\langle
e_{n}^{\ast},e_{i}\rangle|:i\in\omega\\}=1$ for all $n\in\omega$, it follows
that $\\{e_{n}\\}_{n\in\omega}$ is not a $(p,q)$-$(V^{\ast})$ set.

The last assertion follows from the easy fact that any $(p,q)$-limited set is
a $(p,q)$-$(V^{\ast})$ set.

Being motivated by the celebrated Rosenthal’s $\ell_{1}$ theorem, we introduce
the following class of locally convex spaces.

###### Definition 2.7

Let $p\in[1,\infty]$. A locally convex space $E$ is said to have the weak
Cauchy subsequence property of order $p$ (the $wCSP_{p}$ for short) if every
bounded sequence in $E$ has a weakly $p$-Cauchy subsequence. If $p=\infty$, we
shall say simply that $E$ has the $wCSP$.

###### Remark 2.8

(i) By the Rosenthal $\ell_{1}$ theorem, a Banach space $E$ has the $wCSP$ if
and only if it has no an isomorphic copy of $\ell_{1}$.

(ii) It is known (see Corollary 7.3.8 of [45]) that a Banach space $E$ has no
an isomorphic copy of $\ell_{1}$ if and only if the dual Banach space
$E^{\prime}_{\beta}$ has the weak Radon–Nikodym property, and hence if and
only if $E$ has the $wCSP$.

(iii) If $1\leq p<q\leq\infty$ and an lcs $E$ has the $wCSP_{p}$, then $E$ has
the $wCSP_{q}$.

(iv) If an lcs $E$ has the $wCSP$, then $E$ has the Rosenthal property.

(v) A Banach space $E$ has the $wCSP_{p}$ if and only if the identity map
$\mathsf{id}_{E}:E\to E$ is weakly sequentially $p$-precompact.

###### Proposition 2.9

If $1<p<\infty$, then $\ell_{p}$ has the $wCSP_{p}$ if and only if $p\geq 2$.

###### Proof 2

Assume that $1<p<2$. Then $p<p^{\ast}$, and hence, by Proposition 6.5 of [22],
$\ell_{p}$ has the $p$-Schur property. Let $S=\\{e_{n}\\}_{n\in\omega}$ be the
canonical unit basis of $\ell_{p}$. Assuming that $S$ has a weakly $p$-Cauchy
subsequence $\\{e_{n_{k}}\\}_{k\in\omega}$, the $p$-Schur property implies
that it is Cauchy in $\ell_{p}$ which is impossible. Thus $\ell_{p}$ does not
have the $wCSP_{p}$.

Assume that $2\leq p<\infty$. Then, by Proposition 1.4 of [9] (or by Corollary
13.11 of [22]), $B_{\ell_{p}}$ is weakly sequentially $p^{\ast}$-compact.
Since $1<p^{\ast}\leq 2$, we have $p^{\ast}\leq p$. Therefore any weakly
$p^{\ast}$-convergent sequence is also weakly $p$-convergent. Thus
$B_{\ell_{p}}$ is weakly sequentially $p$-compact, and hence $\ell_{p}$ has
the $wCSP_{p}$.

###### Proposition 2.10

Let $p\in[1,\infty]$, and let $\\{E_{i}\\}_{i\in I}$ be a non-empty family of
locally convex spaces.

  1. (i)

If $I=\omega$, then $E=\prod_{i\in\omega}E_{i}$ has the $wCSP_{p}$ if and only
if each factor $E_{i}$ has the $wCSP_{p}$.

  2. (ii)

The space $E=\bigoplus_{i\in I}E_{i}$ has the $wCSP_{p}$ if and only if each
summand $E_{i}$ has the $wCSP_{p}$.

###### Proof 3

(i) Assume that $E$ has the $wCSP_{p}$. Fix $j\in\omega$, and let
$S=\\{x_{n,j}\\}_{n\in\omega}$ be a bounded sequence in $E_{j}$. If $\pi_{j}$
is the natural embedding of $E_{j}$ into $E$, then the sequence $\pi_{j}(S)$
is bounded in $E$ and hence, by the $wCSP_{p}$, it has a weakly $p$-Cauchy
subsequence $\\{\pi_{j}(x_{n_{k},j})\\}_{k\in\omega}$. It is clear that
$\\{x_{n_{k},j}\\}_{k\in\omega}$ is weakly $p$-Cauchy in $E_{j}$. Thus $E_{j}$
has the $wCSP_{p}$.

Conversely, assume that all spaces $E_{i}$ have the $wCSP_{p}$, and let
$\\{x_{n}=(x_{n,i})_{i\in\omega}\\}_{n\in\omega}$ be a sequence in $E$. We
proceed by induction on $i\in\omega$. For $i=0$, the $wCSP_{p}$ of $E_{0}$
implies that there is a sequence $I_{0}$ in $\omega$ such that the sequence
$\\{x_{j,0}\\}_{j\in I_{0}}$ is weakly $p$-Cauchy in $E_{0}$. For $i=1$, the
$wCSP_{p}$ of $E_{1}$ implies that there is a subsequence $I_{1}$ of $I_{0}$
such that the sequence $\\{x_{j,1}\\}_{j\in I_{1}}$ is weakly $p$-Cauchy in
$E_{1}$. Continuing this process we find a sequence $\omega\supseteq
I_{0}\supseteq I_{1}\supseteq\cdots$ such that the sequence
$\\{x_{j,i}\\}_{j\in I_{i}}$ is weakly $p$-Cauchy in $E_{i}$ for every
$i\in\omega$. For every $i\in\omega$, choose $m_{i}\in I_{i}$ such that
$m_{i}<m_{i+1}$ for all $i\in\omega$. We claim that the subsequence
$\\{x_{m_{i}}\\}_{i\in\omega}$ of $\\{x_{n}\\}_{n\in\omega}$ is weakly
$p$-Cauchy in $E$. Indeed, since
$E^{\prime}=\bigoplus_{i\in\omega}E^{\prime}_{i}$, each $\chi\in E^{\prime}$
has a form $\chi=(\chi_{0},\cdots,\chi_{k},0,\dots)$. Then for every strictly
increasing sequence $(i_{n})\subseteq\omega$, we have

$\big{|}\langle\chi,x_{m_{i_{n}}}-x_{m_{i_{n+1}}}\rangle\big{|}\leq\sum_{t=0}^{k}\big{|}\langle\chi_{t},x_{m_{i_{n}},t}-x_{m_{i_{n+1}},t}\rangle\big{|}$
(2.1)

By the choice of $I_{t}$ and $(m_{i})\in\omega$, each sequence
$\big{\\{}x_{m_{i_{n}},t}-x_{m_{i_{n+1}},t}\big{\\}}_{n\in\omega}$ is weakly
$p$-summable. This and (2.1) imply that the sequence
$\big{\\{}x_{m_{i_{n}}}-x_{m_{i_{n+1}}}\big{\\}}_{n\in\omega}$ is weakly
$p$-summable, and hence $\\{x_{m_{i}}\\}_{i\in\omega}$ is weakly $p$-Cauchy in
$E$. Thus $E$ has the $wCSP_{p}$.

(ii) The necessity can be proved repeating word for word the necessity in (i).
The sufficiency follows from (i) because any bounded sequence in $E$ has
finite support.

###### Remark 2.11

The countability of the index set in (i) of Proposition 2.10 is essential.
Indeed, by the example in Lemma 2.2 of [22], the product
$\mathbb{R}^{\mathfrak{c}}$ has a bounded sequence without weak Cauchy
subsequences.

##  3 Permanent properties of Gelfand–Phillips types properties

In this section we study relationships between different Gelfand–Phillips type
properties, stability under taking direct products and direct sums, and a
connection of these properties with the Schur property. We also show that the
class of locally convex spaces with the Gelfand–Phillips property is
sufficiently large.

Recall that a locally convex space $E$ is von Neumann complete if every
precompact subset of $E$ is relatively compact. Following general Definition
11.12 of [22], $E$ is $\mathsf{L}_{(p,q)}$-von Neumann complete (resp.,
$\mathsf{CL}_{(p,q)}$-von Neumann complete) if every closed precompact set in
$\mathsf{L}_{(p,q)}$ (resp., in $\mathsf{CL}_{(p,q)}$) is compact. Recall also
that a locally convex space $E$ is called semi-Montel if every bounded subset
of $E$ is relatively compact, and $E$ is a Montel space if it is a barrelled
semi-Montel space.

###### Lemma 3.1

Let $1\leq p\leq q\leq\infty$, and let $(E,\tau)$ be a locally convex space.

  1. (i)

If $E$ has the $EGP_{(p,q)}$ $($resp., $sEGP_{(p,q)}$$)$ property, then $E$
has the $GP_{(p,q)}$ $($resp., $sGP_{(p,q)}$$)$ property. The converse is true
for $p$-barrelled spaces.

  2. (ii)

If $1\leq p^{\prime}\leq p$ and $q\leq q^{\prime}\leq\infty$ and $E$ has the
$GP_{(p^{\prime},q^{\prime})}$ $($resp., $sGP_{(p^{\prime},q^{\prime})}$,
$EGP_{(p^{\prime},q^{\prime})}$, $sEGP_{(p^{\prime},q^{\prime})}$,
$prGP_{(p^{\prime},q^{\prime})}$ or $spGP_{(p^{\prime},q^{\prime})}$$)$
property, then $E$ has the $GP_{(p,q)}$ $($resp., $sGP_{(p,q)}$,
$EGP_{(p,q)}$, $sEGP_{(p,q)}$, $prGP_{(p,q)}$ or $spGP_{(p,q)}$$)$ property.

  3. (iii)

If the class of relatively compact sets in $E$ coincides with class of
relatively sequentially compact sets, then $E$ has the property $GP_{(p,q)}$
$($resp., $EGP_{(p,q)}$$)$ if and only if it has the property $sGP_{(p,q)}$
$($resp., $sEGP_{(p,q)}$$)$.

  4. (iv)

If $E$ is an angelic $p$-barrelled space, then all the properties
$GP_{(p,q)}$, $EGP_{(p,q)}$, $sGP_{(p,q)}$ and $sEGP_{(p,q)}$ are coincide for
$E$.

  5. (v)

If $E$ has the $GP_{(p,q)}$ property then it has the $prGP_{(p,q)}$ property.
The converse is true if $E$ is $\mathsf{L}_{(p,q)}$-von Neumann complete.

  6. (vi)

If $E$ is sequentially complete, then $E$ has the $spGP_{(p,q)}$ (resp.,
$spEGP_{(p,q)}$) property if and only if it has the $sGP_{(p,q)}$ (resp.,
$sEGP_{(p,q)}$) property.

  7. (vii)

If $E$ is semi-Montel, then $E$ has the $EGP_{(p,q)}$ property.

###### Proof 4

(i) follows from the easy fact that every $(p,q)$-limited set is
$(p,q)$-${\mathcal{E}}$-limited, see Lemma 3.1(i) of [24], and from the
definition of $p$-barrelled spaces.

(ii) follows from the fact that every $(p,q)$-(${\mathcal{E}}$-)limited set is
$(p^{\prime},q^{\prime})$-(${\mathcal{E}}$-)limited, see Lemma 3.1(vi) of
[24].

(iii), (v) and (vii) follow from the corresponding definitions.

(iv) follows from (i) and the fact that for angelic spaces, relatively compact
sets are exactly relatively sequentially compact sets.

(vi) It suffices to prove the necessity. Assume that $E$ has the
$spGP_{(p,q)}$ (resp., $spEGP_{(p,q)}$) property. Let $A$ be a $(p,q)$-limited
(resp., $(p,q)$-${\mathcal{E}}$-limited) set in $E$. Then $A$ is sequentially
precompact. We show that $A$ is even relatively sequentially compact. Indeed,
let $S=\\{x_{n}\\}_{n\in\omega}$ be a sequence in $A$. Since $A$ is
sequentially precompact, $S$ has a Cauchy subsequence
$\\{x_{n_{k}}\\}_{k\in\omega}$. Since $E$ is sequentially complete, there is
$x\in E$ such that $x_{n_{k}}\to x$. Therefore $A$ is relatively sequentially
compact. Thus $E$ has the $sGP_{(p,q)}$ (resp., $sEGP_{(p,q)}$) property.

If the space $E$ carries its weak topology (for example, $E=C_{p}(X)$), we
have the following result.

###### Lemma 3.2

Let $1\leq p\leq q\leq\infty$, and let $(E,\tau)$ be a locally convex space
such that $E=E_{w}$. Then:

  1. (i)

$E$ has the $prEGP_{(p,q)}$ property and hence the $prGP_{(p,q)}$ property;

  2. (ii)

every bounded subset of $E$ is $(p,q)$-${\mathcal{E}}$-limited; consequently,
if $E$ is von Neumann complete, then $E$ has the property $EGP_{(p,q)}$.

###### Proof 5

(i) is trivial because every bounded subset of $E$ is precompact.

(ii) Let $A$ be a bounded subset of $E$, and let
$S=\\{\chi_{n}\\}_{n\in\omega}$ be an equicontinuous weak∗ $p$-summable
sequence in $E^{\prime}$. Since $E$ carries its weak topology it follows that
$S\subseteq F^{\circ\circ}$ for some finite subset
$F=\\{\eta_{1},\dots,\eta_{k}\\}\subseteq E^{\prime}$. Therefore
$S\subseteq\mathsf{span}(F)$, and hence for every $n\in\omega$, there are
scalars $c_{1,n},\dots,c_{k,n}\in\mathbb{F}$ such that

$\chi_{n}=c_{1,n}\eta_{1}+\cdots+c_{k,n}\eta_{k}.$

Since $S$ is weak∗ $p$-summable, we obtain that $(c_{i,n})_{n}\in\ell_{p}$ (or
$\in c_{0}$ if $p=\infty$) for every $i=1,\dots,k$. As $A$ is bounded, for
every $i=1,\dots,k$, one can define $C_{i}:=\sup_{a\in
A}|\langle\eta_{i},a\rangle|$. Then

$\sup_{a\in A}|\langle\chi_{n},a\rangle|\leq\sum_{i=1}^{k}\sup_{a\in
A}|c_{i,n}||\langle\eta_{i},a\rangle|=\sum_{i=1}^{k}C_{i}|c_{i,n}|.$

Therefore $\big{(}\sup_{a\in
A}|\langle\chi_{n},a\rangle|\big{)}_{n}\in\ell_{p}\subseteq\ell_{q}$ (or $\in
c_{0}$ if $p=q=\infty$). Thus $A$ is a $(p,q)$-${\mathcal{E}}$-limited set.

Note that, by Theorem 3.13 of [22], $C_{p}(X)$ is von Neumann complete if and
only if $X$ is discrete.

###### Lemma 3.3

Let $1\leq p\leq q\leq\infty$, and let $\tau$ be a locally convex compatible
topology on a locally convex space $(E,\mathcal{T})$ such that
$\tau\subseteq\mathcal{T}$. If $(E,\mathcal{T})$ has the $prGP_{(p,q)}$
(resp., $spGP_{(p,q)}$, $GP_{(p,q)}$ or $sGP_{(p,q)}$) property, then also
$(E,\tau)$ has the same property.

###### Proof 6

Since $\tau$ and $\mathcal{T}$ are compatible, the spaces $(E,\tau)$ and
$(E,\mathcal{T})$ have the same $(p,q)$-limited subsets by Lemma 3.1(vii) of
[24]. As by assumption all $(p,q)$-limited sets in $(E,\mathcal{T})$ are
precompact (resp., sequentially precompact, relatively compact or relatively
sequentially compact), the inclusion $\tau\subseteq\mathcal{T}$ implies that
so are all $(p,q)$-limited sets also in $(E,\tau)$.

###### Remark 3.4

The converse in Lemma 3.3 is not true in general. Indeed, let
$E=\ell_{\infty}$. Then $E_{w}$ trivially has the precompact $GP$ property.
However, the space $E$ is not a Gelfand–Phillips space by [35]. Consequently,
the property of being a Gelfand–Phillips space is not a property of the dual
pair $(E,E^{\prime})$.

Below we consider relationships between $b$-version of the Gelfand–Phillips
properties and the Gelfand–Phillips properties.

###### Lemma 3.5

Let $1\leq p\leq q\leq\infty$, and let $(E,\tau)$ be a locally convex space.
Then:

  1. (i)

if $E$ has the $b$-$GP_{(p,q)}$ property, then it has the $prGP_{(p,q)}$
property;

  2. (ii)

if $E$ is barrelled, then $E$ has the $b$-$GP_{(p,q)}$ $($resp.,
$b$-$EGP_{(p,q)}$$)$ property if and only if it has the $prGP_{(p,q)}$
$($resp., $prEGP_{(p,q)}$$)$ property;

  3. (iii)

if $E$ is barrelled and von Neumann complete, then $E$ has the
$b$-$GP_{(p,q)}$ $($resp., $b$-$EGP_{(p,q)}$$)$ property if and only if it has
the $GP_{(p,q)}$ $($resp., $EGP_{(p,q)}$$)$ property.

###### Proof 7

(i) follows from the inclusion $\tau\subseteq\beta(E,E^{\prime})$.

(ii) and (iii) follow from the equality $E=E_{\beta}$ and the corresponding
definitions.

We select the next proposition which shows that for wide classes of locally
convex spaces important for applications in fact there is only a unique
Gelfand–Phillips type property (however, in general, these notions are
different, see Examples 3.12 and 3.15 and Theorem 3.14 below).

###### Proposition 3.6

Let $1\leq p\leq q\leq\infty$, and let $E$ be an angelic, complete and
barrelled space (for example, $E$ is a strict $(LF)$-space). Then all the
properties $GP_{(p,q)}$, $EGP_{(p,q)}$, $sGP_{(p,q)}$, $sEGP_{(p,q)}$,
$prGP_{(p,q)}$, $prEGP_{(p,q)}$, $spGP_{(p,q)}$, $spEGP_{(p,q)}$ and
$b$-$GP_{(p,q)}$ are equivalent.

###### Proof 8

The properties $GP_{(p,q)}$, $EGP_{(p,q)}$, $sGP_{(p,q)}$ and $sEGP_{(p,q)}$
are equivalent by (iv) of Lemma 3.1. The properties $GP_{(p,q)}$ and
$prGP_{(p,q)}$ are equivalent by (v) of Lemma 3.1. The properties
$sGP_{(p,q)}$ and $spGP_{(p,q)}$ are equivalent by (vi) of Lemma 3.1. The
properties $prGP_{(p,q)}$ and $b$-$GP_{(p,q)}$ are equivalent by (ii) of Lemma
3.5.

Since $E$ is barrelled, the $(p,q)$-limited sets and the
$(p,q)$-${\mathcal{E}}$-limited sets are the same. Therefore, the properties
$prGP_{(p,q)}$ and $prEGP_{(p,q)}$, as well as the properties $spGP_{(p,q)}$
and $spEGP_{(p,q)}$ are coincide.

By Proposition 4.2 of [24], every $p$-limited subset of $E$ is coarse
$p$-limited. Although every precompact subset of a $c_{0}$-barrelled space is
limited by Corollary 3.7 of [24], non-$c_{0}$-barrelled spaces may contain
even convergent sequences which are not limited, see Example 3.10 in [24]. We
complement these results in the the following assertion. Recall that a subset
$A$ of a topological space $X$ is functionally bounded if $f(A)$ is bounded
for every $f\in C(X)$; clearly, any compact subset of $X$ is functionally
bounded.

###### Proposition 3.7

Let $p\in[1,\infty]$, and let $E$ be a locally convex space. Then:

  1. (i)

each functionally bounded subset $A$ of $E$ is coarse $p$-limited;

  2. (ii)

every limited subset of $E$ is coarse $p$-limited.

###### Proof 9

(i) Let $T:E\to\ell_{p}$ (or $T:E\to c_{0}$ if $p=\infty$) be an operator.
Then $T(A)$ is a functionally bounded subset of the metric space $\ell_{p}$
(or $c_{0}$), and hence $T(A)$ is relatively compact. Thus $A$ is coarse
$p$-limited.

(ii) Let $A$ be a limited subset of $E$, and let $T\in\mathcal{L}(E,\ell_{p})$
(or $T\in\mathcal{L}(E,c_{0})$ if $p=\infty$). Then, by (iv) of Lemma 3.1 of
[24], $T(A)$ is a limited subset of $\ell_{p}$ (or $c_{0}$). Since the Banach
spaces $\ell_{p}$ and $c_{0}$ are separable, they have the Gelfand–Phillips
property by Theorem 1.4. Therefore $T(A)$ is relatively compact in $\ell_{p}$
(or in $c_{0}$). Thus $A$ is a coarse $p$-limited set in $E$.

Below we consider the coarse versions of the Gelfand–Phillips property.

###### Proposition 3.8

Let $1\leq p\leq q\leq\infty$, and let $(E,\tau)$ be a locally convex space.

  1. (i)

If $E$ has the coarse $GP_{p}$ $($resp., coarse $sGP_{p}$ or coarse
$spGP_{p}$$)$ property, then $E$ has the coarse $prGP_{p}$ $($resp., coarse
$spGP_{p}$ or coarse $prGP_{p}$$)$ property.

  2. (ii)

If $E=E_{w}$, then $\mathsf{CL}_{p}(E)=\mathsf{Bo}(E)$ and hence $E$ has the
coarse $prGP_{p}$ property.

  3. (iii)

If $E$ has the coarse $GP_{p}$ (resp., coarse $sGP_{p}$, coarse $prGP_{p}$ or
coarse $spGP_{p}$) property, then $E$ has the $GP_{p}$ (resp., $sGP_{p}$,
$prGP_{p}$ or $spGP_{p}$) property.

  4. (iv)

If $E$ has the coarse $GP_{p}$ property, then $E$ has the $GP$ property. The
converse is not true in general.

  5. (v)

If $E$ is barrelled, then $E$ has the $GP_{(1,\infty)}$ (resp.,
$sGP_{(1,\infty)}$, $prGP_{(1,\infty)}$, or $spGP_{(1,\infty)}$) property if
and only if it has the coarse $GP_{1}$ (resp., coarse $sGP_{1}$, coarse
$prGP_{1}$, or coarse $spGP_{1}$) property.

###### Proof 10

(i) is obvious.

(ii) Since $E=E_{w}$, Lemma 5.12 of [24] implies that each operator from $E$
into any Banach space is finite-dimensional. Therefore any bounded subset of
$E$ is coarse $p$-limited. As any bounded subset of $E$ is precomapct it
follows that $E$ has the coarse $prGP_{p}$ property.

(iii) follows from Proposition 4.2 of [24] which states that every $p$-limited
set is coarse $p$-limited.

(iv) follows from Proposition 3.7. To show that the converse is not true in
general, let $E=c_{0}$ and $1\leq p<\infty$. Since $E$ is separable it has the
$GP$ property by Theorem 1.4. However, $E$ does not have the coarse $GP_{p}$
property because, by the Pitt theorem, $B_{E}$ is a non-compact coarse
$p$-limited set.

(v) follows from Proposition 3.13 of [24] which states that for a barrelled
space $E$, the class of $(1,\infty)$-limited sets coincides with the class of
coarse $1$-limited sets.

It is natural to consider relationships between $V^{\ast}_{(p,q)}$ type
properties of Pełczyński and Gelfand–Phillips type properties. Following [23],
an lcs $E$ has the (weak) Glicksberg property if $E$ and $E_{w}$ have the same
(resp., absolutely convex) compact sets. Note that the weak Glicksberg
property does not imply the Schur property and vise versa.

###### Proposition 3.9

Let $1\leq p\leq q\leq\infty$, and let $(E,\tau)$ be a locally convex space.

  1. (i)

Let $E$ have the weak Glicksberg property. If $E$ has the property
$V^{\ast}_{(p,q)}$ $($resp., $EV^{\ast}_{(p,q)}$$)$, then $E$ has the
$GP_{(p,q)}$ $($resp., $EGP_{(p,q)}$$)$ property.

  2. (ii)

Let $E$ have the Schur property. If $E$ has the property $sV^{\ast}_{(p,q)}$
$($resp., $sEV^{\ast}_{(p,q)}$$)$, then $E$ has the $sGP_{(p,q)}$ $($resp.,
$sEGP_{(p,q)}$$)$ property.

###### Proof 11

(i) Recall that every $(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited)
set $A$ of $E$ is a also a $(p,q)$-$(V^{\ast})$ (resp., $(p,q)$-$(EV^{\ast})$)
set by Lemma 3.1(viii) of [24]. By (iii) of Lemma 7.2 of [22],
$A^{\circ\circ}$ is also a $(p,q)$-$(V^{\ast})$ (resp., $(p,q)$-$(EV^{\ast})$)
set. By the property $V^{\ast}_{(p,q)}$ (resp., $EV^{\ast}_{(p,q)}$),
$A^{\circ\circ}$ is weakly compact, and hence, by the weak Glicksberg
property, $A^{\circ\circ}$ is a compact subset of $E$. Therefore $A$ is
relatively compact. Thus $E$ has the $GP_{(p,q)}$ (resp., $EGP_{(p,q)}$)
property.

(ii) Let $A$ be a $(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited) set
in $E$. Then, by Lemma 3.1(viii) of [24], $A$ is a $(p,q)$-$(V^{\ast})$
(resp., $(p,q)$-$(EV^{\ast})$) set. By the property $sV^{\ast}_{(p,q)}$
(resp., $sEV^{\ast}_{(p,q)}$), $A$ is relatively weakly sequentially compact.
Since $E$ is a Schur space, Lemma 2.1 of [24] implies that $A$ is relatively
sequentially compact in $E$. Thus $E$ has the $sGP_{(p,q)}$ (resp.,
$sEGP_{(p,q)}$) property.

###### Remark 3.10

In both cases (i) and (ii) of Proposition 3.9 the converse is not true in
general. Indeed, let $E=c_{0}$ and $p=q=\infty$. Being separable $c_{0}$ has
the $GP$ property and hence, by Proposition 3.6, the $sGP$ property. On the
other hand, $E$ has neither the property $V^{\ast}_{\infty}$ nor the property
$sV^{\ast}_{\infty}$ because the Schur property of
$E^{\prime}_{\beta}=\ell_{1}$ implies that $B_{E}$ is a non-compact
$\infty$-$(V^{\ast})$ set.

Below we show that the classes of locally convex spaces with Gelfand–Phillips
type properties are stable under taking direct products and direct sums.

###### Proposition 3.11

Let $1\leq p\leq q\leq\infty$, and let $\\{E_{i}\\}_{i\in I}$ be a non-empty
family of locally convex spaces. Then:

  1. (i)

$E=\prod_{i\in I}E_{i}$ has the $GP_{(p,q)}$ $($resp., $EGP_{(p,q)}$,
$prGP_{(p,q)}$, $prEGP_{(p,q)}$, coarse $GP_{p}$, or coarse $prGP_{p}$$)$
property if and only if all spaces $E_{i}$ have the same property;

  2. (ii)

$E=\bigoplus_{i\in I}E_{i}$ has the $GP_{(p,q)}$ $($resp., $EGP_{(p,q)}$,
$sGP_{(p,q)}$, $sEGP_{(p,q)}$, $prGP_{(p,q)}$, $prEGP_{(p,q)}$,
$spGP_{(p,q)}$, $spEGP_{(p,q)}$, coarse $GP_{p}$, coarse $sGP_{p}$, coarse
$spGP_{p}$, or coarse $prGP_{p}$$)$ property if and only if all spaces $E_{i}$
have the same property;

  3. (iii)

if $I=\omega$ is countable, then $E=\prod_{i\in\omega}E_{i}$ has the
$sGP_{(p,q)}$ $($resp., $sEGP_{(p,q)}$, $spGP_{(p,q)}$, $spEGP_{(p,q)}$,
coarse $sGP_{p}$, or coarse $spGP_{p}$$)$ property if and only if all spaces
$E_{i}$ have the same property.

###### Proof 12

We consider only $GP_{(p,q)}$ type properties because the coarse $GP_{p}$ type
properties can be considered analogously using properties of coarse
$p$-limited sets, see Lemma 4.1 and Proposition 4.3 of [24].

To prove the necessity, let $E$ have the $GP_{(p,q)}$ (resp., $EGP_{(p,q)}$,
$sGP_{(p,q)}$, $sEGP_{(p,q)}$, $prGP_{(p,q)}$, $prEGP_{(p,q)}$, $spGP_{(p,q)}$
or $spEGP_{(p,q)}$) property. Fix $j\in I$, and let $A_{j}$ be a
$(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited) set in $E_{j}$. Since
$E_{j}$ is a direct summand of $E$, to show that $A_{j}$ is relatively compact
(resp., precompact, relatively sequentially compact or sequentially
precompact) in $E_{j}$ it suffices to show that the set
$A:=A_{j}\times\prod_{i\in I{\setminus}\\{j\\}}\\{0_{i}\\}$ has the same
property in $E$. In all cases (i)–(iii), by Proposition 3.3 of [24], $A$ is a
$(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited) set in $E$ (also as
the image of $A_{j}$ under the canonical embedding of $E_{j}$ into $E$). Since
$E$ has the $GP_{(p,q)}$ (resp., $EGP_{(p,q)}$, $sGP_{(p,q)}$, $sEGP_{(p,q)}$,
$prGP_{(p,q)}$, $prEGP_{(p,q)}$, $spGP_{(p,q)}$ or $spEGP_{(p,q)}$) property,
it follows that $A$ is relatively compact (resp., precompact, relatively
sequentially compact or sequentially precompact) in $E$, as desired.

To prove the sufficiency, assume that all spaces $E_{i}$ have the $GP_{(p,q)}$
(resp., $EGP_{(p,q)}$, $sGP_{(p,q)}$, $sEGP_{(p,q)}$, $prGP_{(p,q)}$,
$prEGP_{(p,q)}$, $spGP_{(p,q)}$ or $spEGP_{(p,q)}$) property, and let $A$ be a
$(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited) set in $E$. Recall
that Lemma 2.1 of [22] states that a subset of a countable product of
Tychonoff spaces is (relatively) sequentially compact if and only if so are
all its projections. This result and, for the cases (i) and (iii), the fact
that the product of relatively compact sets (resp., precompact sets) is
relatively compact (resp., precompact), and for the case (ii), the fact that
the support of $A$ is finite since $A$ is bounded, to show that $A$ is
relatively compact (resp., precompact, relatively sequentially compact or
sequentially precompact) it suffices to prove that for every $i\in I$, the
projection $A_{i}$ of $A$ onto the $i$th coordinate has the same property in
$E_{i}$. But this condition is satisfied because, by (iv) of Lemma 3.1 of
[24], the projection $A_{i}$ is a $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) set in $E_{i}$ and hence it is relatively
compact (resp., precompact, relatively sequentially compact or sequentially
precompact) in $E_{i}$ by the corresponding property of $E_{i}$.

In (iii) of Proposition 3.11 the countability of the set $I$ is essential.

###### Example 3.12

Let $1\leq p\leq q\leq\infty$. Then the separable space
$\mathbb{R}^{\mathfrak{c}}$ has the $EGP_{(p,q)}$ property and hence the
$GP_{(p,q)}$ property and the coarse $prGP_{p}$ property, but it has neither
the $spGP_{(p,q)}$ property nor the coarse $spGP_{p}$ property.

###### Proof 13

The space $\mathbb{R}^{\mathfrak{c}}$ has the $EGP_{(p,q)}$ property and the
coarse $prGP_{p}$ property by (i) of Proposition 3.11. By (i) of Proposition
3.3 of [24], any bounded subset $A$ of $\mathbb{R}^{\mathfrak{c}}$ is
${(p,q)}$-limited (and hence also ${(p,q)}$-${\mathcal{E}}$-limited) and, by
(i) of Proposition 4.3 of [24], $A$ is also coarse $p$-limited. In particular,
identifying $3^{\omega}$ with $\mathfrak{c}$ we obtain that the sequence
$S=\\{f_{n}\\}_{n\in\omega}$ of functions constructed in Lemma 2.2 of [22] is
a ${(p,q)}$-limited set. However, it is proved in Lemma 2.2 of [22] that $S$
is not sequentially precompact. Thus $\mathbb{R}^{\mathfrak{c}}$ has neither
the $spGP_{(p,q)}$ property nor the coarse $spGP_{p}$ property.

In the next proposition we consider the heredity of Gelfand–Phillips type
properties. Recall that a subspace $Y$ of a Tychonoff space $X$ is
sequentially closed if $Y$ contains all limits of convergent sequences from
$Y$.

###### Proposition 3.13

Let $1\leq p\leq q\leq\infty$, and let $L$ be a subspace of a locally convex
space $E$.

  1. (i)

If $L$ is closed in $E$ and $E$ has the $GP_{(p,q)}$ $($resp., $EGP_{(p,q)}$
or coarse $GP_{p}$$)$ property, then also $L$ has the same property.

  2. (ii)

If $L$ is sequentially closed in $E$ and $E$ has the $sGP_{(p,q)}$ $($resp.,
$sEGP_{(p,q)}$ or coarse $sGP_{p}$$)$ property, then also $L$ has the same
property.

  3. (iii)

If $E$ has the $prGP_{(p,q)}$ $($resp., $prEGP_{(p,q)}$ or coarse
$prGP_{p}$$)$ property, then also $L$ has the same property.

  4. (iv)

If $E$ has the $spGP_{(p,q)}$ $($resp., $spEGP_{(p,q)}$ or coarse
$spGP_{p}$$)$ property, then also $L$ has the same property.

###### Proof 14

Below we consider only $GP_{(p,q)}$ type properties because the coarse
$GP_{p}$ type properties can be considered analogously using properties of
coarse $p$-limited sets, see Lemma 4.1 of [24].

Let $A$ be a $(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited) set in
$L$. Then, by (iv) of Lemma 3.1 of [24] applied to the identity embedding
$\mathsf{id}_{L}:L\to E$, $A$ is a $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) set also in $E$.

(i) By the property $GP_{(p,q)}$ (resp., $EGP_{(p,q)}$) of $E$, the set $A$ is
relatively compact in $E$. Since $L$ is a closed subspace of $E$ it follows
that $A$ is relatively compact in $L$. Thus $L$ has the property $GP_{(p,q)}$
(resp., $EGP_{(p,q)}$).

(ii) By the property $sGP_{(p,q)}$ (resp., $sEGP_{(p,q)}$) of $E$, the set $A$
is relatively sequentially compact in $E$. Since $L$ is a sequentially closed
subspace of $E$ it follows that $A$ is relatively sequentially compact in $L$.
Thus $L$ has the property $sGP_{(p,q)}$ (resp., $sEGP_{(p,q)}$).

(iii) By the property $prGP_{(p,q)}$ $($resp., $prEGP_{(p,q)}$), the set $A$
is precompact in $E$ and hence also in $L$. Thus $L$ has the property
$prGP_{(p,q)}$ $($resp., $prEGP_{(p,q)}$).

(iv) By the property $spGP_{(p,q)}$ (resp., $spEGP_{(p,q)}$) of $E$, the set
$A$ is sequentially precompact in $E$. Let $\\{a_{n}\\}_{n\in\omega}$ be a
sequence in $A$. Take a subsequence $\\{a_{n_{k}}\\}_{k\in\omega}$ of
$\\{a_{n}\\}_{n\in\omega}$ which is Cauchy in $E$. Since $L$ is a subspace of
$E$ it follows that $\\{a_{n_{k}}\\}_{k\in\omega}$ is Cauchy also in $L$.
Therefore $A$ is sequentially precompact in $L$. Thus $L$ has the property
$spGP_{(p,q)}$ (resp., $spEGP_{(p,q)}$).

Below we characterize spaces $C_{p}(X)$ which have one of the equicontinuous
$GP$ properties.

###### Theorem 3.14

Let $1\leq p\leq q\leq\infty$, and let $X$ be a Tychonoff space. Then:

  1. (i)

$C_{p}(X)$ has the $EGP_{(p,q)}$ property if and only if $X$ is discrete;

  2. (ii)

$C_{p}(X)$ has the $b$-$EGP_{(p,q)}$ property if and only if every
functionally bounded subset of $X$ is finite;

  3. (iii)

$C_{p}(X)$ has the $prEGP_{(p,q)}$ property and hence the $prGP_{(p,q)}$
property.

###### Proof 15

(i) Assume that $C_{p}(X)$ has the $EGP_{(p,q)}$ property. Then, by (ii) of
Lemma 3.2, the bounded set $B=\\{f\in C_{p}(X):\|f\|_{X}\leq 1\\}$ is
$(p,q)$-${\mathcal{E}}$-limited. Therefore, by the $EGP_{(p,q)}$ property, $B$
has compact closure in $C_{p}(X)$. As $B$ is dense in the compact space
$\mathbb{D}^{X}$ we obtain that $\mathbb{D}^{X}\subseteq C_{p}(X)$. But this
is possible only if $X$ is discrete.

Conversely, assume that $X$ is discrete. Then clearly every bounded subset of
$C_{p}(X)=\mathbb{F}^{X}$ has compact closure. Thus $C_{p}(X)$ has the
$EGP_{(p,q)}$ property.

(ii) Assume that $C_{p}(X)$ has the $b$-$EGP_{(p,q)}$ property, and let $A$ be
a functionally bounded subset of $X$. Assuming that $A$ is infinite we can
find a sequence $\\{a_{n}\\}\subseteq A$ and a sequence
$\\{U_{n}\\}_{n\in\omega}$ of open subsets of $X$ such that $a_{n}\in U_{n}$
and $U_{n}\cap U_{m}=\emptyset$ for all distinct $n,m\in\omega$. For every
$n\in\omega$, choose $f_{n}\in C_{p}(X)$ such that
$f_{n}(X{\setminus}U_{n})=\\{0\\}$ and $f_{n}(a_{n})=n$. Then the sequence
$S=\\{f_{n}\\}_{n\in\omega}$ is a bounded subset of $C_{p}(X)$ and hence, by
(ii) of Lemma 3.2, $S$ is a ${(p,q)}$-${\mathcal{E}}$-limited set. Therefore
to get a contradiction with the $b$-$EGP_{(p,q)}$ property of $C_{p}(X)$ it
suffices to show that $S$ is not barrel-precompact. To this end, we note that
the functional boundedness of $A$ implies that the set $B:=\\{f\in
C_{p}(X):\|f\|_{A}\leq 1\\}$ is a barrel in $C_{p}(X)$. Since, by
construction, $S\nsubseteq kB$ for every $k\in\omega$ we obtain that $S$ is
not barrel-precompact.

Conversely, assume that every functionally bounded subset of $X$ is finite. We
show that every $(p,q)$-${\mathcal{E}}$-limited subset $B$ of $C_{p}(X)$ is
barrel-precompact. To this end, we note that, by the Buchwalter–Schmetz
theorem, the space $C_{p}(X)$ is barrelled. Now, let $D$ be a barrel in
$C_{p}(X)$. Then $D$ is a neighborhood of zero. Since $C_{p}(X)$ carries its
weak topology and $B$ is its bounded subset, it follows that $B$ is precompact
and hence there is a finite subset $F\subseteq C_{p}(X)$ such that $B\subseteq
F+D$. Thus $B$ is barrel-precompact.

(iii) Since $C_{p}(X)$ carries its weak topology, the assertion follows from
(i) of Lemma 3.2.

In (i) and (ii) of Proposition 3.13 the condition on $L$ of being a
(sequentially) closed subspace of $E$ is essential even for Fréchet spaces as
the next example shows.

###### Example 3.15

Let $1\leq p\leq q\leq\infty$, and let $X$ be a countable non-discrete
Tychonoff space whose compact subsets are finite. Then the dense subspace
$C_{p}(X)$ of $\mathbb{F}^{\omega}$ has no any of the properties $GP_{(p,q)}$,
$EGP_{(p,q)}$, $sGP_{(p,q)}$ or $sEGP_{(p,q)}$, but it has the properties
$prEGP_{(p,q)}$, $prGP_{(p,q)}$, $spEGP_{(p,q)}$ and $spGP_{(p,q)}$.

###### Proof 16

By the Buchwalter–Schmetz theorem, the space $C_{p}(X)$ is barrelled and hence
it is $p$-barrelled. Being metrizable $C_{p}(X)$ is an angelic space. Thus, by
(iv) of Lemma 3.1 and (i) of Theorem 3.14, $C_{p}(X)$ has no any of the
properties $GP_{(p,q)}$, $EGP_{(p,q)}$, $sGP_{(p,q)}$ or $sEGP_{(p,q)}$.

On the other hand, by (iii) of Theorem 3.14 and the metrizability of
$\mathbb{F}^{\omega}$, the space $C_{p}(X)$ has the properties
$prEGP_{(p,q)}$, $prGP_{(p,q)}$, $spEGP_{(p,q)}$ and $spGP_{(p,q)}$.

Our next purpose is to show that any locally convex space is a quotient space
of a locally convex space with the (sequential) $GP_{(p,q)}$ property for all
$1\leq p\leq q\leq\infty$. First we recall some definitions.

The free locally convex space $L(X)$ over a Tychonoff space $X$ is a pair
consisting of a locally convex space $L(X)$ and a continuous map $i:X\to L(X)$
such that every continuous map $f$ from $X$ to a locally convex space $E$
gives rise to a unique continuous linear operator $\Psi_{E}(f):L(X)\to E$ with
$f=\Psi_{E}(f)\circ i$. The free locally convex space $L(X)$ always exists and
is essentially unique. For $\chi=a_{1}x_{1}+\cdots+a_{n}x_{n}\in L(X)$ with
distinct $x_{1},\dots,x_{n}\in X$ and nonzero
$a_{1},\dots,a_{n}\in\mathbb{F}$, we set $\chi[x_{i}]:=a_{i}$ and

$\|\chi\|:=|a_{1}|+\cdots+|a_{n}|,\;\mbox{ and
}\;\mathrm{supp}(\chi):=\\{x_{1},\dots,x_{n}\\}.$

From the definition of $L(X)$ it easily follows the well known fact that the
dual space $L(X)^{\prime}$ of $L(X)$ is linearly isomorphic to the space
$C(X)$ with the pairing

$\langle f,\chi\rangle=a_{1}f(x_{1})+\cdots+a_{n}f(x_{n})\quad\mbox{ for $f\in
C(X)$}.$

For every subset $A\subseteq L(X)$, let $\mathrm{supp}(A):=\bigcup_{\chi\in
A}\mathrm{supp}(\chi)$ and $C_{A}:=\sup\big{(}\\{\|\chi\|:\chi\in
A\\}\cup\\{0\\}\big{)}$.

###### Theorem 3.16

Let $1\leq p\leq q\leq\infty$, and let $L(X)$ be the free locally convex space
over a Tychonoff space $X$. Then:

  1. (i)

each $(p,q)$-limited subset of $L(X)$ is finite-dimensional;

  2. (ii)

$L(X)$ has the $GP_{(p,q)}$ property, the sequential $GP_{(p,q)}$ property and
the $b$-$GP_{(p,q)}$ property;

  3. (iii)

every locally convex space $E$ is a quotient space of $L(E)$.

###### Proof 17

(i) Suppose for a contradiction that there is a one-to-one infinite-
dimensional sequence $A=\\{\chi_{n}\\}_{n\in\omega}$ in
$L(X){\setminus}\\{0\\}$ which is a $(p,q)$-limited set. Since the support of
any $\chi\in L(X)$ is finite and $A$ is infinite-dimensional, without loss of
generality we can assume that $A$ satisfies the following condition:

  1. (a)

for every $n\in\omega$, there is
$x_{n+1}\in\mathrm{supp}(\chi_{n+1}){\setminus}\bigcup_{i\leq
n}\mathrm{supp}(\chi_{i})$.

Fix an arbitrary $x_{0}\in\mathrm{supp}(\chi_{0})$. Passing to a subsequence
of $A$ if needed, we can assume also that there is a sequence
$\\{U_{n}\\}_{n\in\omega}$ of open subsets of $X$ such that

  1. (b)

$U_{n}\cap\mathrm{supp}(\chi_{n})=\\{x_{n}\\}$ for every $n\in\omega$;

  2. (c)

$U_{n}\cap U_{m}=\emptyset$ for all distinct $n,m\in\omega$.

For every $n\in\omega$ and taking into account (b), choose $f_{n}\in C(X)$
such that

  1. (d)

$f_{n}\big{(}X{\setminus}U_{n}\big{)}=\\{0\\}$;

  2. (e)

$\langle f_{n},\chi_{n}\rangle=f_{n}(x_{n})\cdot\chi_{n}[x_{n}]=1$.

Since the support of any $\chi\in L(X)$ is finite, (c) and (d) imply that the
sequence $\\{f_{n}\\}_{n\in\omega}$ is weak∗ $p$-summable in
$C(X)=L(X)^{\prime}$ for every $p\in[1,\infty]$. On the other hand, (e)
implies

$\sup_{\chi\in A}|\langle f_{n},\chi\rangle|\geq\sup_{i\in\omega}|\langle
f_{n},\chi_{i}\rangle|\geq\langle f_{n},\chi_{n}\rangle=1\not\to 0,$

which means that $A$ is not a $(p,q)$-limited set, a contradiction.

(ii) immediately follows from (i).

(iii) By the definition of $L(E)$, the identity map $\mathsf{id}_{E}:E\to E$
can be extended to an operator $I:L(E)\to E$. Since $E$ is a closed subspace
of $L(E)$, it is clear that $I$ is a quotient map.

For the coarse $p$-limited subsets of $L(X)$ the situation is antipodal to the
$(p,q)$-limited subsets of $L(X)$.

###### Theorem 3.17

Let $p\in[1,\infty]$, and let $L(X)$ be the free locally convex space over a
Tychonoff space $X$. Then:

  1. (i)

each bounded subset $B$ of $L(X)$ is coarse $p$-limited;

  2. (ii)

$L(X)$ has the coarse $GP_{p}$ property if and only if each functionally
bounded subset of $X$ is finite.

###### Proof 18

(i) Recall that Proposition 2.7 of [20] states that a subset $A$ of $L(X)$ is
bounded if and only if $\mathrm{supp}(A)$ is functionally bounded in $X$ and
$C_{A}$ is finite. Let $B$ be a bounded subset of $L(X)$ and let
$T\in\mathcal{L}\big{(}L(X),\ell_{p}\big{)}$ (or
$T\in\mathcal{L}\big{(}L(X),c_{0}\big{)}$ if $p=\infty$). Then
$\mathrm{supp}(B)$ is functionally bounded in $X$, and hence
$T(\mathrm{supp}(B))$ is functionally bounded in the Banach space $\ell_{p}$
(or in $c_{0}$). Since any metric space is a $\mu$-space, the closure $K$ of
$T(\mathrm{supp}(B))$ is compact. Now the inclusion $T(B)\subseteq
C_{B}\cdot\overline{\mathrm{acx}}(K)$ implies that $T(B)$ is a relatively
compact subset of $\ell_{p}$ (or of $c_{0}$). Thus $B$ is a coarse $p$-limited
subset of $L(X)$.

(ii) Assume that $L(X)$ has the coarse $GP_{p}$ property. Then, by (i), every
bounded subset $B$ of $L(X)$ has compact closure. In particular, $L(X)$ is
quasi-complete. Therefore, by Theorem 3.8 of [24], each functionally bounded
subset of $X$ is finite. Conversely, if all functionally bounded subsets of
$X$ are finite, then, by Proposition 2.7 of [20], any bounded subset of $L(X)$
is finite-dimensional and hence relatively compact. Thus $L(X)$ trivially has
the coarse $GP_{p}$ property.

Theorems 3.16 and 3.17 suggest the following problem.

###### Problem 3.18

Let $1\leq p\leq q\leq\infty$. Characterize Tychonoff spaces $X$ for which
$L(X)$ has one of the $V^{\ast}_{(p,q)}$ type properties defined in Definition
2.5.

We know from (ii) of Lemma 3.1 that the case $(1,\infty)$ is the strongest one
in the sense that if $E$ has some $GP_{(1,\infty)}$ type property, then it has
the same $GP_{(p,q)}$ type property for all $1\leq p\leq q\leq\infty$. This
result and (v) of Proposition 3.8 motivate the problem of whether, for
example, the $spGP_{(1,\infty)}$ property implies some addition conditions on
the space $E$. We answer this problem in the affirmative in the rest of this
section.

###### Theorem 3.19

Let $E$ be a strict $(LF)$-space. If $E$ does not contain an isomorphic copy
of $\ell_{1}$ which is complemented in $E$, then the following assertions are
equivalent:

  1. (i)

$E$ has the $GP_{(1,\infty)}$ property;

  2. (ii)

$E$ has the coarse $GP_{1}$ property;

  3. (iii)

$E$ is a Montel space.

If (i)–(iii) hold, then $E$ is separable.

###### Proof 19

Since strict $(LF)$-spaces are barrelled by Proposition 11.3.1 of [32], (i)
and (ii) are equivalent by (v) of Proposition 3.8. Corollary 5.16 of [24]
states that a subset of $E$ is bounded if and only if it is a coarse
$1$-limited set. Therefore $E$ has the coarse $GP_{1}$ property if and only if
every bounded subset of $E$ is relatively compact, i.e. $E$ is a Montel space.

To prove that $E$ is separable, let $E:=\underrightarrow{\mbox{
s-$\mathsf{ind}$}}_{n}\,E_{n}$, where all $E_{n}$ are closed Fréchet subspaces
of $E$. By Proposition 11.5.4(b) of [32] and (iii), all spaces $E_{n}$ are
Montel. Therefore, by Theorem 11.6.2 of [32], all spaces $E_{n}$ are
separable. Thus also $E=\bigcup_{n\in\omega}E_{n}$ is separable.

###### Remark 3.20

Let $\Gamma$ be an uncountable set and let $1<p<\infty$. Then, by Theorem
3.19, the reflexive non-separable Banach space $\ell_{p}(\Gamma)$ has neither
the $GP_{(1,\infty)}$ property nor the coarse $GP_{1}$ property. However,
$\ell_{p}(\Gamma)$ has the coarse $GP_{p}$ property by Remark 3(2) of [27],
and it has the Gelfand–Phillips property by Proposition 3.8.

###### Proposition 3.21

Let $E$ be a sequentially complete barrelled locally convex space. If $E$ has
the $spGP_{(1,\infty)}$ property, then $E$ has the Schur property.

###### Proof 20

Let $S=\\{x_{n}\\}_{n\in\omega}$ be a weakly null sequence in $E$. Then $S$ is
weakly sequentially precompact. Therefore, by Corollary 3.14 of [24], $S$ is a
$(1,\infty)$-limited set. Hence, by the $spGP_{(1,\infty)}$ property, $S$ is
sequentially precompact. Thus, by Lemma 2.1, $x_{n}\to 0$ in $E$ and hence $E$
has the Schur property.

In general, the converse in Proposition 3.21 is not true as Example 3.12
shows. Nevertheless, in some important cases the converse holds true.

###### Proposition 3.22

Let $E$ be a sequentially complete barrelled locally convex space whose
bounded subsets are weakly sequentially precompact. Then $E$ has the
$spGP_{(1,\infty)}$ property if and only if it has the Schur property.

###### Proof 21

The necessity follows from Proposition 3.21. To prove the sufficiency, assume
that $E$ has the Schur property and let $A$ be a $(1,\infty)$-limited subset
of $E$. Since $A$ is bounded it is weakly sequentially precompact. Then, by
Lemma 2.2 of [24], $A$ is sequentially precompact. Thus $E$ has the
$spGP_{(1,\infty)}$ property.

###### Corollary 3.23

Let $E$ be a strict $(LF)$-space that does not contain an isomorphic copy of
$\ell_{1}$. Then $E$ has the Schur property if and only if it has one (and
hence all) of the properties $GP_{(1,\infty)}$, $EGP_{(1,\infty)}$,
$sGP_{(1,\infty)}$, $sEGP_{(1,\infty)}$, $prGP_{(1,\infty)}$,
$spGP_{(1,\infty)}$, $prEGP_{(1,\infty)}$, $spEGP_{(1,\infty)}$ and
$b$-$GP_{(1,\infty)}$.

###### Proof 22

By Proposition 3.6, it suffices to consider only the $spGP_{(1,\infty)}$
property. It is well known that $E$ is a complete barrelled space. By the
Ruess Theorem [40] mentioned before Theorem 2.6, $E$ has the Rosenthal
property. Since $E$ has no an isomorphic copy of $\ell_{1}$, it follows that
every bounded sequence has a weakly Cauchy subsequence (i.e. every bounded
subset is weakly sequentially precompact). Now Proposition 3.22 applies.

##  4 Characterizations of Gelfand–Phillips types properties

In this section we characterize locally convex spaces with Gelfand–Phillips
types properties, in particular, in some important partial cases.

Let $p,q\in[1,\infty]$, and let $E$ and $L$ be locally convex spaces.
Generalizing the notions of limited, limited completely continuous and limited
$p$-convergent operators between Banach spaces introduced in [6], [41] and
[18], respectively, and following [25], a linear map $T:E\to L$ is called
$(p,q)$-limited if $T(U)$ is a $(p,q)$-limited subset of $L$ for some
$U\in\mathcal{N}_{0}(E)$; if $p=q$ or $p=q=\infty$ we shall say that $T$ is
$p$-limited or limited, respectively.

###### Theorem 4.1

Let $1\leq p\leq q\leq\infty$, and let $E$ be a locally convex space. Then the
following assertions are equivalent:

  1. (i)

$E$ has the $prGP_{(p,q)}$ (resp., $GP_{(p,q)}$, $sGP_{(p,q)}$ or
$spGP_{(p,q)}$) property;

  2. (ii)

for every locally convex space $L$, if an operator $T:L\to E$ transforms
bounded sets of $L$ to $(p,q)$-limited sets of $E$, then $T$ transforms
bounded sets of $L$ to precompact (resp., relatively compact, relatively
sequentially compact or sequentially precompact) subsets of $E$;

  3. (iii)

every $(p,q)$-limited operator $T:L\to E$ from a locally convex space $L$ to
$E$ is precompact (resp., compact, sequentially compact or sequentially
precompact);

  4. (iv)

every $(p,q)$-limited operator $T:L\to E$ from a normed space $L$ to $E$ is
precompact (resp., compact, sequentially compact or sequentially precompact).

If $E$ is sequentially complete, then (i)–(iv) are equivalent to the following

  1. (v)

every $(p,q)$-limited operator $T:L\to E$ from a Banach space $L$ to $E$ is
precompact (resp., compact, sequentially compact or sequentially precompact).

###### Proof 23

(i)$\Rightarrow$(ii) Let $T:L\to E$ be an operator which transforms bounded
sets of $L$ to $(p,q)$-limited sets of $E$, and let $B$ be a bounded subset of
$L$. By assumption $T(B)$ is a $(p,q)$-limited subset of $E$. Then, by (i),
$T(B)$ is a precompact (resp., relatively compact, relatively sequentially
compact or sequentially precompact) subset of $E$, as desired.

The implications (ii)$\Rightarrow$(iii)$\Rightarrow$(iv) are trivial.

(iv)$\Rightarrow$(i) Fix a $(p,q)$-limited subset $B$ of $E$. It is clear that
the closed absolutely convex hull $D$ of the set $B$ in $E$ is also
$(p,q)$-limited. Let $L$ be the linear hull of $D$. Since $D$ is bounded in
$E$, the function $\|\cdot\|:L\to[0,\infty)$, $\|\cdot\|:x\mapsto\inf\\{r\geq
0:x\in rD\\}$ is a well-defined norm on the linear space $L$ and the set $D$
coincides with the closed unit ball $B_{L}$ of the normed space
$(L,\|\cdot\|)$. Since the identity inclusion $T:(L,\|\cdot\|)\to E$ is
continuous and the set $D=T(B_{L})$ is $(p,q)$-limited in $E$, the operator
$T$ is $(p,q)$-limited. By (iv), the set $D=T(B_{L})$ is precompact (resp.,
relatively compact, relatively sequentially compact or sequentially
precompact) in $E$, and hence so is the set $B\subseteq D$. Thus $E$ has the
$prGP_{(p,q)}$ (resp., $GP_{(p,q)}$, $sGP_{(p,q)}$ or $spGP_{(p,q)}$)
property.

Assume that $E$ is sequentially complete. Then the implication
(iv)$\Rightarrow$(v) is trivial. To prove the implication
(v)$\Rightarrow$(iv), let $T:L\to E$ be a $(p,q)$-limited operator from a
normed space $L$ to $E$. Then, by Proposition 3.7 of [22], $T$ can be extended
to a bounded operator ${\bar{T}}$ from the completion $\overline{L}$ of $L$ to
$E$. Observe that ${\bar{T}}(B_{\overline{L}})\subseteq\overline{T(B_{L})}$
and hence ${\bar{T}}$ is also $(p,q)$-limited. Thus, by (v), the operator
${\bar{T}}$ and hence also $T$ are precompact (resp., compact, sequentially
compact or sequentially precompact).

Let $1\leq p\leq q\leq\infty$, and let $E$ and $L$ be locally convex spaces.
Following Definition 16.1 of [22], a linear map $T:E\to L$ is called
$(q,p)$-convergent if it sends weakly $p$-summable sequences in $E$ to
strongly $q$-summable sequences in $L$. Recall (see § 19.4 of [32]) that a
sequence $\\{x_{n}\\}_{n\in\omega}$ in an lcs $L$ is called strongly
$p$-summable if $\big{(}q_{U}(x_{n})\big{)}\in\ell_{p}$ (or
$\big{(}q_{U}(x_{n})\big{)}\in c_{0}$ if $p=\infty$) for every
$U\in\mathcal{N}_{0}^{c}(E)$, where as usual $q_{U}$ denotes the gauge
functional of $U$.

###### Theorem 4.2

Let $1\leq p\leq q\leq\infty$, and let $E$ be a locally convex space. Then the
following assertions are equivalent:

  1. (i)

if $L$ is a locally convex space and $T:L\to E$ is an operator such that
$T^{\ast}:E^{\prime}_{w^{\ast}}\to L^{\prime}_{\beta}$ is $(q,p)$-convergent,
then $T$ transforms bounded sets of $L$ to relatively sequentially compact
(resp., sequentially precompact) subsets of $E$;

  2. (ii)

if $L$ is a normed space and $T:L\to E$ is an operator such that
$T^{\ast}:E^{\prime}_{w^{\ast}}\to L^{\prime}_{\beta}$ is $(q,p)$-convergent,
then $T$ is sequentially compact (resp., sequentially precompact);

  3. (iii)

the same as (ii) with $L=\ell_{1}^{0}$;

  4. (iv)

$E$ has the $sGP_{(p,q)}$ property (resp., the $spGP_{(p,q)}$ property).

Moreover, if $E$ is locally complete, then (i)-(iv) are equivalent to

  1. (v)

the same as (ii) with $L=\ell_{1}$.

###### Proof 24

The implications (i)$\Rightarrow$(ii)$\Rightarrow$(iii) and
(ii)$\Rightarrow$(v) are trivial.

(iii)$\Rightarrow$(iv) and (v)$\Rightarrow$(iv): Let $A$ be a $(p,q)$-limited
subset of $E$. Fix an arbitrary sequence $S=\\{x_{n}\\}_{n\in\omega}$ in $A$,
so $S$ is a bounded subset of $E$. Therefore, by Proposition 5.9 of [24], the
linear map $T:\ell_{1}^{0}\to E$ (or $T:\ell_{1}\to E$ if $E$ is locally
complete) defined by

$T(a_{0}e_{0}+\cdots+a_{n}e_{n}):=a_{0}x_{0}+\cdots+a_{n}x_{n}\quad(n\in\omega,\;a_{0},\dots,a_{n}\in\mathbb{F}).$

is continuous. For every $n\in\omega$ and each $\chi\in E^{\prime}$, we have
$\langle
T^{\ast}(\chi),e_{n}\rangle=\langle\chi,T(e_{n})\rangle=\langle\chi,x_{n}\rangle$
and hence
$T^{\ast}(\chi)=\big{(}\langle\chi,x_{n}\rangle\big{)}_{n}\in\ell_{\infty}$.
In particular,
$\|T^{\ast}(\chi)\|_{\ell_{\infty}}=\sup_{n\in\omega}|\langle\chi,x_{n}\rangle|$.

Let now $\\{\chi_{n}\\}_{n\in\omega}$ be a weak∗ $p$-summable sequence in
$E^{\prime}_{w^{\ast}}$. Since $A$ and hence also $S$ are $(p,q)$-limited sets
we obtain
$\big{(}\|T^{\ast}(\chi_{n})\|_{\ell_{\infty}}\big{)}=\big{(}\sup_{i\in\omega}|\langle\chi_{n},x_{i}\rangle|\big{)}\in\ell_{q}$
(or $\in c_{0}$ if $q=\infty$). Therefore $T^{\ast}$ is $(q,p)$-convergent,
and hence, by (iii) or (v), the operator $T$ is sequentially compact (resp.,
sequentially precompact). Therefore $S=\\{T(e_{n})\\}_{n\in\omega}$ has a
convergent (resp., Cauchy) subsequence in $E$. Hence $A$ is a relatively
sequentially compact (resp., sequentially precompact) subset of $E$. Thus $E$
has the $sGP_{(p,q)}$ property (resp., the $spGP_{(p,q)}$ property).

(iv)$\Rightarrow$(i) Let $T:L\to E$ be an operator from a locally convex space
$(L,\tau)$ such that $T^{\ast}:E^{\prime}_{w^{\ast}}\to L^{\prime}_{\beta}$ is
$(q,p)$-convergent. Fix an arbitrary bounded subset $B$ of $L$. Then
$U:=B^{\circ}$ is a neighbourhood of zero of $L^{\prime}_{\beta}$. Let
$\\{\chi_{n}\\}_{n\in\omega}$ be a weakly $p$-summable sequence in
$E^{\prime}_{w^{\ast}}$. By assumption $T^{\ast}:E^{\prime}_{w^{\ast}}\to
L^{\prime}_{\beta}$ is $(q,p)$-convergent, and hence

$\big{(}q_{U}(T^{\ast}(\chi_{n}))\big{)}\in\ell_{q}\quad\mbox{(or $\in c_{0}$
if $q=\infty$)},$ (4.1)

where $q_{U}$ is the gauge functional of $U$. For every $\chi\in E^{\prime}$,
we have

$\displaystyle q_{U}(T^{\ast}(\chi))$
$\displaystyle=\inf\big{\\{}\lambda>0:T^{\ast}(\chi)\in\lambda
B^{\circ}\big{\\}}=\inf\big{\\{}\lambda>0:\sup_{x\in B}|\langle
T^{\ast}(\chi),x\rangle|\leq\lambda\big{\\}}$ $\displaystyle=\sup_{x\in
B}|\langle T^{\ast}(\chi),x\rangle|=\sup_{x\in B}|\langle\chi,T(x)\rangle|.$

Therefore, (4.1) means that the set $T(B)$ is a $(p,q)$-limited subset of $E$.
By (iv), we obtain that $T(B)$ is a relatively sequentially compact (resp.,
sequentially precompact) subsets of $E$.

Let $p,q,q^{\prime}\in[1,\infty]$, $q^{\prime}\leq q$, and let $E$ and $L$ be
locally convex spaces. Following [25], a linear map $T:E\to L$ is called
$(q^{\prime},q)$-limited $p$-convergent if $T(x_{n})\to 0$ for every weakly
$p$-summable sequence $\\{x_{n}\\}_{n\in\omega}$ in $E$ which is a
$(q^{\prime},q)$-limited subset of $E$.

The next our purpose is to give another characterization of locally convex
spaces $E$ with the sequentially precompact $GP_{(q^{\prime},q)}$ property
under an additional assumption that $E$ has the $wCSP$. First we prove the
following lemma.

###### Lemma 4.3

Let $1\leq p\leq q\leq\infty$, and let $E$ be a locally convex space. If $E$
has the $spGP_{(p,q)}$ property, then the identity map $\mathsf{id}_{E}:E\to
E$ is $(p,q)$-limited $\infty$-convergent.

###### Proof 25

Fix a $(p,q)$-limited weakly null sequence $S=\\{x_{n}\\}_{n\in\omega}$ in
$E$, and suppose for a contradiction that $x_{n}\not\to 0$ in $E$. Passing to
a subsequence if needed we can assume that $S\cap U=\emptyset$ for some
neighborhood $U$ of zero in $E$. Since $S$ is a $(p,q)$-limited set and $E$
has the $spGP_{(p,q)}$ property, $S$ is sequentially precompact hence
precompact in $E$. As $S$ is also weakly null, Lemma 2.1, implies that
$x_{n}\to 0$. But this is impossible since $S\cap U=\emptyset$.

###### Theorem 4.4

Let $1\leq p\leq q\leq\infty$, and let $E$ be a locally convex space with the
$wCSP$. Then $E$ has the $spGP_{(p,q)}$ property if and only if the identity
map $\mathsf{id}_{E}:E\to E$ is $(p,q)$-limited $\infty$-convergent.

###### Proof 26

The necessity immediately follows from Lemma 4.3. To prove the sufficiency
assume that the identity map $\mathsf{id}_{E}:E\to E$ is $(p,q)$-limited
$\infty$-convergent, and let $A$ be a $(p,q)$-limited subset of $E$. We have
to show that any sequence $S=\\{x_{n}\\}_{n\in\omega}$ in $A$ contains a
Cauchy subsequence (in $E$). Since $E$ has the $wCSP$, $S$ has a subsequence
$S^{\prime}=\\{x_{n_{k}}\\}_{k\in\omega}$ which is weakly Cauchy. Let
$\\{k_{i}\\}_{i\in\omega}$ be a strictly increasing sequence in $\omega$. Then
$\\{x_{n_{k_{i}}}-x_{n_{k_{i+1}}}\\}_{k\in\omega}$ is weakly null. As
$\\{x_{n_{k_{i}}}-x_{n_{k_{i+1}}}\\}_{k\in\omega}$ is also a $(p,q)$-limited
set and $\mathsf{id}_{E}$ is $(p,q)$-limited $\infty$-convergent, it follows
that $x_{n_{k_{i}}}-x_{n_{k_{i+1}}}\to 0$ in $E$. Thus $S^{\prime}$ is Cauchy
in $E$.

We know from Theorem 1.3 that a Banach space $E$ has the Gelfand–Phillips
property if and only if every limited weakly null sequence in $E$ is norm
null. An analogue assertion for the coarse $GP_{p}$ property (and $2\leq
p<\infty$) was obtained in Theorem 3 of [27]. Below we generalize both these
results.

###### Theorem 4.5

Let $1\leq p\leq q\leq\infty$, and let $(E,\tau)$ be a locally convex space.

  1. (i)

If $E$ has the $prGP_{(p,q)}$ property (the $spGP_{(p,q)}$ property), then
every weakly null $(p,q)$-limited sequence in $E$ is $\tau$-null.

  2. (ii)

If $E$ has the coarse $prGP_{p}$ property (the coarse $spGP_{p}$ property),
then every weakly null coarse $p$-limited sequence in $E$ is $\tau$-null.

  3. (iii)

Assume additionally that $p\geq 2$ and $E$ has the Rosenthal property. Then
$E$ has the $prGP_{(p,q)}$ property if and only if every weakly null
$(p,q)$-limited sequence in $E$ is $\tau$-null.

  4. (iv)

Assume additionally that $p\geq 2$ and $E$ has the Rosenthal property. Then
$E$ has the coarse $prGP_{p}$ property if and only if every weakly null coarse
$p$-limited sequence in $E$ is $\tau$-null.

###### Proof 27

(i) and (ii): Let $S=\\{x_{n}\\}_{n\in\omega}$ be a weakly null
$(p,q)$-limited (resp., coarse $p$-limited) sequence in $E$. By the
$prGP_{(p,q)}$ property (resp., the $spGP_{(p,q)}$ property, the coarse
$prGP_{p}$ property or the coarse $spGP_{p}$ property), $S$ is a precompact
subset of $E$. Thus, by Lemma 2.1, $S$ is $\tau$-null.

(iii) and (iv): Assume that $p\geq 2$ and $E$ has the Rosenthal property. The
necessity is proved in (i) and (ii). To prove the sufficiency, let every
weakly null $(p,q)$-limited (resp., coarse $p$-limited) sequence in $E$ is
$\tau$-null. Let $A$ be a $(p,q)$-limited (resp., coarse $p$-limited) subset
of $E$, and suppose for a contradiction that $A$ is not $\tau$-precompact.
Then there are a sequence $\\{a_{n}\\}_{n\in\omega}$ in $A$ and
$U\in\mathcal{N}_{0}(E)$ such that $a_{n}-a_{m}\not\in U$ for all distinct
$n,m\in\omega$. By Theorem 5.21 of [24] and Theorem 2.6, we obtain that $A$ is
weakly sequentially precompact. Therefore, passing to a subsequence if needed
we can assume that $\\{a_{n}\\}_{n\in\omega}$ is weakly Cauchy. In particular,
the sequence $\\{a_{n}-a_{n+1}\\}_{n\in\omega}$ is weakly null. Since
$\\{a_{n}-a_{n+1}\\}_{n\in\omega}$ is also a $(p,q)$-limited (resp., coarse
$p$-limited) subset of $E$ (see Lemmas 3.1 and 4.1 of [24]), it follows that
$a_{n}-a_{n+1}\to 0$ in $E$ which contradicts the condition
$a_{n}-a_{n+1}\not\in U$ for all $n\in\omega$. Thus $A$ is $\tau$-precompact
and hence $E$ has the $prGP_{(p,q)}$ property (resp., the coarse $prGP_{p}$
property).

Let $D$ be a subset of a locally convex space $E$. Recall that a subset $A$ of
$E$ is called $D$-separated if $a-a^{\prime}\not\in D$ for all distinct
$a,a^{\prime}\in A$, and $A$ is separated if $A$ is $U$-separated for some
neighborhood $U$ of zero. We also denote by $(\ell_{r})_{p}$ and $(c_{0})_{p}$
the Banach spaces $\ell_{r}$ and $c_{0}$ endowed with the topology induced
from $\mathbb{F}^{\omega}$.

Since any limited set and each $(p,q)$-limited set are $(p,\infty)$-limited,
the case $q=\infty$ is of interest. Below we give necessary conditions to have
the precompact $GP_{(p,\infty)}$ property.

###### Proposition 4.6

Let $p\in[1,\infty]$, and let $E$ be a locally convex space. Consider the
following assertions:

  1. (i)

$E$ has the precompact $GP_{(p,\infty)}$ property;

  2. (ii)

For every bounded non-precompact set $B\subseteq E$, there is a weak∗
$p$-summable sequence $\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$ such that
$\|\chi_{n}\|_{B}\not\to 0$.

  3. (iii)

For any infinite bounded separated subset $D$ of $E$ and every $\delta>0$
there exist a sequence $\\{x_{n}\\}_{n\in\omega}$ in $D$ and a weak∗
$p$-summable sequence $\\{f_{n}\\}_{n\in\omega}$ in $E^{\prime}$ such that
$|f_{n}(x_{k})|<\delta$ and $|f_{n}(x_{n})|>1+\delta$ for all natural numbers
$k<n$.

  4. (iv)

For any infinite bounded separated set $D$ in $E$ there exists a continuous
operator $T:E\to(\ell_{p})_{p}$ (or $T:E\to(c_{0})_{p}$ if $p=\infty$) such
that $T(D)$ is not precompact in the Banach space $\ell_{p}$ (or in $c_{0}$ if
$p=\infty$).

Then (i)$\Rightarrow$(ii)$\Rightarrow$(iii)$\Rightarrow$(iv).

###### Proof 28

(i)$\Rightarrow$(ii) Fix any bounded non-precompact set $B\subseteq E$. By
(i), $E$ has the precompact $GP_{(p,\infty)}$ property and hence $B$ is not
$(p,\infty)$-limited. Therefore there exists a weak∗ $p$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$ such that
$\|\chi_{n}\|_{B}\not\to 0$, as desired.

(ii)$\Rightarrow$(iii) Fix any $\delta>0$ and any infinite bounded separated
set $D$ in $E$. Choose $U\in\mathcal{N}_{0}(E)$ such that the set $D$ is
$U$-separated. Observe that $D$ is not precompact because $D$ is $U$-separated
and infinite. By (ii), there is a weak∗ $p$-summable sequence
$\\{g_{n}\\}_{n\in\omega}$ in $E^{\prime}$ such that $\|g_{n}\|_{D}\not\to 0$.
Passing to a subsequence if needed we can assume that $\|g_{n}\|_{D}>a>0$ for
all $n\in\omega$. Further, multiplying each functional $g_{n}$ by
$(1+\delta)/a$ we can assume that

$\|g_{n}\|_{D}=\sup_{d\in D}|g_{n}(d)|>1+\delta\quad\mbox{ for every
}\;n\in\omega.$ (4.2)

Now, choose an arbitrary $x_{0}\in D$ such that $|g_{0}(x_{0})|>1+\delta$.
Assume that, for $k\in\omega$, we found $x_{0},\dots,x_{k}\in D$ and a
sequence $0=n_{0}<n_{1}<\cdots<n_{k}$ of natural numbers such that

$|g_{n_{j}}(x_{i})|<\delta\;\mbox{ and
}\;|g_{n_{j}}(x_{j})|>1+\delta\quad\mbox{ for every }\;0\leq i<j\leq k.$

Since $g_{n}\to 0$ in the weak∗ topology $\sigma(E^{\prime},E)$, (4.2) implies
that there are $n_{k+1}>n_{k}$ and $x_{k+1}\in
D{\setminus}\\{x_{0},\dots,x_{k}\\}$ such that
$|g_{n_{k+1}}(x_{k+1})|>1+\delta$ and $|g_{n_{k+1}}(x_{i})|<\delta$ for every
$i\leq k$. For every $k\in\omega$, put $f_{k}:=g_{n_{k}}$, and observe that
the subsequence $\\{f_{k}\\}_{k\in\omega}$ of $\\{g_{n}\\}_{n\in\omega}$ is
also weak∗ $p$-summable in $E^{\prime}$ and

$|f_{k}(x_{i})|=|g_{n_{k}}(x_{i})|<\delta\;\mbox{ and
}\;|f_{k}(x_{k})|=|g_{n_{k}}(x_{k})|>1+\delta$

for any numbers $i<k$, as desired.

(iii)$\Rightarrow$(iv) Let $D$ be any bounded separated set in $E$. By (iii)
applied for $\delta=\tfrac{1}{2}$, there exist a sequence
$\\{x_{n}\\}_{n\in\omega}$ in $D$ and a weak∗ $p$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$ such that
$|\chi_{m}(x_{n})|<\frac{1}{2}$ and $|\chi_{m}(x_{m})|>\frac{3}{2}$ for all
$n<m$. Then
$|\chi_{m}(x_{m})-\chi_{m}(x_{n})|\geq|\chi_{m}(x_{m})|-|\chi_{m}(x_{n})|>\frac{3}{2}-\frac{1}{2}=1$
for every $n<m$. It follows that the operator

$T:E\to(\ell_{p})_{p}\;\;(\mbox{or }T:E\to(c_{0})_{p}\mbox{ if
$p=\infty$}),\quad T(x):=\big{(}\chi_{n}(x)\big{)}_{n\in\omega},$

is well-defined and continuous. Since

$\|T(x_{n})-T(x_{m})\|\geq|\chi_{m}(x_{m})-\chi_{m}(x_{n})|>1\;\;\mbox{ for
every $n<m$},$

the set $T(D)\supseteq\\{T(x_{n})\\}_{n\in\omega}$ is not precompact in
$\ell_{p}$ (or in $c_{0}$ if $p=\infty$).

In two extreme cases when $p=1$ or $p=\infty$ we can reverse implications in
Proposition 4.6, cf. Theorem 2.2 of [5].

###### Theorem 4.7

Let $p\in\\{1,\infty\\}$. For a locally convex space $E$ the following
assertions are equivalent:

  1. (i)

$E$ has the precompact $GP_{(p,\infty)}$ property;

  2. (ii)

For every bounded non-precompact set $B\subseteq E$, there is a weak∗
$p$-summable sequence $\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$ such that
$\|\chi_{n}\|_{B}\not\to 0$.

  3. (iii)

For any infinite bounded separated subset $D$ of $E$ and every $\delta>0$
there exist a sequence $\\{x_{n}\\}_{n\in\omega}$ in $D$ and a weak∗
$p$-summable sequence $\\{f_{n}\\}_{n\in\omega}$ in $E^{\prime}$ such that
$|f_{n}(x_{k})|<\delta$ and $|f_{n}(x_{n})|>1+\delta$ for all natural numbers
$k<n$.

  4. (iv)

For any infinite bounded separated set $D$ in $E$ there exists a continuous
operator $T:E\to(\ell_{1})_{p}$ (or $T:E\to(c_{0})_{p}$ if $p=\infty$) such
that $T(D)$ is not precompact in the Banach space $\ell_{1}$ (or in $c_{0}$ if
$p=\infty$).

###### Proof 29

The implications (i)$\Rightarrow$(ii)$\Rightarrow$(iii)$\Rightarrow$(iv) are
proved in Proposition 4.6.

(iv)$\Rightarrow$(i) To show that $E$ has the precompact $GP_{(p,\infty)}$
property we show that any non-precompact subset of $E$ is not
$(p,\infty)$-limited. Fix a bounded non-precompact subset $P\subseteq E$. Then
there exists $U\in\mathcal{N}_{0}(E)$ such that $L\not\subseteq F+U$ for any
finite subset $F\subseteq E$. For every $n\in\omega$, choose inductively a
point $z_{n}\in P$ so that $z_{n}\notin\bigcup_{k<n}(z_{k}+U)$. Observe that
the set $D:=\\{z_{n}:n\in\omega\\}$ is infinite, bounded and $U$-separated, so
non-precompact. By (iv), there exists a continuous operator
$T:E\to(\ell_{1})_{p}$ (or $T:E\to(c_{0})_{p}$ if $p=\infty$) such that the
set $T(D)$ is not precompact in the Banach space $\ell_{1}$ (or in $c_{0}$ if
$p=\infty$). We distinguish between two cases.

Case 1. Assume that $p=\infty$. We follow the idea of Theorem 2.2 of [5].
Since $c_{0}$ is a Banach space and $T(D)$ is not precompact, there exist a
sequence $\\{a_{n}\\}_{n\in\omega}$ in $D$ and $\delta>0$ such that
$\|T(a_{n})-T(a_{m})\|_{c_{0}}\geq\delta$ for all distinct $n,m\in\omega$ (see
also (ii) of Proposition 2.3). Observe that the sequence
$\\{T(a_{n})\\}_{n\in\omega}$ is bounded in the Banach space $c_{0}$.
Therefore there are two sequences $0\leq n_{0}<n_{1}<\cdots$ and $0\leq
m_{0}<m_{1}<\cdots$ of natural numbers such that

$\big{|}\langle
e^{\prime}_{m_{k}},T(a_{n_{k}})\rangle\big{|}>\tfrac{\delta}{2}\quad\mbox{ for
every }\;k\in\omega,$

where $e^{\prime}_{n}:(c_{0})_{p}\to\mathbb{F}$ is the $n$th coordinate
functional. For every $k\in\omega$, set $f_{k}:=e^{\prime}_{m_{k}}\circ T$. It
follows that $\\{f_{k}\\}_{k\in\omega}$ is a weak∗ null in $E^{\prime}$ and

$\|f_{k}\|_{P}=\sup_{x\in
P}|f_{k}(x)|\geq|f_{k}(a_{n_{k}})|>\tfrac{\delta}{2}$

for every $k\in\omega$, witnessing that the set $P$ is not limited.

Case 2. Assume that $p=1$. Recall that, by (i) of Proposition 2.3, a bounded
subset $A$ of $\ell_{1}$ is precompact if and only if

$\lim_{m\to\infty}\sup\Big{\\{}\sum_{m\leq n}|x_{n}|:x=(x_{n})\in
A\Big{\\}}=0.$ (4.3)

For every $k\in\omega$, set
$y_{k}:=T(z_{k})=(a_{n,k})_{n\in\omega}\in\ell_{1}$. Since $T(D)$ is non-
precompact, (4.3) implies that there is $\varepsilon>0$ such that

$\sup\Big{\\{}\sum_{m\leq n}|a_{n,k}|:y_{k}=(a_{n,k})\in
T(D)\Big{\\}}>10\varepsilon\quad\mbox{ for every $m\in\omega$}.$ (4.4)

Let $i=0$. Set $m_{0}:=0$. By (4.4), choose $k_{0},r_{0}\in\omega$ such that
$m_{0}<r_{0}$ and

$\sum_{n=m_{0}}^{r_{0}}|a_{n,k_{0}}|>8\varepsilon.$ (4.5)

Choose $m_{1}>r_{0}$ such that

$\sum_{n\geq m_{1}}|a_{n,k}|<\varepsilon\quad\mbox{ for every }\;k\leq k_{0}.$
(4.6)

For $i=1$, (4.4)–(4.6) imply that there are $k_{1},r_{1}\in\omega$ such that
$k_{1}>k_{0}$, $m_{1}<r_{1}$ and

$\sum_{n=m_{1}}^{r_{1}}|a_{n,k_{1}}|>8\varepsilon.$ (4.7)

Choose $m_{2}>r_{1}$ such that

$\sum_{n\geq m_{2}}|a_{n,k}|<\varepsilon\quad\mbox{ for every }\;k\leq k_{1}.$

Continuing this process we find sequences $0=m_{0}<r_{0}<m_{1}<r_{1}<\cdots$
and $k_{0}<k_{1}<\cdots$ such that for every $i\in\omega$, we have

$\sum_{n=m_{i}}^{r_{i}}|a_{n,k_{i}}|>8\varepsilon\;\;\mbox{ and
}\;\;\sum_{n\geq m_{i+1}}|a_{n,k_{i}}|<\varepsilon.$ (4.8)

For every $i\in\omega$, by Lemma 6.3 of [38], there is a subset $F_{i}$ of
$[m_{i},r_{i}]$ such that

$\Big{|}\sum_{n\in F_{i}}a_{n,k_{i}}\Big{|}>2\varepsilon.$ (4.9)

For every $n\in\omega$, let $e^{\ast}_{n}$ be the $n$th coordinate functional
of the dual space $(\ell_{1})_{p}^{\prime}$ of $(\ell_{1})_{p}$. For every
$i\in\omega$, set

$\chi_{i}:=\sum_{n\in F_{i}}e^{\ast}_{n}.$

Observe that the sequence $\\{\chi_{i}\\}_{i\in\omega}$ is weak∗ $1$-summable
in $(\ell_{1})_{p}^{\prime}$ because if $y=(a_{n})\in\ell_{1}$, then the
inequalities $m_{i}<r_{i}<m_{i+1}$ imply that all $F_{i}$ are pairwise
disjoint and hence

$\sum_{i\in\omega}|\langle\chi_{i},y\rangle|=\sum_{i\in\omega}\Big{|}\sum_{n\in
F_{i}}a_{n}\Big{|}\leq\sum_{i\in\omega}\sum_{n\in
F_{i}}|a_{n}|\leq\sum_{n\in\omega}|a_{n}|=\|y\|_{\ell_{1}}<\infty.$

Therefore the sequence $\\{T^{\ast}(\chi_{i})\\}_{i\in\omega}$ is weak∗
$1$-summable in $E^{\prime}$. For every $i\in\omega$, the inequality (4.9)
implies

$\sup_{x\in P}|\langle
T^{\ast}(\chi_{i}),x\rangle|\geq|\langle\chi_{i},T(z_{k_{i}})\rangle|=|\langle\chi_{i},y_{k_{i}}\rangle|=\Big{|}\sum_{n\in
F_{i}}a_{n,k_{i}}\Big{|}>2\varepsilon,$

which means that the set $P$ is not $(1,\infty)$-limited.

The next theorem immediately follows from Corollary 3.18 of [24] and, under
some conditions, it characterizes the (precompact) $GP_{(p,\infty)}$ property.

###### Theorem 4.8

Let $E$ be a $p$-barrelled Mackey space such that $E^{\prime}_{w^{\ast}}$ is a
weakly $p$-angelic space. Then $E$ has the precompact $GP_{(p,\infty)}$
property. If in addition $E$ is von Neumann complete, then $E$ has the
$GP_{(p,\infty)}$ property.

Since the case $p=\infty$ is of independent interest, we select the next
corollary which follows from Corollary 3.19 of [24] (and in fact from Theorem
4.8).

###### Corollary 4.9

Let a locally convex space $E$ satisfy one of the following conditions:

  1. (i)

$E$ is a $c_{0}$-barrelled Mackey space such that $E^{\prime}_{w^{\ast}}$ is a
weakly angelic space;

  2. (ii)

$E$ is a reflexive space such that $E^{\prime}_{\beta}$ is a weakly angelic
space;

  3. (iii)

$E$ is a separable $c_{0}$-barrelled Mackey space.

Then $E$ has the precompact $GP$ property. If in addition $E$ is von Neumann
complete, then $E$ has the $GP$ property.

Let $p\in[1,\infty]$, and let $E$ and $L$ be locally convex spaces. Following
[25], an operator $T\in\mathcal{L}(E,L)$ is called coarse $p$-limited if there
is $U\in\mathcal{N}_{0}(E)$ such that $T(U)$ is a coarse $p$-limited subset of
$L$.

Banach spaces with the coarse $p$-Gelfand–Phillips property are characterized
in Proposition 13 of [27]. The next theorem generalizes that result (we use in
the proof the easy fact that a subset $A$ of $E$ is precompact if and only if
each sequence in $A$ is precompact).

###### Theorem 4.10

For $p\in[1,\infty]$ and a locally convex space $E$, the following assertions
are equivalent:

  1. (i)

$E$ has the coarse $sGP_{p}$ property (resp., the coarse $spGP_{p}$ property
or the coarse $prGP_{p}$ property);

  2. (ii)

for every locally convex space $Y$, if an operator $T:Y\to E$ transforms
bounded sets of $Y$ to coarse $p$-limited sets of $E$, then $T$ transforms
bounded sets of $Y$ to relatively sequentially compact (resp., sequentially
precompact or precompact) subsets of $E$;

  3. (iii)

for every normed space $Y$, each coarse $p$-limited operator $T:Y\to E$ is
sequentially compact (resp., sequentially precompact or precompact);

  4. (iv)

each coarse $p$-limited operator $T:\ell_{1}^{0}\to E$ is sequentially compact
(resp., sequentially precompact or precompact).

If in addition $E$ is locally complete, then (i)-(iv) are equivalent to

  1. (v)

each coarse $p$-limited operator $T:\ell_{1}\to E$ is sequentially compact
(resp., sequentially precompact or precompact).

###### Proof 30

The implications (i)$\Rightarrow$(ii)$\Rightarrow$(iii)$\Rightarrow$(iv) and
(iii)$\Rightarrow$(v) are trivial.

(iv)$\Rightarrow$(i) and (v)$\Rightarrow$(i): Let $A$ be a coarse $p$-limited
subset of $E$. To show that $A$ is relatively sequentially compact (resp.,
sequentially precompact or precompact), let $\\{x_{n}\\}_{n\in\omega}$ be a
sequence in $A$. Then, by Proposition 14.9 of [22], the linear map
$T:\ell_{1}^{0}\to E$ (or $T:\ell_{1}\to E$ if $E$ is locally complete)
defined by

$T(a_{0}e_{0}+\cdots+a_{n}e_{n}):=a_{0}x_{0}+\cdots+a_{n}x_{n}\quad(n\in\omega,\;a_{0},\dots,a_{n}\in\mathbb{F}).$

is continuous. It is clear that
$T(B_{\ell_{1}^{0}})\subseteq\overline{\mathrm{acx}}\big{(}\\{x_{n}\\}_{n\in\omega}\big{)}$
(or
$T(B_{\ell_{1}})\subseteq\overline{\mathrm{acx}}\big{(}\\{x_{n}\\}_{n\in\omega}\big{)}$
if $E$ is locally complete). Since
$\overline{\mathrm{acx}}\big{(}\\{x_{n}\\}_{n\in\omega}\big{)}\subseteq\overline{\mathrm{acx}}(A)$
is a coarse $p$-limited set (see (ii) of Lemma 4.1 of [24]), it follows that
the operator $T$ is coarse $p$-limited. Therefore, by (iv) or (v), the
sequence $\\{x_{n}\\}_{n\in\omega}\subseteq T(B_{\ell_{1}^{0}})$ has a
convergent subsequence (resp., a Cauchy subsequence or
$T\big{(}\\{e_{n}\\}_{n\in\omega}\big{)}=\\{x_{n}\\}_{n\in\omega}$ is
precompact). Thus $A$ is relatively sequentially compact (resp., sequentially
precompact or precompact), as desired.

##  5 $p$-Gelfand–Phillips sequentially compact property of order
$(q^{\prime},q)$

We start this section with the next assertion. Recall (see Definition 1.13)
that an lcs $(E,\tau)$ is said to have the $p$-$GPscP_{(q^{\prime},q)}$ if
every weakly $p$-summable sequence in $E$ which is also a
$(q^{\prime},q)$-limited set is $\tau$-null.

###### Proposition 5.1

Let $1\leq q^{\prime}\leq q\leq\infty$, and let $E$ be a locally convex space.

  1. (i)

If $E$ has the $prGP_{(q^{\prime},q)}$, then it has the
$p$-$GPscP_{(q^{\prime},q)}$ for every $p\in[1,\infty]$.

  2. (ii)

If $q^{\prime}\geq 2$ and $E$ has the Rosenthal property, then $E$ has the
$prGP_{(q^{\prime},q)}$ if and only if it has the $GPscP_{(q^{\prime},q)}$.

###### Proof 31

(i) follows from (i) of Theorem 4.5, and (ii) follows from (iii) of Theorem
4.5.

###### Remark 5.2

The converse in (i) of Proposition 5.1 is not true in general. Indeed, let
$1<r,p<2$ and $E=\ell_{r}$. By Corollary 3.23, $\ell_{r}$ does not have the
$prGP_{(1,\infty)}$. On the other hand, let $\\{x_{n}\\}_{n\in\omega}$ be a
weakly $p$-summable sequence in $E$ which is also a $(q^{\prime},q)$-limited
set, and suppose for a contradiction that $x_{n}\not\to 0$. Passing to a
subsequence we can assume that $1/c\leq\|x_{n}\|\leq c$ for some $c>1$ and all
$n\in\omega$. Dividing $x_{n}$ by its norm we can assume also that
$\\{x_{n}\\}_{n\in\omega}$ is a normalized sequence. Taking into account that
$\\{x_{n}\\}_{n\in\omega}$ is weakly null, Proposition 2.1.3 of [1] implies
that there is a subsequence $\\{x_{n_{k}}\\}_{k\in\omega}$ which is a basic
sequence equivalent to the canonical basis of $\ell_{r}$ and such that
$\overline{\mathsf{span}}\big{(}\\{x_{n_{k}}\\}_{k\in\omega}\big{)}$ is
complemented in $\ell_{r}$. It follows that the canonical basis
$\\{e_{n}\\}_{n\in\omega}$ of $\ell_{r}$ is weakly $p$-summable that
contradicts Example 4.4 of [22] since $p<r^{\ast}$. Thus $x_{n}\to 0$ and
$\ell_{r}$ has the $p$-$GPscP_{(q^{\prime},q)}$ for all $1\leq q^{\prime}\leq
q\leq\infty$.

Let $p,q\in[1,\infty]$. Following [23], a locally convex space $E$ is said to
have the sequential Dunford–Pettis property of order $(p,q)$ (the sequential
$DP_{(p,q)}$ property) if $\lim_{n\to\infty}\langle\chi_{n},x_{n}\rangle=0$
for every weakly $p$-summable sequence $\\{x_{n}\\}_{n\in\omega}$ in $E$ and
each weakly $q$-summable sequence $\\{\chi_{n}\\}_{n\in\omega}$ in
$E^{\prime}_{\beta}$. Following [25], a linear map $T$ from $E$ to a locally
convex space $L$ is called weakly $(p,q)$-convergent if
$\lim_{n\to\infty}\langle\eta_{n},T(x_{n})\rangle=0$ for every weakly
$q$-summable sequence $\\{\eta_{n}\\}_{n\in\omega}$ in $L^{\prime}_{\beta}$
and each weakly $p$-summable sequence $\\{x_{n}\\}_{n\in\omega}$ in $E$. The
following proposition, which will be used below, shows that the sequential
$DP_{(p,q)}$ property of the range implies some strong additional properties
of operators.

###### Proposition 5.3

Let $p,q\in[1,\infty]$, and let $E$ and $L$ be locally convex spaces. If $L$
has the sequential $DP_{(p,q)}$ property (for instance, $L$ is a
quasibarrelled locally complete space with the Dunford–Pettis property, e.g.
$L=\ell_{\infty}$), then each operator $T\in\mathcal{L}(E,L)$ is weakly
$(p,q)$-convergent.

###### Proof 32

Let $\\{\eta_{n}\\}_{n\in\omega}$ be a weakly $q$-summable sequence in
$L^{\prime}_{\beta}$, and let $\\{x_{n}\\}_{n\in\omega}$ be a weakly
$p$-summable sequence in $E$. Then $\\{T(x_{n})\\}_{n\in\omega}$ is a weakly
$p$-summable sequence in $L$. Now the sequential $DP_{(p,q)}$ property of $L$
implies $\langle\eta_{n},T(x_{n})\rangle\to 0$. Thus $T$ is weakly
$(p,q)$-convergent.

If $L$ is a quasibarrelled locally complete space with the Dunford–Pettis
property, Corollary 5.13 of [23] implies that $L$ has the sequential
$DP_{(p,q)}$ property.

The following theorem characterizes locally convex spaces $E$ with the
$p$-$GPscP_{(q^{\prime},q)}$ and explains our usage of “sequentially compact”
in the notion of the $p$-$GPscP_{(q^{\prime},q)}$.

###### Theorem 5.4

Let $p,q,q^{\prime}\in[1,\infty]$, $q^{\prime}\leq q$. Then for a locally
convex space $(E,\tau)$, the following assertions are equivalent:

  1. (i)

$E$ has the $p$-$GPscP_{(q^{\prime},q)}$;

  2. (ii)

each operator $T\in\mathcal{L}(E,\ell_{\infty})$ is $(q^{\prime},q)$-limited
$p$-convergent;

  3. (iii)

each weakly $(p,q)$-convergent operator $T\in\mathcal{L}(E,\ell_{\infty})$ is
$(q^{\prime},q)$-limited $p$-convergent;

  4. (iv)

each weakly sequentially $p$-compact subset $A$ of $E$ which is a
$(q^{\prime},q)$-limited set is relatively sequentially compact in $E$;

  5. (v)

each weakly sequentially $p$-precompact subset $A$ of $E$ which is a
$(q^{\prime},q)$-limited set is sequentially precompact in $E$.

###### Proof 33

(i)$\Rightarrow$(ii) Assume that $E$ has the $p$-$GPscP_{(q^{\prime},q)}$, and
let $\\{x_{n}\\}_{n\in\omega}$ be weakly $p$-summable sequence in $E$ which is
a $(q^{\prime},q)$-limited set. By the $p$-$GPscP_{(q^{\prime},q)}$, we have
$x_{n}\to 0$ in $E$. Therefore also $T(x_{n})\to 0$ in $\ell_{\infty}$. Thus
$T$ is a $(q^{\prime},q)$-limited $p$-convergent operator.

(ii)$\Rightarrow$(i) Assume that each operator
$T\in\mathcal{L}(E,\ell_{\infty})$ is $(q^{\prime},q)$-limited $p$-convergent.
To show that $E$ has the $p$-$GPscP_{(q^{\prime},q)}$, let
$\\{x_{n}\\}_{n\in\omega}$ be a weakly $p$-summable sequence in $E$ which is a
$(q^{\prime},q)$-limited set. Assuming that $x_{n}\not\to 0$ in $E$ and
passing to a subsequence if needed we can assume that there is
$U\in\mathcal{N}_{0}^{c}(E)$ such that $x_{n}\not\in U$ for all $n\in\omega$.
For every $n\in\omega$, choose $\chi_{n}\in U^{\circ}$ such that
$\langle\chi_{n},x_{n}\rangle>1$. Define a linear map $T:E\to\ell_{\infty}$ by

$T(x):=\big{(}\langle\chi_{n},x\rangle\big{)}_{n}\quad(x\in E).$

Since $T(U)\subseteq B_{\ell_{\infty}}$, the map $T$ is continuous. As
$\|T(x_{n})\|_{\ell_{\infty}}\geq\langle\chi_{n},x_{n}\rangle>1\not\to 0$, it
follows that $T$ is not $(q^{\prime},q)$-limited $p$-convergent. This is a
contradiction.

The equivalence (ii)$\Leftrightarrow$(iii) immediately follows from
Proposition 5.3.

(i)$\Rightarrow$(iv) To show that $A$ is relatively sequentially compact in
$E$, take an arbitrary sequence $S=\\{a_{n}\\}_{n\in\omega}$ in $A$. Since $A$
is weakly sequentially $p$-compact, passing to a subsequence we can assume
that $S$ weakly $p$-converges to some point $a\in A$. So the sequence
$\\{a_{n}-a\\}_{n\in\omega}$ is weakly $p$-summable and also a
$(q^{\prime},q)$-limited set. Therefore, by the $p$-$GPscP_{(q^{\prime},q)}$,
we have $a_{n}-a\to 0$. Thus $A$ is a relatively sequentially compact subset
of $E$.

(iv)$\Rightarrow$(i) If $S=\\{a_{n}\\}_{n\in\omega}$ is a weakly $p$-summable
sequence in $E$ which is a $(q^{\prime},q)$-limited set, then $S$ is
relatively sequentially compact in $E$. Being also weakly null $S$ is a null
sequence in $E$ by Lemma 2.1.

(i)$\Rightarrow$(v) To show that $A$ is sequentially precompact in $E$, take
an arbitrary sequence $S=\\{a_{n}\\}_{n\in\omega}$ in $A$. Since $A$ is weakly
sequentially $p$-precompact, we can assume that $S$ is weakly $p$-Cauchy. For
every strictly increasing sequence $(n_{k})$ in $\omega$, it follows that
$\\{a_{n_{k}}-a_{n_{k+1}}\\}_{k\in\omega}$ is weakly $p$-summable and also a
$(q^{\prime},q)$-limited set. Therefore, by the $p$-$GPscP_{(q^{\prime},q)}$,
we have $a_{n_{k}}-a_{n_{k+1}}\to 0$ and hence $S$ is Cauchy in $E$. Thus $A$
is a sequentially precompact subset of $E$.

(v)$\Rightarrow$(i) If $S=\\{a_{n}\\}_{n\in\omega}$ is a weakly $p$-summable
sequence in $E$ which is a $(q^{\prime},q)$-limited set, then $S$ is
sequentially precompact in $E$. As $S$ is also weakly null, we apply Lemma 2.1
to get that $S$ is a null sequence in $E$.

Theorem 5.4 motivates the problem to characterize $(q^{\prime},q)$-limited
$p$-convergent operators. Under some additional restrictions this is done in
the next assertion. If $E$ and $L$ are Banach spaces and
$q^{\prime}=q=\infty$, the following theorem is proved in Theorem 2.1 of [41]
and Theorem 1.1 of [46].

###### Theorem 5.5

Let $2\leq q^{\prime}\leq q\leq\infty$, and let $E$ be a locally convex space
with the Rosenthal property. Then for an operator $T$ from $E$ to a locally
convex space $L$, the following assertions are equivalent:

  1. (i)

$T$ is $(q^{\prime},q)$-limited $\infty$-convergent;

  2. (ii)

for each $(q^{\prime},q)$-limited set $A\subseteq E$, the image $T(A)$ is
sequentially precompact in $L$;

  3. (iii)

for every locally convex (the same, normed) space $H$ and each
$(q^{\prime},q)$-limited operator $R:H\to E$, the operator $T\circ R$ is
sequentially precompact;

  4. (iv)

for each $(q^{\prime},q)$-limited operator $R:\ell_{1}^{0}\to E$, the operator
$T\circ R$ is sequentially precompact.

If in addition $E$ is locally complete, then (i)–(iv) are equivalent to the
following:

  1. (v)

for each $(q^{\prime},q)$-limited operator $R:\ell_{1}\to E$, the operator
$T\circ R$ is sequentially precompact.

###### Proof 34

(i)$\Rightarrow$(ii) Assume that $T\in\mathcal{L}(E,L)$ is
$(q^{\prime},q)$-limited $\infty$-convergent, and let $A$ be a
$(q^{\prime},q)$-limited subset of $E$. To show that $T(A)$ is sequentially
precompact in $L$, let $\\{a_{n}\\}_{n\in\omega}$ be a sequence in $A$. Since,
by Theorem 2.6, $A$ is weakly sequentially precompact we can assume that
$\\{a_{n}\\}_{n\in\omega}$ is weakly Cauchy. Therefore, for every strictly
increasing sequence $\\{n_{k}\\}_{k\in\omega}$ in $\omega$, the sequence
$\\{a_{n_{k+1}}-a_{n_{k}}\\}_{k\in\omega}$ is weakly null and also a
$(q^{\prime},q)$-limited set. Since $T$ is $(q^{\prime},q)$-limited
$\infty$-convergent it follows that $T(a_{n_{k+1}})-T(a_{n_{k}})\to 0$ in $L$,
and hence the sequence $\\{T(a_{n_{k}})\\}_{k\in\omega}$ is Cauchy in $L$.
Thus $T(A)$ is sequentially precompact in $E$.

(ii)$\Rightarrow$(iii) Take $U\in\mathcal{N}_{0}(H)$ such that $R(U)$ is a
$(q^{\prime},q)$-limited subset of $E$. By (ii), we have $T(R(U))$ is
sequentially precompact in $L$. Thus $T\circ R$ is a sequentially precompact
operator.

(iii)$\Rightarrow$(iv) and (iii)$\Rightarrow$(v) are obvious.

(iv)$\Rightarrow$(i) and (v)$\Rightarrow$(i): Let $S=\\{x_{n}\\}_{n\in\omega}$
be a weakly null sequence in $E$ which is a $(q^{\prime},q)$-limited set.
Since $S$ is bounded, Proposition 14.9 of [22] implies that the linear map
$R:\ell_{1}^{0}\to E$ (or $R:\ell_{1}\to E$ if $E$ is locally complete)
defined by

$R(a_{0}e_{0}+\cdots+a_{n}e_{n}):=a_{0}x_{0}+\cdots+a_{n}x_{n}\quad(n\in\omega,\;a_{0},\dots,a_{n}\in\mathbb{F}).$

is continuous. It is clear that
$R(B_{\ell_{1}^{0}})\subseteq\overline{\mathrm{acx}}\big{(}S\big{)}$ (or
$R(B_{\ell_{1}})\subseteq\overline{\mathrm{acx}}\big{(}S\big{)}$ if $E$ is
locally complete). Observe that $\overline{\mathrm{acx}}\big{(}S\big{)}$ is
also a $(q^{\prime},q)$-limited set. Therefore the operator $R$ is
$(q^{\prime},q)$-limited, and hence, by (iv) or (v), $T\circ R$ is
sequentially precompact. In particular, the weakly null sequence $T(S)$ is
(sequentially) precompact in $L$. Therefore, by Lemma 2.1, $T(x_{n})\to 0$ in
$L$. Thus $T$ is $(q^{\prime},q)$-limited $\infty$-convergent.

Below we obtain a sufficient condition on locally convex spaces to have the
$p$-$GPscP_{(q,\infty)}$.

###### Proposition 5.6

Let $p,q\in[1,\infty]$, and let $E$ be a locally convex space. Then $E$ has
the $p$-$GPscP_{(q,\infty)}$ if one of the the following conditions holds:

  1. (i)

for every locally convex space $X$ and for each $T\in\mathcal{L}(X,E)$ whose
adjoint $T^{\ast}:E^{\prime}_{w^{\ast}}\to X^{\prime}_{\beta}$ is
$q$-convergent, the operator $T$ transforms the bounded subsets of $X$ into
sequentially precompact subsets of $E$;

  2. (ii)

for every normed space $X$ and for each $T\in\mathcal{L}(X,E)$ whose adjoint
$T^{\ast}:E^{\prime}_{w^{\ast}}\to X^{\prime}_{\beta}$ is $q$-convergent, the
operator $T$ is sequentially precompact;

  3. (iii)

the same as (ii) with $X=\ell_{1}^{0}$;

  4. (iv)

if $E$ is locally complete, the same as (ii) with $X$ a Banach space;

  5. (v)

if $E$ is locally complete, the same as (ii) with $X=\ell_{1}$;

###### Proof 35

Since the implications (i)$\Rightarrow$(ii)$\Rightarrow$(iii) and
(ii)$\Rightarrow$(iv)$\Rightarrow$(v) are obvious, we shall prove that (iii)
and (v) imply the $p$-$GPscP_{(q,\infty)}$. Suppose for a contradiction that
$E$ has no the $p$-$GPscP_{(q,\infty)}$. Then there is a weakly $p$-summable
sequence $S=\\{x_{n}\\}_{n\in\omega}$ in $E$ which is a $(q,\infty)$-limited
set such that $x_{n}\not\to 0$ in $E$. Without loss of generality we assume
that $x_{n}\not\in U$ for some $U\in\mathcal{N}_{0}(E)$ and all $n\in\omega$.

Since $S$ is a $(q,\infty)$-limited set, Proposition 5.9 of [24] implies that
the linear map $T:\ell_{1}^{0}\to E$ (or $T:\ell_{1}\to E$ if $E$ is locally
complete) defined by

$T(a_{0}e_{0}+\cdots+a_{n}e_{n}):=a_{0}x_{0}+\cdots+a_{n}x_{n}\quad(n\in\omega,\;a_{0},\dots,a_{n}\in\mathbb{F}),$

is continuous and its adjoint $T^{\ast}:E^{\prime}_{w^{\ast}}\to\ell_{\infty}$
is $q$-convergent. Therefore, by (iii) or (v), the operator $T$ is
sequentially precompact. In particular, the set
$T(\\{e_{n}\\}_{n\in\omega})=S$ is (sequentially) precompact in $E$. Since $S$
is also weakly null, Lemma 2.1 implies that $x_{n}\to 0$ in $E$. But this
contradicts the choice of $S$.

For an important case which covers all strict $(LF)$ spaces we have the
following characterization the $GPscP_{(q,\infty)}$.

###### Theorem 5.7

Let $2\leq q\leq\infty$. Then for a locally convex space $E$ with the
Rosenthal property, the following assertions are equivalent:

  1. (i)

$E$ has the $GPscP_{(q,\infty)}$;

  2. (ii)

for every locally convex space $X$ and for each $T\in\mathcal{L}(X,E)$ whose
adjoint $T^{\ast}:E^{\prime}_{w^{\ast}}\to X^{\prime}_{\beta}$ is
$q$-convergent, the operator $T$ transforms the bounded subsets of $X$ into
sequentially precompact subsets of $E$;

  3. (iii)

for every normed space $X$ and for each $T\in\mathcal{L}(X,E)$ whose adjoint
$T^{\ast}:E^{\prime}_{w^{\ast}}\to X^{\prime}_{\beta}$ is $q$-convergent, the
operator $T$ is sequentially precompact;

  4. (iv)

the same as (ii) with $X=\ell_{1}^{0}$;

If in addition $E$ is locally complete, then (i)–(iv) are equivalent to

  1. (v)

the same as (iii) with $X$ a Banach space;

  2. (vi)

the same as (iii) with $X=\ell_{1}$;

###### Proof 36

By (the proof of) Proposition 5.6, we have to prove only the implication
(i)$\Rightarrow$(ii). So, let $X$ be a locally convex space, and let
$T\in\mathcal{L}(X,E)$ be such that the adjoint
$T^{\ast}:E^{\prime}_{w^{\ast}}\to X^{\prime}_{\beta}$ is $q$-convergent. We
have to show that for every bounded subset $B$ of $X$, the set $T(B)$ is
sequentially precompact in $E$. To this end, let $\\{\chi_{n}\\}_{n\in\omega}$
be a weak∗ $q$-summable sequence in $E^{\prime}$. As $T^{\ast}$ is
$q$-convergent, we have $T^{\ast}(\chi_{n})\to 0$ in $X^{\prime}_{\beta}$.
Since $B$ is bounded, the set $B^{\circ}$ is a neighborhood of zero in
$X^{\prime}_{\beta}$. Therefore, for every $\varepsilon>0$ there exists
$N\in\omega$ such that $T^{\ast}(\chi_{n})\in\varepsilon B^{\circ}$ for all
$n\geq N$. Now for every $y\in B$, we have

$|\langle\chi_{n},T(y)\rangle|=|\langle
T^{\ast}(\chi_{n}),y\rangle|\leq\varepsilon\;\;\mbox{ for all $n\geq N$}.$

Therefore $T(B)$ is a $(q,\infty)$-limited set in $E$. Since $q\geq 2$ and $E$
has the Rosenthal property, Theorem 2.6 implies that $T(B)$ is weakly
sequentially precompact. To show that $T(B)$ is sequentially precompact in $E$
it suffices to prove that for any sequence $S=\\{x_{n}\\}_{n\in\omega}$ in
$B$, the image $T(S)$ has a Cauchy subsequence. To this end, taking into
account that $T(B)$ is weakly sequentially precompact, we can assume that
$T(S)$ is weakly Cauchy. For any strictly increasing sequence
$\\{n_{k}\\}_{k\in\omega}$ in $\omega$, the sequence
$\\{T(x_{n_{k}})-T(x_{n_{k+1}})\\}_{k\in\omega}$ is weakly null and a
$(q,\infty)$-limited set. Therefore, by the $GPscP_{(q,\infty)}$ of $E$, we
obtain $T(x_{n_{k}})-T(x_{n_{k+1}})\to 0$ in $E$. So
$\\{T(x_{n_{k}})\\}_{k\in\omega}$ is Cauchy in $E$.

Now we show that the class of spaces with the $p$-$GPscP_{(q^{\prime},q)}$ has
nice stability properties.

###### Proposition 5.8

Let $p,q,q^{\prime}\in[1,\infty]$, $q^{\prime}\leq q$, and let $H$ be a
subspace of a locally convex space $E$. If $E$ has the
$p$-$GPscP_{(q^{\prime},q)}$, then also $H$ has the
$p$-$GPscP_{(q^{\prime},q)}$.

###### Proof 37

Let $S=\\{h_{n}\\}_{n\in\omega}$ be a weakly $p$-summable sequence in $H$
which is also a $(q^{\prime},q)$-limited set. Then, by (vi) of Lemma 4.6 of
[22], $S$ is weakly $p$-summable also in $E$, and, by (iv) of Lemma 3.1 of
[24], $S$ is a $(q^{\prime},q)$-limited subset of $E$. By the
$p$-$GPscP_{(q^{\prime},q)}$ of $E$, we have $h_{n}\to 0$ in $E$, and hence
$h_{n}\to 0$ also in $H$. Thus $H$ has the $p$-$GPscP_{(q^{\prime},q)}$.

###### Proposition 5.9

Let $p,q,q^{\prime}\in[1,\infty]$, $q^{\prime}\leq q$, and let
$\\{E_{i}\\}_{i\in I}$ be a mom-empty family of locally convex spaces. Then
the following assertions are equivalent:

  1. (i)

the space $E=\prod_{i\in I}E_{i}$ has the $p$-$GPscP_{(q^{\prime},q)}$;

  2. (ii)

the space $L=\bigoplus_{i\in I}E_{i}$ has the $p$-$GPscP_{(q^{\prime},q)}$;

  3. (iii)

for every $i\in I$, the space $E_{i}$ has the $p$-$GPscP_{(q^{\prime},q)}$.

###### Proof 38

(i)$\Rightarrow$(ii) Let $S=\\{x_{n}\\}_{n\in\omega}$ be a weakly $p$-summable
sequence in $L$ which is also a $(q^{\prime},q)$-limited set. Since $S$ is
bounded, there is a finite subset $F$ of $I$ such that $S\subseteq\prod_{i\in
F}E_{i}\times\prod_{i\in I{\setminus}F}\\{0_{i}\\}$. Since $\prod_{i\in
F}E_{i}$ is a subspace of $E$, Proposition 5.8 implies that $x_{n}\to 0$ in
$L$. Thus $L$ has the $p$-$GPscP_{(q^{\prime},q)}$.

(ii)$\Rightarrow$(iii) follows from Proposition 5.8 because any $E_{i}$ can be
considered as a subspace of $L$.

(iii)$\Rightarrow$(i) Let $S=\\{x_{n}=(x_{i,n})\\}_{n\in\omega}$ be a weakly
$p$-summable sequence in $E$ which is also a $(q^{\prime},q)$-limited set.
Observe that $x_{n}\to 0$ in $E$ if and only if $x_{i,n}\to 0_{i}$ for every
$i\in I$. For every $i\in I$, (ii) of Lemma 4.25 of [22] and (i) of
Proposition 3.3 of [24] imply that the sequence $\\{x_{i,n}\\}_{n\in\omega}$
is weakly $p$-summable in $E_{i}$ which is also a $(q^{\prime},q)$-limited
set. Therefore, by the $p$-$GPscP_{(q^{\prime},q)}$ of $E_{i}$, we obtain
$x_{i,n}\to 0_{i}$ in $E_{i}$, as desired.

##  6 Strong versions of the $V^{\ast}$ type properties of Pełczyński

If in Definition 2.5 of $V^{\ast}$ type sets to ask that the
$(p,q)$-$(V^{\ast})$ sets are relatively compact in the original topology
instead of the weak topology we obtain “strong” versions of $V^{\ast}_{(p,q)}$
properties.

###### Definition 6.1

Let $1\leq p\leq q\leq\infty$. A locally convex space $(E,\tau)$ is said to
have

  1. $\bullet$

the strong $V^{\ast}_{(p,q)}$ property (resp., strong $EV^{\ast}_{(p,q)}$
property) if every $(p,q)$-$(V^{\ast})$ (resp., $(p,q)$-$(EV^{\ast})$) subset
of $E$ is relatively compact;

  2. $\bullet$

the strong sequential $V^{\ast}_{(p,q)}$ property (resp., strong sequential
$EV^{\ast}_{(p,q)}$ property) if every $(p,q)$-$(V^{\ast})$ (resp.,
$(p,q)$-$(EV^{\ast})$) subset of $E$ is relatively sequentially compact; for
short, the strong $sV^{\ast}_{(p,q)}$ property or the strong
$sEV^{\ast}_{(p,q)}$ property;

  3. $\bullet$

the strong precompact $V^{\ast}_{(p,q)}$ property (resp., strong precompact
$EV^{\ast}_{(p,q)}$ property) if every $(p,q)$-$(V^{\ast})$ (resp.,
$(p,q)$-$(EV^{\ast})$) subset of $E$ is $\tau$-precompact; for short, the
strong $prV^{\ast}_{(p,q)}$ property or the strong $prEV^{\ast}_{(p,q)}$
property;

  4. $\bullet$

the strong sequentially precompact $V^{\ast}_{(p,q)}$ property (resp., strong
sequentially precompact $EV^{\ast}_{(p,q)}$ property) if every
$(p,q)$-$(V^{\ast})$ (resp., $(p,q)$-$(EV^{\ast})$) subset of $E$ is
sequentially precompact in $\tau$; for short, the strong $spV^{\ast}_{(p,q)}$
property or the strong $spEV^{\ast}_{(p,q)}$ property.

If $q=\infty$, we shall say that $E$ has the strong (resp., strong sequential,