# Limited type subsets 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
limited, $p$-limited and coarse $p$-limited subsets of a Banach space, we
introduce and study $(p,q)$-limited subsets and their equicontinuous versions
and coarse $p$-limited subsets of an arbitrary locally convex space $E$.
Operator characterizations of these classes are given. We compare these
classes with the classes of bounded, (pre)compact, weakly (pre)compact and
relatively weakly sequentially (pre)compact sets. If $E$ is a Banach space, we
show that the class of coarse $1$-limited subsets of $E$ coincides with the
class of $(1,\infty)$-limited sets, and if $1<p<\infty$, then the class of
coarse $p$-limited sets in $E$ coincides with the class of $p$-$(V^{\ast})$
sets of Pełczyński. We also generalize a known theorem of Grothendieck.

###### keywords:

$(p,q)$-limited set , coarse $p$-limited set , $p$-$(V^{\ast})$ set ,
$p$-convergent operator , $p$-barrelled space

###### MSC:

[2010] 46A3 , 46E10

##  1 Introduction

Let $E$ be a locally convex space (lcs for short), and let $E^{\prime}$ denote
the topological dual of $E$. For a bounded subset $A\subseteq E$ and a
functional $\chi\in E^{\prime}$, we put

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

###### Definition 1.1

A bounded subset $A$ 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 $A$, that is $\lim_{n\to\infty}\|\chi_{n}\|_{A}=0$. Denote by
$\mathsf{L}(E)$ the family of all limited subsets of $E$.

Limited sets in Banach spaces were systematically studied by Bourgain and
Diestel [5], see also Schlumprecht [39]. Among other things they proved the
following result (all relevant definitions are given in Section 2).

######  Theorem 1.2 ([5])

Let $E$ be a Banach space. Then:

  1. (i)

$\mathsf{L}(E)$ is closed under taking subsets, finite sums and absolutely
convex hulls;

  2. (ii)

if $E$ contains no copy of $\ell_{1}$, then each $A\in\mathsf{L}(E)$ is
relatively weakly compact;

  3. (iii)

every $A\in\mathsf{L}(E)$ is weakly sequentially precompact;

  4. (iv)

if $E$ is separable or reflexive, then each $A\in\mathsf{L}(E)$ is relatively
compact.

Let $E$ and $H$ be locally convex spaces. Denote by $\mathcal{L}(E,H)$ the
family of all operators from $E$ to $H$. If $p\in[1,\infty]$, 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 $p\in[1,\infty)$, or $(\langle\chi,x_{n}\rangle)\in c_{0}$ if $p=\infty$.
The family $\ell_{p}^{w}(E)$ (or $c_{0}^{w}(E)$ if $p=\infty$) of all weakly
$p$-summable sequences in $E$ is a vector space which admits a natural locally
convex vector topology such that it is complete if so is $E$, for details see
Section 19.4 in [28] or Section 4 in [16].

Let $p\in[1,\infty]$, and let $X$ and $Y$ be Banach spaces. Generalizing the
notion of completely continuous operators Castillo and Sánchez defined in [6]
an operator $T:X\to Y$ to be $p$-convergent if $T$ sends weakly $p$-summable
sequences of $X$ into norm null-sequences of $Y$. The influential article of
Castillo and Sánchez [6] 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 [29] and Galindo
and Miranda [20].

###### Definition 1.3

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

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

for every $(\chi_{n})\in\ell_{p}^{w}(X^{\ast})$ (or $(\chi_{n})\in
c_{0}^{w}(X^{\ast})$ if $p=\infty$) ([29]);

  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 ([20]).

It turns out that the family $\mathsf{L}_{p}(X)$ of all $p$-limited subsets of
$X$ and the family $\mathsf{CL}_{p}(X)$ of all coarse $p$-limited subsets of
$X$ have similar properties as $\mathsf{L}(X)$ described in Theorem 1.2, see
[29] and [20], respectively.

Limited sets in Fréchet spaces were studied by Alonso [1]. The notion of a
limited set in general locally convex spaces was introduced by Lindström and
Schlumprecht in [31] and independently by Banakh and Gabriyelyan in [3]. Since
limited sets in the sense of [31] are defined using equicontinuity, to
distinguish both notions we called them in [3] by ${\mathcal{E}}$-limited
sets.

###### Definition 1.4

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 $E^{\prime}$ ([31]);

  2. (ii)

limited  if $\|\chi_{n}\|_{A}\to 0$ for every weak∗ null sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$ ([3]).

It is clear that if $E$ is a $c_{0}$-barrelled (for example, Banach) space,
then $A$ is limited if and only if it is ${\mathcal{E}}$-limited.

Definitions 1.1, 1.3 and 1.4 and the notions of $(p,q)$-$(V^{\ast})$ subsets
and $(p,q)$-$(EV^{\ast})$ subsets of a locally convex space $E$ introduced and
studied in [16] motivate the following notions.

###### Definition 1.5

Let $p,q\in[1,\infty]$. A non-empty subset $A$ of a separated topological
vector 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 $E^{\prime}$. 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. $(p,p)$-limited sets and $(\infty,\infty)$-limited sets
will be called simply $p$-limited sets and limited sets, respectively.

  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. The family of all coarse $p$-limited sets is denoted by
$\mathsf{CL}_{p}(E)$.

The purpose of the article is to study $(p,q)$-limited subsets and coarse
$p$-limited subsets of locally convex spaces in the spirit of Theorem 1.2 and
the articles [29] and [20].

Now we describe the content of the article. In Section 2 we fix the main
notions and some auxiliary results used in what follows.

In Section 3 we study the classes $\mathsf{L}_{(p,q)}(E)$ and
$\mathsf{EL}_{(p,q)}(E)$. In Lemma 3.1 we generalize (i) of Theorem 1.2 and
show that $\mathsf{L}_{(p,q)}(E)=\mathsf{EL}_{(p,q)}(E)=\\{0\\}$ if $q<p$. In
Proposition 3.3 we characterize $(p,q)$-limited subsets and
$(p,q)$-${\mathcal{E}}$-limited subsets in products and direct sums of locally
convex spaces. In Theorem 3.4 we give an operator characterization of
$(p,q)$-limited and $(p,q)$-${\mathcal{E}}$-limited subsets of the locally
convex space $E$. The following diagram easily follows from Definition 1.5
(see also (vi) of Lemma 3.1)

$\textstyle{\left.{\begin{subarray}{c}\mbox{limited}\\\
\mbox{$(p,q)$-limited}\end{subarray}}\right\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$(p,\infty)$-limited$\textstyle{\mbox{$(1,\infty)$-limited}.}$

This diagram motivates the study of $(p,\infty)$-limited sets and
$(1,\infty)$-limited sets. It is well known that any (pre)compact subset of a
Banach space is limited. In Proposition 3.6 we generalize this useful result
by showing that each precompact subset of an lcs $E$ is
$(p,\infty)$-${\mathcal{E}}$-limited, and if in addition $E$ is $p$-barrelled,
then every precompact subset of $E$ is $(p,\infty)$-limited. Consequently,
each precompact subset of a $c_{0}$-barrelled space is limited. In Theorem 3.8
we show that every precompact subset of $C_{p}(X)$ (= the space $C(X)$ of all
continuous functions over a Tychonoff space $X$ endowed with the pointwise
topology) is $(p,q)$-limited if and only if $X$ has no infinite functionally
bounded subsets. As a corollary (see Example 3.10) we obtain that the
metrizable space $C_{p}([0,\omega])$ has even compact subsets which are not
limited. Being motivated by (iv) of Theorem 1.2 it is natural to consider the
case when every $(p,q)$-limited set is precompact. This problem is solved in
Theorem 3.11. In Proposition 3.13 we characterize $(1,\infty)$-limited subsets
of barrelled locally convex spaces. In [24] (see also Theorem 3.11 of [26])
Grothendieck proved that if $E$ is a Banach space, then a bounded subset $B$
of $\big{(}E^{\prime},\mu(E^{\prime},E)\big{)}$ is precompact if and only if
it is limited. In Theorem 3.16 we generalize this result.

In Section 4 we study coarse $p$-limited subsets of locally convex spaces.
Generalizing Proposition 2 of [20] we show in Lemma 4.1 that the family
$\mathsf{CL}_{p}(E)$ of all coarse $p$-limited sets in $E$ is closed under
taking subsets, finite unions, closed absolutely convex hulls, and continuous
linear images. In Proposition 4.2 we show that every $p$-limited subset of $E$
is coarse $p$-limited (this generalizes Proposition 1 of [20]), and under
addition assumption we prove that even every $(p,p)$-$(V^{\ast})$ subset of
$E$ is coarse $p$-limited. A description of coarse $p$-limited subsets of
direct products and direct sums is given in Proposition 4.3.

The following class of linear maps is defined and studied in Section 16 of
[16].

###### Definition 1.6

Let $1\leq p\leq q\leq\infty$, and let $E$ and $L$ be locally convex spaces. 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 $L$.

Let $1\leq p\leq q\leq\infty$. In Section 16 of [16] we naturally extend the
notion of $p$-convergent operators between Banach spaces to the general case
saying that a linear map $T:E\to L$ between locally convex spaces $E$ and $L$
is $(q,p)$-convergent if it sends weakly $p$-summable sequences in $E$ to
strongly $q$-summable sequences in $L$ (so $(\infty,p)$-convergent operators
are exactly $p$-convergent operators). The notion of $(q,p)$-convergent
operators is useful to solve the following general problem: Characterize those
operators $T$ which map all bounded  sets into $(p,q)$-limited sets (or into
coarse $p$-limited sets). If $E$ and $L$ are Banach spaces and $p=q$, a
partial answer to this problem is given by Ghenciu, see Theorem 14 of [23]. In
Section 5 we give a complete answer to this problem, see Theorem 5.5. The
clauses (ii)-(iv) of Theorem 1.2 motivate the problem of finding conditions on
a space $E$ under which $(p,q)$-limited sets and coarse $p$-limited sets have
additional topological properties. For $p$-limited subsets of Banach spaces
this problem was considered by Ghenciu, see Theorem 15 of [23]. In Theorem
5.10 we essentially generalize Ghenciu’s result. In Theorem 5.15 we
characterize coarse $1$-limited sets. As a consequence of the obtained results
we show in Corollary 5.19 that: (1) if $p=1$, then the class of coarse
$1$-limited subsets of a Banach space $E$ coincides with the class of
$(1,\infty)$-limited sets, and (2) if $1<p<\infty$, then the class of coarse
$p$-limited sets in $E$ coincides with the class of $p$-$(V^{\ast})$ sets. It
should be mentioned that $p$-$(V^{\ast})$ sets in Banach spaces were defined
and study by Chen, Chávez-Domínguez and Li in [7] and [30]. Using the idea of
the proof of (iii) of Theorem 1.2, Galindo and Miranda proved in Proposition 3
of [20] that if $2\leq p<\infty$, then every coarse $p$-limited set is weakly
sequentially precompact. In Theorem 5.21 we extend this result to locally
convex spaces with the Rosenthal property.

The clause (iv) of Theorem 1.2 implies that each separable or reflexive Banach
space has the Gelfand–Phillips property. By this reason generalizations of
this clause will be given in the forthcoming article [18].

##  2 Preliminaries results

We start with some necessary definitions and notations used in the article.
Set $\omega:=\\{0,1,2,\dots\\}$. 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 $\overline{A}$, $\overline{A}^{X}$ or
$\mathrm{cl}_{X}(A)$. A function $f:X\to Y$ between topological spaces $X$ and
$Y$ is called sequentially continuous if for any convergent sequence
$\\{x_{n}\\}_{n\in\omega}\subseteq X$, the sequence
$\\{f(x_{n})\\}_{n\in\omega}$ converges in $Y$ and
$\lim_{n}f(x_{n})=f(\lim_{n}x_{n})$. A subset $A$ of a topological space $X$
is called functionally bounded in $X$ if every $f\in C(X)$ is bounded on $A$.

All topological vector spaces are over the field $\mathbb{F}$ of real or
complex numbers. The closed unit ball of the field $\mathbb{F}$ is denoted by
$\mathbb{D}$.

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 $n,m\geq N$. It is easy to see that a sequence
$\\{x_{n}\\}_{n\in\omega}$ in $E$ is Cauchy if and only if
$x_{n_{k}}-x_{n_{k+1}}\to 0$ for every (strictly) increasing sequence
$(n_{k})$ in $\omega$. If $E$ is a normed space, $B_{E}$ denotes the closed
unit ball of $E$.

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 closure of a subset $A$ in the weak
topology is denoted by $\overline{A}^{\,w}$ or
$\overline{A}^{\,\sigma(E,E^{\prime})}$, and $\overline{B}^{\,w^{\ast}}$ (or
$\overline{B}^{\,\sigma(E^{\prime},E)}$) denotes the closure of $B\subseteq
E^{\prime}$ in the weak∗ topology. 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, by the
converse is not true in general, see Lemma 2.2 of [16].

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

  1. $\bullet$

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

  2. $\bullet$

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

  3. $\bullet$

is 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
$X$.

Recall that a locally convex space $(E,\tau)$ has the Schur property (resp.,
the Glicksberg property) if $E$ and $E_{w}$ have the same convergent sequences
(resp., the same compact sets). If an lcs $E$ has the Glicksberg property,
then it has the Schur property. The converse is true for strict $(LF)$-spaces
(in particular, for Banach spaces), but not in general, see Corollary 2.13 and
Proposition 3.5 of [14]. We shall use the next two lemmas.

###### Lemma 2.1

A locally convex space $(E,\tau)$ has the Schur property if and only if $E$
and $E_{w}$ have the same relatively sequentially compact sets.

###### Proof 1

Assume that $(E,\tau)$ has the Schur property. If $A$ is a relatively
sequentially compact subset of $E$, then evidently $A$ is relatively
sequentially compact in $E_{w}$. Conversely, let $A$ be a relatively
sequentially compact subset of $E_{w}$. Take a sequence
$S=\\{a_{n}\\}_{n\in\omega}$ in $A$. Then $S$ has a subsequence
$\\{a_{n_{k}}\\}_{k\in\omega}$ weakly converging to a point $x\in E$. By the
Schur property $a_{n_{k}}\to x$ in $\tau$. Hence $A$ is a relatively
sequentially compact subset of $E$. Thus $E$ and $E_{w}$ have the same
relatively sequentially compact sets.

Assume that $E$ and $E_{w}$ have the same relatively sequentially compact
sets. To show that $E$ has the Schur property, let
$S=\\{a_{n}\\}_{n\in\omega}$ be a weakly null sequence. Then $S$ is relatively
sequentially compact in $E_{w}$ and hence also in $E$. We show that $a_{n}\to
0$ also in $E$. Suppose for a contradiction that there is a
$U\in\mathcal{N}_{0}(E)$ such that $a_{n}\not\in U$ for each $n\in I$ for some
infinite $I\subseteq\omega$. Since $S$ is relatively sequentially compact in
$E$, the sequence $S^{\prime}=\\{a_{n}\\}_{n\in I}$ has a subsequence
$\\{b_{n}\\}_{n\in\omega}$ converging to some point $b\in E$. As
$\\{b_{n}\\}_{n\in\omega}$ is also weakly null we have $b=0$. But since
$\\{b_{n}\\}_{n\in\omega}\subseteq S^{\prime}$ it follows that $b_{n}\not\in
U$ for every $n\in\omega$; so $b_{n}\not\to b$, a contradiction.

###### Lemma 2.2

Every weakly sequentially precompact subset $A$ of a Schur space $E$ is
sequentially precompact. If in addition $E$ is sequentially complete, then
$E_{w}$ is sequentially complete.

###### Proof 2

Let $\\{x_{n}\\}_{n\in\omega}$ be a sequence in $A$. As $A$ is weakly
sequentially precompact, there is a subsequence $\\{y_{n}\\}_{n\in\omega}$ of
$\\{x_{n}\\}_{n\in\omega}$ which is weakly Cauchy. Let $(n_{k})$ be a strictly
increasing sequence in $\omega$. Then
$\\{y_{n_{k+1}}-y_{n_{k}}\\}_{n\in\omega}$ is weakly null. By the Schur
property of $E$, we obtain that $y_{n_{k+1}}-y_{n_{k}}\to 0$ in $E$. Thus
$\\{y_{n}\\}_{n\in\omega}$ is a Cauchy sequence in $E$, and hence $A$ is
sequentially precompact.

Assume that $E$ is in addition a sequentially complete space. Let
$S=\\{x_{n}\\}_{n\in\omega}$ be a weakly Cauchy sequence. As we proved above
$S$ is a Cauchy sequence in $E$. Since $E$ is sequentially complete, there is
$a\in E$ such that $x_{n}\to a$. Thus $x_{n}$ converges to $a$ also in the
weak topology.

Two vector topologies $\tau$ and $\mathcal{T}$ on a vector space $L$ are
called compatible if $(L,\tau)^{\prime}=(L,\mathcal{T})^{\prime}$
algebraically. If $(E,\tau)$ is a locally convex space, then there is a finest
locally convex vector topology $\mu(E,E^{\prime})$ compatible with $\tau$. The
topology $\mu(E,E^{\prime})$ is called the Mackey topology, and if
$\tau=\mu(E,E^{\prime})$, the space $E$ is called a Mackey space. Set
$E_{\mu}:=\big{(}E,\mu(E,E^{\prime})\big{)}$. It is well known that any
quasibarrelled space is Mackey.

Recall that an lcs $E$ is called semi-reflexive if the canonical map
$J_{E}:E\to E^{\prime\prime}=(E^{\prime}_{\beta})^{\prime}_{\beta}$ defined by
$\langle J_{E}(x),\chi\rangle:=\langle\chi,x\rangle$ ($\chi\in E^{\prime}$) is
an isomorphism; if in addition $J_{E}$ is a topological isomorphism, the space
$E$ is called reflexive. Each reflexive space is barrelled.

We 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
$\mathbf{x}(i):=x_{i}$.

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

###### Definition 2.3

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
$\\{e_{n}^{\ast}\\}_{n\in\omega}$. Denote by $\ell_{p}^{0}$ and $c_{0}^{0}$
the linear span of $\\{e_{n}\\}_{n\in\omega}$ in $\ell_{p}$ or $c_{0}$ endowed
with the induced norm topology, respectively. We shall use repeatedly the
following well known description of relatively compact subsets of $\ell_{p}$
and $c_{0}$, see [8, p. 6].

###### Proposition 2.4

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

One of the most important classes of locally convex spaces is the class of
free locally convex spaces introduced by Markov in [32]. 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, and $X$ is the Hamel basis of $L(X)$. So, each nonzero
$\chi\in L(X)$ has a unique decomposition
$\chi=a_{1}i(x_{1})+\cdots+a_{n}i(x_{n})$, where all $a_{k}$ are nonzero and
$x_{k}$ are distinct. The set $\mathrm{supp}(\chi):=\\{x_{1},\dots,x_{n}\\}$
is called the support of $\chi$. In what follows we shall identify $i(x)$ with
$x$ and consider $i(x)$ as the Dirac measure $\delta_{x}$ at the point $x\in
X$. We also recall that $C_{p}(X)^{\prime}=L(X)$ and $L(X)^{\prime}=C(X)$. It
is worth mentioning that $L(X)$ has the Glicksberg property for every
Tychonoff space $X$, and if $X$ is non-discrete, then $L(X)$ is not a Mackey
space, see [15] and [12], respectively.

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

  1. $\bullet$

weakly $p$-summable if for every $\chi\in E^{\prime}$, it follows

$(\langle\chi,x_{n}\rangle)_{n\in\omega}\in\ell_{p}$ if $p<\infty$, and
$(\langle\chi,x_{n}\rangle)_{n\in\omega}\in c_{0}$ if $p=\infty$;

  2. $\bullet$

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

  3. $\bullet$

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 $E^{\prime}_{w^{\ast}}$.

The following weak barrelledness conditions introduced and studied in [16]
will play a considerable role in the article. Let $p\in[1,\infty]$. A locally
convex space $E$ is called

  1. $\bullet$

$p$-barrelled  if every weakly $p$-summable sequence in
$E^{\prime}_{w^{\ast}}$ is equicontinuous;

  2. $\bullet$

$p$-quasibarrelled  if every weakly $p$-summable sequence in
$E^{\prime}_{\beta}$ is equicontinuous.

We shall consider also the following linear map introduced in [16]

$S_{p}:\mathcal{L}(E,\ell_{p})\to\ell_{p}^{w}(E^{\prime}_{w^{\ast}})\quad\big{(}\mbox{or
$S_{\infty}:\mathcal{L}(E,c_{0})\to c_{0}^{w}(E^{\prime}_{w^{\ast}})$ if
$p=\infty$}\big{)}$

defined by $S_{p}(T):=\big{(}T^{\ast}(e^{\ast}_{n})\big{)}_{n\in\omega}$.

The following class of subsets of an lcs $E$ was introduced and studied in
[16], and it generalizes the notion of $p$-$(V^{\ast})$ subsets of Banach
spaces defined in [7].

###### Definition 2.5

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.
Analogously, $(p,\infty)$-$(EV^{\ast})$ sets and $(1,\infty)$-$(EV^{\ast})$
sets will be called $p$-$(EV^{\ast})$ sets and $(EV^{\ast})$ sets,
respectively.

The family of all $(p,q)$-$(V^{\ast})$ sets (resp. $p$-$(V^{\ast})$ sets,
$(p,q)$-$(EV^{\ast})$ sets, $(V^{\ast})$ sets etc.) of an lcs $E$ is denoted
by $\mathsf{V}^{\ast}_{(p,q)}(E)$ (resp. $\mathsf{V}^{\ast}_{p}(E)$,
$\mathsf{EV}^{\ast}_{(p,q)}(E)$, $\mathsf{V}^{\ast}(E)$ etc.).

Following [16], a non-empty subset $B$ of $E^{\prime}$ is called a
$(p,q)$-$(V)$ set if

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

for every weakly $p$-summable sequence $\\{x_{n}\\}_{n\in\omega}$ in $E$.
$(p,\infty)$-$(V)$ sets and $(1,\infty)$-$(V)$ sets will be called simply
$p$-$(V)$ sets and $(V)$ sets, respectively.

Let $1\leq p\leq q\leq\infty$, and let $E$ and $L$ be locally convex spaces.
Following [16], 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 $L$.

The following $p$-versions of weakly compact-type properties are defined in
[16] generalizing the corresponding notions in the class of Banach spaces
introduced in [6] and [23]. 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.

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 $p\in[1,\infty]$. Following [17], a locally convex space $E$ 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.

##  3 Limited-type sets in locally convex spaces

In the next lemma we summarize some basic elementary properties of
$(p,q)$-limited sets, cf. (i) of Theorem 1.2.

###### Lemma 3.1

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

  1. (i)

every $(p,q)$-limited set is $(p,q)$-${\mathcal{E}}$-limited; the converse is
true if $E$ is a $p$-barrelled space;

  2. (ii)

every $(p,q)$-${\mathcal{E}}$-limited set in $E$ is bounded;

  3. (iii)

the family of all $(p,q)$-limited $($resp., $(p,q)$-${\mathcal{E}}$-limited$)$
sets in $E$ is closed under taking subsets, finite unions and sums, and closed
absolutely convex hulls;

  4. (iv)

the family of all $(p,q)$-limited $($resp., $(p,q)$-${\mathcal{E}}$-limited$)$
sets in $E$ is closed under taking continuous linear images; in particular, if
$H$ is a subspace of $E$, then every $(p,q)$-limited $($resp.,
$(p,q)$-${\mathcal{E}}$-limited$)$ set in $H$ is $(p,q)$-limited $($resp.,
$(p,q)$-${\mathcal{E}}$-limited$)$ in $E$;

  5. (v)

a subset $A$ of $E$ is a $(p,q)$-limited $($resp.,
$(p,q)$-${\mathcal{E}}$-limited$)$ set if and only if every countable subset
of $A$ is a $(p,q)$-limited $($resp., $(p,q)$-${\mathcal{E}}$-limited$)$ set;

  6. (vi)

if $p^{\prime},q^{\prime}\in[1,\infty]$ are such that $p^{\prime}\leq p$ and
$q\leq q^{\prime}$, then every $(p,q)$-limited $($resp.,
$(p,q)$-${\mathcal{E}}$-limited$)$ set in $E$ is also
$(p^{\prime},q^{\prime})$-limited $($resp.,
$(p^{\prime},q^{\prime})$-${\mathcal{E}}$-limited$)$; in particular, any
$(p,q)$-limited $($resp., $(p,q)$-${\mathcal{E}}$-limited$)$ set is
$(1,\infty)$-limited $($resp., $(1,\infty)$-${\mathcal{E}}$-limited$)$;

  7. (vii)

the property of being a $(p,q)$-limited set depends only on the duality
$(E,E^{\prime})$, i.e., if $\mathcal{T}$ is a locally convex vector topology
on $E$ compatible with the topology $\tau$ of $E$, then the $(p,q)$-limited
sets of $(E,\mathcal{T})$ are exactly the $(p,q)$-limited sets of $(E,\tau)$;

  8. (viii)

every $(p,q)$-limited $($resp., $(p,q)$-${\mathcal{E}}$-limited$)$ set in $E$
is a $(p,q)$-$(V^{\ast})$ $($resp., $(p,q)$-$(EV^{\ast})$$)$ set; the converse
is true for semi-reflexive spaces;

  9. (ix)

every $(p,q)$-limited subset of $E^{\prime}_{\beta}$ is a $(p,q)$-$(V)$ set;

  10. (x)

if $q<p$ and $A$ is a $(p,q)$-${\mathcal{E}}$-limited subset of $E$, then
$A=\\{0\\}$;

  11. (xi)

if $q\geq p$, then any finite subset of $E$ is $(p,q)$-limited;

  12. (xii)

a bounded subset $A$ of $E$ is $(p,q)$-limited $($resp.,
$(p,q)$-${\mathcal{E}}$-limited$)$ if and only if for every sequence
$\\{x_{n}\\}_{n\in\omega}$ in $A$ and each $($resp., equicontinuous$)$ weak∗
$p$-summable sequence $\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$, it
follows $\big{(}|\langle\chi_{n},x_{n}\rangle|\big{)}\in\ell_{q}$ $($or $\in
c_{0}$ if $p=\infty$$)$.

###### Proof 3

(i) and (iii) are clear, and (viii) follows from Definitions 1.5 and 2.5 and
the trivial fact that every (equicontinuous) weakly $p$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}_{\beta}$ is (resp.,
equicontinuous) weak∗ $p$-summable in $E^{\prime}$. The clause (ii) follows
from (viii) and (ii) of Lemma 7.2 of [16] (which states that every
$(p,q)$-$(EV^{\ast})$ set is bounded). (vii) follows from the definition of
$(p,q)$-limited sets, and (ix) follows from the easy fact that for every
weakly $p$-summable sequence $\\{x_{n}\\}_{n\in\omega}$ in $E$, the sequence
$\\{J_{E}(x_{n})\\}_{n\in\omega}$ is weak∗ $p$-summable in $E^{\prime\prime}$.

(iv) Let $T:E\to L$ be an operator from $E$ to an lcs $L$, and let $A$ be a
$(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited) set in $E$. Observe
that the adjoint map $T^{\ast}:L^{\prime}_{w^{\ast}}\to E^{\prime}_{w^{\ast}}$
is continuous. Fix a (resp., equicontinuous) weak∗ $p$-summable sequence
$S=\\{\chi_{n}\\}_{n\in\omega}$ in $L^{\prime}$. It is easily seen (see Lemma
4.5 of [16]) that the sequence $\\{T^{\ast}(\chi_{n})\\}$ is weak∗
$p$-summable in $E^{\prime}$. If in addition the sequence $S$ is
equicontinuous, then its image $T^{\ast}(S)$ is equicontinuous as well.
Therefore

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

Thus $T(A)$ is a $(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited) set
in $L$.

The last assertion follows from the proved one applied to the identity
embedding $T:H\to E$.

(v) The necessity follows from (iii). To prove the sufficiency suppose for a
contradiction that $A$ is not a $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) set in $E$. Then there is a (resp.,
equicontinuous) weak∗ $p$-summable sequence $\\{\chi_{n}\\}_{n\in\omega}$ in
$E^{\prime}$ such that

$\Big{(}\sup_{a\in A}|\langle\chi_{n},a\rangle|\Big{)}\not\in\ell_{q}\;\mbox{
if $q<\infty$, }\;\mbox{ or }\;\;\Big{(}\sup_{a\in
A}|\langle\chi_{n},a\rangle|\Big{)}\not\in c_{0}\;\mbox{ if $q=\infty$}.$

Assume that $q<\infty$ (the case $q=\infty$ can be considered analogously).
For every $n\in\omega$, choose $a_{n}\in A$ such that
$|\langle\chi_{n},a_{n}\rangle|\geq\tfrac{1}{2}\cdot\sup_{a\in
A}|\langle\chi_{n},a\rangle|$. Then

$\sum_{n\in\omega}|\langle\chi_{n},a_{n}\rangle|^{q}\geq\tfrac{1}{2^{q}}\sum_{n\in\omega}\big{(}\sup_{a\in
A}|\langle\chi_{n},a\rangle|\big{)}^{q}=\infty.$

Thus the countable subset $\\{a_{n}\\}_{n\in\omega}$ of $A$ is not a
$(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited) set in $E$, a
contradiction.

(vi) Take any (resp., equicontinuous) weak∗ $p^{\prime}$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$. Since $p^{\prime}\leq p$,
$\\{\chi_{n}\\}_{n\in\omega}$ is also (resp., equicontinuous) weak∗
$p$-summable and hence $\big{(}\sup_{a\in
A}|\langle\chi_{n},a\rangle|\big{)}\in\ell_{q}$ (or $\in c_{0}$ if
$q=\infty$). It remains to note that $\ell_{q}\subseteq\ell_{q^{\prime}}$
because $q\leq q^{\prime}$.

(x) Let $q<p$ and $A$ be a $(p,q)$-${\mathcal{E}}$-limited subset of $E$.
Then, by (viii), $A$ is a $(p,q)$-$(EV^{\ast})$ set. Therefore, by Proposition
7.5 of [16], $A=\\{0\\}$.

(xi) By (iii) it suffices to show that $A=\\{x\\}$ is a $(p,q)$-limited set
for every $x\in E$. Let $\\{\chi_{n}\\}_{n\in\omega}$ be a weak∗ $p$-summable
sequence in $E^{\prime}_{\beta}$. Then
$\big{(}\langle\chi_{n},x\rangle\big{)}\in\ell_{p}$ (or $\in c_{0}$ if
$p=\infty$). Since $p\leq q$ it follows that $\big{(}\sup_{x\in
A}|\langle\chi_{n},x\rangle|\big{)}\in\ell_{q}$ (or $\in c_{0}$ if
$q=\infty$). Thus $A$ is a $(p,q)$-limited set.

(xii) The necessity is clear. To prove the sufficiency, for every
$n\in\omega$, choose $x_{n}\in A$ such that
$|\langle\chi_{n},x_{n}\rangle|\geq\tfrac{1}{2}\sup_{a\in
A}|\langle\chi_{n},a\rangle|$. By assumption,
$\big{(}|\langle\chi_{n},x_{n}\rangle|\big{)}\in\ell_{q}$ (or $\in c_{0}$ if
$q=\infty$). Therefore also $\big{(}\sup_{a\in
A}|\langle\chi_{n},a\rangle|\big{)}\in\ell_{q}$ (or $\in c_{0}$ if
$q=\infty$). Thus $A$ is a $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) set.

It follows from (x) and (xi) that there is sense to consider only the case
when $1\leq p\leq q\leq\infty$.

###### Notation 3.2

The family of all $(p,q)$-limited (resp., $p$-limited, limited,
$(p,q)$-${\mathcal{E}}$-limited, $p$-${\mathcal{E}}$-limited, or
${\mathcal{E}}$-limited) sets of an lcs $E$ is denoted by
$\mathsf{L}_{(p,q)}(E)$ (resp., $\mathsf{L}_{p}(E)$, $\mathsf{L}(E)$,
$\mathsf{EL}_{(p,q)}(E)$, $\mathsf{EL}_{p}(E)$, or $\mathsf{EL}(E)$).

Below we characterize $(p,q)$-limited sets in products and direct sums.

###### Proposition 3.3

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)

a subset $K$ of $E=\prod_{i\in I}E_{i}$ is a $(p,q)$-limited $($resp.,
$(p,q)$-${\mathcal{E}}$-limited$)$ set if and only if so are all its
coordinate projections;

  2. (ii)

a subset $K$ of $E=\bigoplus_{i\in I}E_{i}$ is a $(p,q)$-limited $($resp.,
$(p,q)$-${\mathcal{E}}$-limited$)$ set if and only if so are all its
coordinate projections and the support of $K$ is finite.

###### Proof 4

The necessity follows from (iv) of Lemma 3.1 because $E_{i}$ is a direct
summand of $E$ and, for the case (ii), the well known fact that any bounded
subset of a locally convex direct sum has finite support.

To prove the sufficiency, let $K$ be a subset of $E$ such that each projection
$K_{i}$ of $K$ is a $(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited)
set in $E_{i}$, and, for the case (ii), $K_{i}=\\{0\\}$ for all but finitely
many indices $i\in I$. We distinguish between the cases (i) and (ii).

(i) Take an arbitrary (resp., equicontinuous) weak∗ $p$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$, where
$\chi_{n}=(\chi_{i,n})_{i\in I}$. By Lemma 4.18 of [16], the sequence
$\\{\chi_{n}\\}_{n\in\omega}$ has finite support $F\subseteq I$ (i.e.,
$\chi_{i,n}=0$ for all $n\in\omega$ and $i\in I{\setminus}F$) and for every
$i\in F$, each sequence $\\{\chi_{i,n}\\}_{n\in\omega}$ is weak∗ $p$-summable
in $E^{\prime}_{i}$. If in addition $\\{\chi_{n}\\}_{n\in\omega}$ is
equicontinuous, then for every $i\in F$, the sequence
$\\{\chi_{i,n}\\}_{n\in\omega}\subseteq E^{\prime}_{i}$ is also equicontinuous
(indeed, if $T_{i}:E_{i}\to E$ is the identity embedding, then
$\\{\chi_{i,n}\\}_{n}=\\{T^{\ast}_{i}(\chi_{n})\\}_{n}$ is equicontinuous).
Then

$\sup_{x\in K}|\langle\chi_{n},x\rangle|=\sup_{x\in K}\big{|}\sum_{i\in
F}\langle\chi_{i,n},x(i)\rangle\big{|}\leq\sum_{i\in F}\sup_{x(i)\in
K_{i}}|\langle\chi_{i,n},x(i)\rangle|.$

Since all $K_{i}$ are $(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited)
sets, we have $\big{(}\sup_{x(i)\in
K_{i}}|\langle\chi_{i,n},x(i)\rangle|\big{)}\in\ell_{q}$ (or $\in c_{0}$ if
$q=\infty$). Therefore also $\big{(}\sup_{x\in
K}|\langle\chi_{n},x\rangle|\big{)}\in\ell_{q}$ (or $\in c_{0}$ if
$q=\infty$). Thus $K$ is a $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) set in $E$.

(ii) Let $F\subseteq I$ be the finite support of $K$. Take an arbitrary
(resp., equicontinuous) weak∗ $p$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}_{\beta}$, where
$\chi_{n}=(\chi_{i,n})_{i\in I}$ with $\chi_{i,n}\in E^{\prime}_{i}$. As in
(i) above, if $\\{\chi_{n}\\}_{n\in\omega}$ is equicontinuous, then for every
$i\in F$, the sequence $\\{\chi_{i,n}\\}_{n\in\omega}\subseteq E^{\prime}_{i}$
is also equicontinuous. Then, by Lemma 4.18 of [16], for every $i\in F$, the
sequence $\\{\chi_{i,n}\\}_{n\in\omega}$ is weak∗ $p$-summable in
$E^{\prime}_{i}$ and hence

$\sup_{x\in K}|\langle\chi_{n},x\rangle|=\sup_{x\in K}\big{|}\sum_{i\in
F}\langle\chi_{i,n},x(i)\rangle\big{|}\leq\sum_{i\in F}\sup_{x(i)\in
K_{i}}|\langle\chi_{i,n},x(i)\rangle|.$

Since all $K_{i}$ are $(p,q)$-limited (resp., $(p,q)$-${\mathcal{E}}$-limited)
sets, we have $\big{(}\sup_{x(i)\in
K_{i}}|\langle\chi_{i,n},x(i)\rangle|\big{)}\in\ell_{q}$ (or $\in c_{0}$ if
$q=\infty$). Therefore also $\big{(}\sup_{x\in
K}|\langle\chi_{n},x\rangle|\big{)}\in\ell_{q}$ (or $\in c_{0}$ if
$q=\infty$). Thus $K$ is a $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) set in $E^{\prime}$.

Let $A$ be a bounded subset of a locally convex space $E$. Then, by
Proposition 16.10 of [16], the map $T_{A}:\ell_{1}^{0}(A)\to E$ defined by

$T_{A}(\lambda_{0}a_{0}+\cdots+\lambda_{n}a_{n}):=\lambda_{0}a_{0}+\cdots+\lambda_{n}a_{n}\quad(n\in\omega,\;\lambda_{0},\dots,\lambda_{n}\in\mathbb{F},\;a_{0},\dots,a_{n}\in
A),$ (3.1)

is an operator. Now we characterize $(p,q)$-limited sets.

###### Theorem 3.4

Let $1\leq p\leq q\leq\infty$, and let $E$ be a (resp., $p$-barrelled) locally
convex space. Then a bounded subset $A$ of $E$ is a $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) set if and only if the adjoint operator
$T_{A}^{\ast}:E^{\prime}_{w^{\ast}}\to\ell_{\infty}(A)$ is $(q,p)$-convergent.

###### Proof 5

Consider an operator $T_{A}:\ell_{1}^{0}(A)\to E$ defined in (3.1). Observe
that for each $\chi\in E^{\prime}$, the $a$th coordinate
$T_{A}^{\ast}(\chi)(a)$ of $T_{A}^{\ast}(\chi)$ is

$T_{A}^{\ast}(\chi)(a)=\langle
T_{A}^{\ast}(\chi),a\rangle=\langle\chi,T_{A}(a)\rangle=\langle\chi,a\rangle,$

and hence

$\|T^{\ast}_{A}(\chi)\|_{\ell_{\infty}(A)}=\sup_{a\in
A}|T^{\ast}_{A}(\chi)(a)|=\sup_{a\in A}|\langle\chi,a\rangle|$ (3.2)

Now, by definition, a subset $A$ of $E$ is a $(p,q)$-limited set if and only
if $\big{(}\sup_{a\in A}|\langle\chi_{n},a\rangle|\big{)}\in\ell_{q}$ (or $\in
c_{0}$ if $q=\infty$) for every weak∗ $p$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$, and hence, by (3.2), if and
only if
$\big{(}\|T^{\ast}_{A}(\chi_{n})\|_{\ell_{\infty}(A)}\big{)}\in\ell_{q}$ (or
$\in c_{0}$ if $q=\infty$) for every weakly $p$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}_{w^{\ast}}$, i.e., $T^{\ast}_{A}$
is a $(q,p)$-convergent linear map.

The case when $E$ is $p$-barrelled follows from the fact that $(p,q)$-limited
subsets of $E$ are exactly $(p,q)$-${\mathcal{E}}$-limited (see (i) of Lemma
3.1).

We select the next theorem.

###### Theorem 3.5

Let $1\leq p\leq q\leq\infty$, $E$ be a locally convex space, and let $T$ be
an operator from a normed space $L$ to $E$. Then $T(B_{L})$ is a
$(p,q)$-limited subset of $E$ if and only if
$T^{\ast}:E^{\prime}_{w^{\ast}}\to L^{\prime}_{\beta}$ is $(q,p)$-convergent.

###### Proof 6

Observe that for every $\chi\in E^{\prime}$, we have

$\|T^{\ast}(\chi)\|_{L^{\prime}_{\beta}}=\sup_{y\in B_{L}}|\langle
T^{\ast}(\chi),y\rangle|=\sup_{y\in B_{L}}|\langle\chi,T(y)\rangle|.$ (3.3)

Let $\\{\chi_{n}\\}_{n\in\omega}$ be a weak∗ $p$-summable sequence in
$E^{\prime}$. Then, by (3.3), we have
$\big{(}\|T^{\ast}(\chi_{n})\|_{L^{\prime}_{\beta}}\big{)}_{n\in\omega}=\big{(}\sup_{y\in
B_{L}}|\langle\chi_{n},T(y)\rangle|\big{)}_{n\in\omega}$. Now the theorem
follows from the definition of $(p,q)$-limited sets and the definition of
$(q,p)$-convergent linear map.

It is natural to find some classes of subsets which are $(p,q)$-limited.
Below, under additional assumption on an lcs $E$, we show that any precompact
subset $A$ of $E$ is $(p,\infty)$-limited.

###### Proposition 3.6

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

  1. (i)

Every precompact subset $A$ of $E$ is $(p,\infty)$-${\mathcal{E}}$-limited.

  2. (ii)

If $E$ is $p$-barrelled, then every precompact subset $A$ of $E$ is
$(p,\infty)$-limited.

###### Proof 7

Let $S=\\{\chi_{n}\\}_{n\in\omega}$ be a (resp., equicontinuous) weak∗
$p$-summable sequence in $E^{\prime}$. If $E$ is $p$-barrelled, then $S$ is
equicontinuous. Therefore in both cases (i) and (ii) we can assume that $S$ is
equicontinuous. Hence, by Proposition 3.9.8 of [27], the weak∗ topology
$\sigma(E^{\prime},E)$ and the topology $\tau_{pc}$ of uniform convergence on
precompact subsets of $E$ coincide on $S$. Since $S$ is weak∗ $p$-summable, it
is a weak∗ null-sequence. Therefore $\chi_{n}\to 0$ also in $\tau_{pc}$. As
$A$ is precompact, we obtain $\sup_{x\in A}|\langle\chi_{n},x\rangle|\to 0$.
Thus $A$ is a $(p,\infty)$-limited set (resp., a
$(p,\infty)$-${\mathcal{E}}$-limited set).

Since, by definition, $\infty$-barrelled spaces are exactly $c_{0}$-barrelled,
setting $p=\infty$ in (ii) of Proposition 3.6 we obtain the next assertion.

###### Corollary 3.7

If $E$ is a $c_{0}$-barrelled space, then every precompact subset of $E$ is
limited.

The condition in (ii) of Proposition 3.6 that $E$ is $p$-barrelled is
essential as the following theorem shows. Moreover, it may happen that a
non-$p$-barrelled space contains even compact sets which are not limited, see
Example 3.10 below.

###### Theorem 3.8

Let $p\in[1,\infty]$, $X$ be a Tychonoff space, and let $\mathcal{T}$ be a
locally convex vector topology on $L(X)$ compatible with the duality
$(L(X),C(X))$. Then the following assertions are equivalent:

  1. (i)

the space $C_{p}(X)$ is $p$-barrelled;

  2. (ii)

every precompact (= bounded) subset of $C_{p}(X)$ is $(p,\infty)$-limited;

  3. (iii)

$X$ has no infinite functionally bounded subsets;

  4. (iv)

each bounded subset of $C_{p}(X)$ is $(p,q)$-limited for some (every) $p\leq
q\leq\infty$;

  5. (v)

$L_{\mathcal{T}}(X)$ is quasi-complete;

  6. (vi)

$L_{\mathcal{T}}(X)$ is sequentially complete;

  7. (vii)

$L_{\mathcal{T}}(X)$ is locally complete.

###### Proof 8

(i)$\Rightarrow$(ii) follows from (ii) of Proposition 3.6.

(ii)$\Rightarrow$(iii) Assume that every bounded subset of $C_{p}(X)$ is
$(p,\infty)$-limited, and suppose for a contradiction that $X$ has an infinite
functionally bounded subset $A$. Then one can find a sequence
$\\{x_{n}\\}_{n\in\omega}$ in $A$ and a sequence $\\{U_{n}\\}_{n\in\omega}$ of
open subsets of $X$ such that $x_{n}\in U_{n}$ and $U_{n}\cap U_{m}=\emptyset$
for all distinct $n,m\in\omega$. Set

$B:=\Big{\\{}f\in C_{p}(X):f(U_{n})\subseteq[0,2^{n+1}]\mbox{ for all
$n\in\omega$, and
}f\big{(}X{\setminus}\bigcup_{n\in\omega}U_{n}\big{)}\subseteq\\{0\\}\Big{\\}}.$

Then $B$ is a bounded subset of $C_{p}(X)$, and hence $B$ is
$(p,\infty)$-limited. For every $n\in\omega$, set
$\chi_{n}:=\tfrac{1}{2^{n}}\delta_{x_{n}}$. Since $A$ is functionally bounded,
we obtain that the sequence $S=\\{\chi_{n}\\}_{n\in\omega}$ is weak∗
$p$-summable in the dual space $C_{p}(X)^{\prime}$. For every $n\in\omega$,
take a continuous function $g_{n}:X\to[0,2^{n}]$ such that
$g_{n}(X{\setminus}U_{n})\subseteq\\{0\\}$ and $g_{n}(x_{n})=2^{n}$. It is
clear that $g_{n}\in B$ for all $n\in\omega$. However, since

$\sup_{f\in
B}|\langle\chi_{n},f\rangle|\geq|\langle\chi_{n},g_{n}\rangle|=1\not\to 0,$

we obtain that $B$ is not $(p,\infty)$-limited, a contradiction.

(iii)$\Rightarrow$(i) Assume that $X$ has no infinite functionally bounded
subsets. By the Buchwalter–Schmets theorem, the space $C_{p}(X)$ is barrelled
and hence it is $p$-barrelled.

(iii)$\Rightarrow$(iv) Fix $p\leq q\leq\infty$, and let $B$ be a bounded
subset of $C_{p}(X)$. Take an arbitrary weak∗ $p$-summable sequence
$S=\\{\chi_{n}\\}_{n\in\omega}$ in $C_{p}(X)^{\prime}=L(X)$. Since $S$ is
weak∗ bounded and the topology of the free lcs $L(X)$ is compatible with
$\sigma\big{(}L(X),C_{p}(X)\big{)}$ it follows that $S$ is a bounded subset of
$L(X)$. As all functionally bounded subsets of $X$ are finite, Proposition 2.7
of [13] implies that $S$ is finite-dimensional. By Lemma 4.6 of [16], there
are linearly independent elements $\eta_{1},\dots,\eta_{s}\in L(X)$ and
sequences $(a_{1,n}),\dots,(a_{s,n})\in\ell_{p}$ (or $\in c_{0}$ if
$p=\infty$) such that

$\chi_{n}=a_{1,n}\eta_{1}+\cdots+a_{s,n}\eta_{s}\;\;\mbox{ for every
$n\in\omega$}.$

Now, since $B$ is a bounded subset of $C_{p}(X)$ we obtain

$\sup_{f\in
B}|\langle\chi_{n},f\rangle|\leq\sum_{i=1}^{s}|a_{i,n}|\cdot\sup_{f\in
B}|\langle\eta_{i},f\rangle|$

and hence the inequality $p\leq q$ implies $\big{(}\sup_{f\in
B}|\langle\chi_{n},f\rangle|\big{)}_{n}\in\ell_{q}$ (or $\in c_{0}$ if
$q=\infty$). Therefore $B$ is a $(p,q)$-limited set, as desired.

(iv)$\Rightarrow$(ii) Assume that each bounded subset of $C_{p}(X)$ is
$(p,q)$-limited for some $p\leq q\leq\infty$. Then, by (vi) of Lemma 3.1,
every bounded subset of $C_{p}(X)$ is $(p,\infty)$-limited.

(iii)$\Rightarrow$(v) Since all functionally bounded subsets of $X$ are
finite, Proposition 2.7 of [13] implies that any bounded subset of
$L_{\mathcal{T}}(X)$ is finite-dimensional. Thus $L_{\mathcal{T}}(X)$ is
quasi-complete.

The implication (v)$\Rightarrow$(vi) and (vi)$\Rightarrow$(vii) hold true for
any lcs.

(vii)$\Rightarrow$(i) Assume that $L_{\mathcal{T}}(X)$ is locally complete.
Since $(L(X),C(X))$ is a dual pair and $L_{\mathcal{T}}(X)$ is locally
complete, it follows that $L(X)_{w^{\ast}}$ is also locally complete. As
$C_{p}(X)$ is quasibarrelled hence Mackey, Theorem 5.6 of [16], implies that
$C_{p}(X)$ is $p$-barrelled.

For numerous other equivalent conditions to (i)-(vii) of Theorem 3.8 see
Theorem 3.5 of [2].

According to (viii) of Lemma 3.1, every $(p,q)$-limited set is a
$(p,q)$-$(V^{\ast})$ set, but the converse is not true in general as the
following corollary shows.

###### Corollary 3.9

Let $1\leq p\leq q\leq\infty$, and let $X$ be a Tychonoff space which has
infinite functionally bounded subsets. Then $C_{p}(X)$ contains
$(p,q)$-$(V^{\ast})$ sets which are not $(p,q)$-limited.

###### Proof 9

By Corollary 7.11 of [16], for every Tychonoff space $X$ we have
$\mathsf{V}_{(p,q)}^{\ast}\big{(}C_{p}(X)\big{)}=\mathsf{Bo}\big{(}C_{p}(X)\big{)}$.
Now the assertion follows from Theorem 3.8.

For a better understanding it is convenient to have a concrete example of a
compact subset which is not limited. Denote by $\mathbf{s}=[0,\omega]$ a
convergent sequence.

###### Example 3.10

There are compact subsets of $C_{p}(\mathbf{s})$ which are not limited.

###### Proof 10

For every $n\in\omega$, let $f_{n}=1_{\\{n\\}}$ be the characteristic function
of the set $\\{n\\}$ and let $\chi_{n}:=\delta_{n}-\delta_{n+1}$, where
$\delta_{x}$ denoted the Dirac measure at the point $x$. Evidently, the
sequence $S=\\{f_{n}\\}_{n\in\omega}$ is a null sequence in
$C_{p}(\mathbf{s})$, and the sequence $\\{\chi_{n}\\}_{n\in\omega}$ is weak∗
null. Since
$\sup\\{|\langle\chi_{n},f_{i}\rangle|:i\in\omega\\}\geq|\langle\chi_{n},f_{n}\rangle|=1\not\to
0$ it follows that $S$ is not limited.

Proposition 3.6 motivates the following inverse problem: Characterize locally
convex spaces whose $(p,q)$-limited subset (resp.,
$(p,q)$-${\mathcal{E}}$-limited subset) of $E$ are precompact. We solve this
problem in the next theorem.

###### Theorem 3.11

Let $1\leq p\leq q\leq\infty$. For a locally convex space $E$ the following
assertions are equivalent:

  1. (i)

every $(p,q)$-limited subset (resp., $(p,q)$-${\mathcal{E}}$-limited subset)
of $E$ is precompact;

  2. (ii)

each operator $T:L\to E$ from an lcs $L$ to $E$ which transforms bounded
subsets of $L$ to $(p,q)$-limited subsets (resp.,
$(p,q)$-${\mathcal{E}}$-limited subset) of $E$, transforms bounded subsets of
$L$ to precompact subsets of $E$;

  3. (iii)

as in (ii) with a normed space $L$.

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

  1. (iv)

as in (ii) with a Banach space $L$.

###### Proof 11

(i)$\Rightarrow$(ii) Let $T:L\to E$ be an operator which transforms bounded
subsets of an lcs $L$ to $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) subsets of $E$. Let $A$ be a bounded subset
of $L$. Then $T(A)$ is a $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) subset of $E$, and hence, by (i), $T(A)$ is
precompact. Thus $T$ transforms bounded subsets of $L$ to precompact subsets
of $E$.

(ii)$\Rightarrow$(iii) and (ii)$\Rightarrow$(iv) are trivial.

(iii)$\Rightarrow$(i) and (iv)$\Rightarrow$(i): Fix a $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) subset $A$ of $E$. By (iii) of Lemma 3.1,
without loss of generality we can assume that $A=A^{\circ\circ}$. Consider the
normed space $E_{A}$ (if $E$ is locally complete, then $E_{A}$ is a Banach
space), where the norm on $E_{A}$ is defined by the gauge of $A$, and recall
that the closed unit ball $B_{A}$ of $E_{A}$ is exactly $A$. By Propositions
3.2.2 and 5.1.6 of [35], the identity inclusion $T:E_{A}\to E$ is continuous
and the set $T(B_{A})=A$ is a $(p,q)$-limited (resp.,
$(p,q)$-${\mathcal{E}}$-limited) set. Since any bounded subset of $E_{A}$ is
contained in some $aB_{A}$, $a>0$, Lemma 3.1 implies that $T$ transforms
bounded subsets of the normed (resp., Banach) space $E_{A}$ to $(p,q)$-limited
(resp., $(p,q)$-${\mathcal{E}}$-limited) subsets of $E$. Therefore, by (iii)
or (iv), the set $A=T(B)$ is precompact.

By (vi) of Lemma 3.1, every $(p,q)$-limited set is $(1,\infty)$-limited.
Therefore to characterize $(1,\infty)$-limited sets in locally convex spaces
is an important problem. For barrelled spaces we solve this problem in
Proposition 3.13 below. Our proof is similar to the proof of Proposition 1.1
of [4], where it is obtained a characterization of $(V^{\ast})$ sets in Banach
spaces. First we prove the next lemma.

###### Lemma 3.12

If a subset $A$ of a locally convex space $E$ is a $(1,\infty)$-limited set,
then $T(A)$ is relatively compact for every operator $T:E\to\ell_{1}$.

###### Proof 12

Suppose for a contradiction that $T(A)$ is not relatively compact in
$\ell_{1}$ for some operator $T:E\to\ell_{1}$. By (i) of Proposition 4.17 of
[16], $T(x)=(\langle\chi_{n},x\rangle)_{n\in\omega}$ for some equicontinuous
weak∗ $1$-summable sequence $\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$.
Then Proposition 2.4 implies that there are $\varepsilon>0$, a sequence
$r_{0}<s_{0}<r_{1}<s_{1}<\cdots$ in $\omega$, and a sequence
$\\{a_{j}\\}_{j\in\omega}$ in $A$ such that

$\sum_{n=r_{j}}^{s_{j}}|\langle\chi_{n},a_{j}\rangle|>\varepsilon\;\;\mbox{
for every }\;j\in\omega.$

For every $j\in\omega$, by Lemma 6.3 of [37], there is a subset $F_{j}$ of
$[r_{j},s_{j}]$ such that

$\Big{|}\sum_{n\in
F_{j}}\langle\chi_{n},a_{j}\rangle\Big{|}>\tfrac{\varepsilon}{4}.$ (3.4)

For every $j\in\omega$, set $\eta_{j}:=\sum_{n\in F_{j}}\chi_{n}$. Then the
sequence $\\{\eta_{j}\\}_{j\in\omega}$ is weak∗ $1$-summable in $E^{\prime}$.
By (3.4), we have

$\sup_{a\in
A}|\langle\eta_{j},a\rangle|\geq|\langle\eta_{j},a_{j}\rangle|>\tfrac{\varepsilon}{4},\quad\mbox{
for every }\;j\in\omega,$

and hence $A$ is not a $(1,\infty)$-limited set, a contradiction.

###### Proposition 3.13

For a bounded subset $A$ of a barrelled locally convex space $E$ the following
assertions are equivalent:

  1. (i)

$A$ is a $(1,\infty)$-limited set;

  2. (ii)

$T(A)$ is relatively compact for every operator $T:E\to\ell_{1}$;

  3. (iii)

for any weak∗ $1$-summable sequence $\\{\chi_{n}\\}_{n\in\omega}$ in
$E^{\prime}$, it follows

$\lim_{m\to\infty}\sup\Big{\\{}\sum_{m\leq n}|\langle\chi_{n},x\rangle|:x\in
A\Big{\\}}=0.$

###### Proof 13

(i)$\Rightarrow$(ii) follows from Lemma 3.12.

(ii)$\Rightarrow$(iii) Let $\\{\chi_{n}\\}_{n\in\omega}$ be a weak∗
$1$-summable sequence in $E^{\prime}$. Then, by Proposition 4.19 of [16],
there is an operator $T:E\to\ell_{1}$ such that
$T(x):=(\langle\chi_{n},x\rangle)$ for every $x\in E$. By (ii), the set $T(A)$
is relatively compact in $\ell_{1}$. Therefore, by Proposition 2.4, we obtain

$\lim_{m\to\infty}\sup\Big{\\{}\sum_{m\leq n}|y_{n}|:y=(y_{n})\in
T(A)\Big{\\}}=0.$

It remains to note that if $y=(y_{n})=T(x)$ for some $x\in A$, then
$y_{n}=\langle\chi_{n},x\rangle$ for all $n\in\omega$.

(iii) implies (i) since $\sup_{a\in
A}|\langle\chi_{m},a\rangle|\leq\sup\big{\\{}\sum_{m\leq
n}|\langle\chi_{n},x\rangle|:x\in A\big{\\}}\to 0$ for every weak∗
$1$-summable sequence $\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$.

###### Corollary 3.14

Weakly sequentially precompact subsets and precompact subsets of a barrelled
locally convex space $E$ are $(1,\infty)$-limited.

###### Proof 14

Let $A$ be a weakly sequentially precompact subset of $E$ or a precompact
subset of $E$. Then for every operator $T:E\to\ell_{1}$, the image $T(A)$ is
also weakly sequentially precompact or precompact in $\ell_{1}$. By Lemma 2.2,
$T(A)$ is (sequentially) precompact and hence relatively compact in
$\ell_{1}$. Thus, by Proposition 3.13, $A$ is a $(1,\infty)$-limited set.

It is natural to characterize spaces for which all relatively weakly
sequentially $p$-compact sets are $(q,\infty)$-limited. For the case when $E$
is a Banach space and $q=\infty$, the next proposition gives
$(a)\Leftrightarrow(b)$ of Proposition 2.10 of [11].

###### Proposition 3.15

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

  1. (i)

all relatively weakly sequentially $p$-compact subsets of $E$ are
$(q,\infty)$-limited (resp., $(q,\infty)$-${\mathcal{E}}$-limited);

  2. (ii)

every weakly $p$-summable sequence in $E$ is $(q,\infty)$-limited (resp.,
$(q,\infty)$-${\mathcal{E}}$-limited).

###### Proof 15

(i)$\Rightarrow$(ii) is clear because every weakly $p$-summable sequence is
relatively weakly sequentially $p$-compact.

(ii)$\Rightarrow$(i) Suppose for a contradiction that there is a relatively
weakly sequentially $p$-compact subset $A$ of $E$ which is not
$(q,\infty)$-limited (resp., $(q,\infty)$-${\mathcal{E}}$-limited). Then there
are a weak∗ (resp., equicontinuous) $q$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $E^{\prime}$ and $\varepsilon>0$ such that
$\sup_{a\in A}|\langle\chi_{n},a\rangle|\geq\varepsilon$ for every
$n\in\omega$. For every $n\in\omega$, choose $a_{n}\in A$ such that

$|\langle\chi_{n},a_{n}\rangle|\geq\tfrac{\varepsilon}{2}.$ (3.5)

Since $A$ is relatively weakly sequentially $p$-compact, there are $a\in E$
and a subsequence $\\{a_{n_{k}}\\}_{k\in\omega}$ of $\\{a_{n}\\}_{n\in\omega}$
such that $\\{a_{n_{k}}-a\\}_{n\in\omega}$ is weakly $p$-summable. Taking into
account that $\\{\chi_{n_{k}}\\}_{k\in\omega}$ is also weak∗ null, (ii) and
(3.5) imply

$\tfrac{\varepsilon}{2}\leq|\langle\chi_{n_{k}},a_{n_{k}}\rangle|\leq|\langle\chi_{n_{k}},a_{n_{k}}-a\rangle|+|\langle\chi_{n_{k}},a\rangle|\leq\sup_{i\in\omega}|\langle\chi_{n_{k}},a_{n_{i}}-a\rangle|+|\langle\chi_{n_{k}},a\rangle|\to
0,$

a contradiction.

If $E$ is a Banach space and $p=\infty$, the next theorem was proved by
Grothendieck in [24]. Note that Example 3.5 of [17] shows that our result is
indeed more general than the Grothenfieck theorem.

###### Theorem 3.16

Let $p\in[1,\infty]$, $E$ be a weakly $p$-angelic and locally complete space,
and let $H:=\big{(}E^{\prime},\mu(E^{\prime},E)\big{)}$. Then a bounded subset
$B$ of $H$ is precompact if and only if it is a $(p,\infty)$-limited set.

###### Proof 16

Assume that $B$ is a $\mu(E^{\prime},E)$-precompact subset of $E^{\prime}$.
Let $S=\\{x_{n}\\}_{n\in\omega}$ be a weak∗ $p$-summable sequence in
$H^{\prime}=E$. Then $S$ is weakly $p$-summable in $E$. Since $E$ is locally
complete, the set $K:=\overline{\mathrm{acx}}(S)$ is weakly compact, and
hence, by the Mackey–Arens theorem, $K^{\circ}$ is a neighborhood of zero in
$H$. Consequently, $K=K^{\circ\circ}$ and hence also $S$ are
$\mu(E^{\prime},E)$-equicontinuous. The weak∗ $p$-summability of $S$ implies
that $S$ is weak∗ null. Hence, by Proposition 3.9.8 of [27], $x_{n}\to 0$ in
the topology of uniform convergence on precompact subsets of $H$, in
particular, $x_{n}\to 0$ uniformly on $B$. Since $S$ was arbitrary, by
definition this means that $B$ is a $(p,\infty)$-limited subset of $H$.

Conversely, assume that $B$ is a $(p,\infty)$-limited subset of $H$. Let
$u=\mathsf{id}_{E}:E\to E$ be the identity map, $\mathfrak{S}$ be the family
of all absolutely convex weakly compact subsets of $E$, $\mathfrak{T}$ be the
family of all $(p,\infty)$-limited subsets of $H$. Then the equivalence of (1)
and (1’) in Theorem 12 of [25, p. 91] can be formulated as follows: the
$(p,\infty)$-limited sets of $H$ are precompact if and only if any set
$K\in\mathfrak{S}$ is precompact for the topology $\mathcal{T}$ of uniform
convergence on all $(p,\infty)$-limited sets of $H$. Therefore to prove that
$B$ is precompact it suffices to show that any $K\in\mathfrak{S}$ is
$\mathcal{T}$-precompact. To this end, fix a $K\in\mathfrak{S}$.

We claim that the topology $\mathcal{T}$ and the weak topology
$\sigma(E,E^{\prime})$ coincide on $K$. By (xi) of Lemma 3.1, we have
$\sigma(E,E^{\prime})\subseteq\mathcal{T}$. Therefore to prove the claim we
have to show only that any $\mathcal{T}$-closed subset $A$ of $K$ is also
weakly closed. Let $z\in\overline{A}^{\,\sigma(E,E^{\prime})}$. Since $E$ is a
weakly $p$-angelic space and $K$ is weakly compact, (i) of Lemma 3.6 of [17]
implies that there is a sequence $\\{x_{n}\\}_{n\in\omega}$ in $A$ which
weakly $p$-converges to $z$, i.e., the sequence $\\{x_{n}-z\\}_{n\in\omega}$
is weakly $p$-summable. By the definition of $(p,\infty)$-limited sets we
obtain that $x_{n}\to z$ in the topology $\mathcal{T}$ and hence $z\in A$.
Thus $A$ is weakly closed. The claim is proved.

Since $K$ is weakly compact, the claim implies that also $K$ is compact for
the topology $\mathcal{T}$.

It is convenient to formulate Theorem 3.16 in a dual form.

###### Corollary 3.17

Let $p\in[1,\infty]$, and let $E$ be a Mackey space. If the space
$E^{\prime}_{w^{\ast}}$ is weakly $p$-angelic and locally complete, then a
bounded subset $A$ of $E$ is precompact if and only if it is a
$(p,\infty)$-limited set.

###### Proof 17

Set $E_{1}:=E^{\prime}_{w^{\ast}}$ (so $E_{1}$ carries its weak topology) and
$H_{1}:=\big{(}E_{1}^{\prime},\mu(E^{\prime}_{1},E_{1})\big{)}$. Then
$E^{\prime}_{1}=E$ algebraically. Since, by the Mackey–Arens theorem, the
polars of the weak∗ compact absolutely convex subsets of
$E^{\prime}_{w^{\ast}}=E_{1}$ define the Mackey topology on $E$ and the Mackey
topology $\mu(E^{\prime}_{1},E_{1})$ on $E^{\prime}_{1}=E$, we obtain
$\mu(E^{\prime}_{1},E_{1})=\mu(E,E^{\prime})$. As $E$ is a Mackey space it
follows $H_{1}=E$. Now Theorem 3.16 applies.

Theorem 5.6 of [16] (which states that a Mackey space $E$ is $p$-barrelled if
and only if $E^{\prime}_{w^{\ast}}$ is locally complete) and Corollary 3.17
imply

###### Corollary 3.18

Let $p\in[1,\infty]$, and let $E$ be a Mackey $p$-barrelled space (for
example, $E$ is barrelled). If $E^{\prime}_{w^{\ast}}$ is a weakly $p$-angelic
space, then a bounded subset $A$ of $E$ is precompact if and only if it is a
$(p,\infty)$-limited set.

The case $p=\infty$ is of independent interest.

###### Corollary 3.19

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

  1. (i)

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

Then a bounded subset $A$ of $E$ is precompact if and only if it is a limited
set. Moreover, if in addition $E$ is von Neumann complete, then a bounded
subset $A$ of $E$ is relatively compact if and only if it is a limited set.

###### Proof 18

(i) Proposition 3.4 of [17] states that every weakly angelic space $E$ is
weakly $\infty$-angelic. Now Corollary 3.18 applies.

(ii) Recall that any reflexive space is barrelled, see Proposition 11.4.2 of
[28]. Since $E$ is semi-reflexive, $E^{\prime}_{\beta}$ is a weakly angelic
space if and only if so is $E^{\prime}_{w^{\ast}}$. Now (i) applies.

(iii) Since $E$ is separable, the space $E^{\prime}_{w^{\ast}}$ admits a
weaker metrizable locally convex topology $\mathcal{T}$. Therefore
$(E^{\prime},\mathcal{T})$ and hence also $E^{\prime}_{w^{\ast}}$ are even
(weakly) angelic spaces. By Proposition 3.4 of [17], the space
$E^{\prime}_{w^{\ast}}$ is weakly (sequentially) $\infty$-angelic. Now (i)
applies.

The last assertion follows from the fact that if in addition $E$ is von
Neumann complete, then any precompact subset of $E$ is relatively compact.

##  4 Coarse $p$-limited sets

Below we summarize some basic properties of coarse $p$-limited sets, cf.
Proposition 2 of [20].

###### Lemma 4.1

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

  1. (i)

every coarse $p$-limited subset of $E$ is bounded;

  2. (ii)

the family $\mathsf{CL}_{p}(E)$ of all coarse $p$-limited sets in $E$ is
closed under taking subsets, finite unions and sums, and closed absolutely
convex hulls;

  3. (iii)

if $L$ is a locally convex space and $T\in\mathcal{L}(E,L)$ and if $A\subseteq
E$ is coarse $p$-limited, then $T(A)$ is a coarse $p$-limited subset of $L$;

  4. (iv)

a subset $A$ of $E$ is a coarse $p$-limited set if and only if every countable
subset of $A$ is a coarse $p$-limited set;

  5. (v)

([20, Remark 2]) in general, even for Banach spaces there is no inclusion
relationships between the class of coarse $p$-limited sets and the class of
coarse $q$-limited sets for $p\not=q$;

  6. (vi)

in general the property of being a coarse $p$-limited set is not the property
of the duality $(E,E^{\prime})$.

###### Proof 19

The clauses (i)-(iii) are clear.

(iv) The necessity follows from (ii). To prove the sufficiency we consider the
case $1\leq p<\infty$ since the case $p=\infty$ can be considered analogously.
Suppose for a contradiction that $A$ is not a coarse $p$-limited set in $E$.
Then there is an operator $T:E\to\ell_{p}$ such that $T(A)$ is not relatively
compact in $\ell_{p}$. Then, by Proposition 2.4, there is $\varepsilon>0$ such
that

$\sup_{a\in A}\Big{\\{}\sum_{m\leq n}|\langle
T^{\ast}(e^{\ast}_{n}),a\rangle|^{p}\Big{\\}}\geq\varepsilon\;\;\mbox{ for
every }m\in\omega.$

For every $m\in\omega$, choose $a_{m}\in A$ such that $\sum_{m\leq n}|\langle
T^{\ast}(e^{\ast}_{n}),a_{m}\rangle|^{p}>\tfrac{\varepsilon}{2}$. By
assumption the sequence $\\{a_{m}\\}_{m\in\omega}$ is a coarse $p$-limited
set. Therefore, by Proposition 2.4 and the choice of $a_{m}$, we have

$\tfrac{\varepsilon}{2}<\sup_{m\in\omega}\Big{\\{}\sum_{m\leq n}|\langle
T^{\ast}(e^{\ast}_{n}),a_{m}\rangle|^{p}\Big{\\}}\to 0,$

a contradiction.

(vi) Let $1\leq p<\infty$, and let $E=\ell_{p}$ (for $p=\infty$, one can
consider $E=c_{0}$). Then the unit ball $B_{E}$ is not a coarse $p$-limited
set in $E$ (if $\mathsf{id}:E\to\ell_{p}$ is the identity map then
$\mathsf{id}(B_{E})$ is not relatively compact in $\ell_{p}$). However, since
every $T\in\mathcal{L}(E_{w},\ell_{p})$ is finite-dimensional by Lemma 17.18
of [16] it follows that $B_{E}$ is a coarse $p$-limited set in $E_{w}$.

If $1\leq p<\infty$, Proposition 1 of [20] states that every $p$-limited
subset of a Banach space is a coarse $p$-limited set. Below we generalize this
result.

###### Proposition 4.2

Let $E$ be a locally convex space. Then:

  1. (i)

if $p\in[1,\infty]$ and
$S_{p}\big{(}\mathcal{L}(E,\ell_{p})\big{)}\subseteq\ell_{p}^{w}(E^{\prime}_{\beta})$,
then every $(p,p)$-$(V^{\ast})$ subset of $E$ is a coarse $p$-limited set;

  2. (ii)

if $1<p<\infty$, then every $(p,p)$-$(V^{\ast})$ subset $A$ of $E$ is a coarse
$p$-limited set;

  3. (iii)

if $p=\infty$ and
$S_{\infty}\big{(}\mathcal{L}(E,c_{0})\big{)}=c_{0}^{w}(E^{\prime}_{\beta})$,
then the class of $\infty$-$(V^{\ast})$ subsets of $E$ coincides with the
class of coarse $\infty$-limited subsets of $E$;

  4. (iv)

if $p\in[1,\infty]$, then every $p$-limited subset of $E$ is coarse
$p$-limited; in particular, every finite subset of $E$ is coarse $p$-limited.

###### Proof 20

(i) Let $A$ be a $(p,p)$-$(V^{\ast})$ subset of $E$, and let $T:E\to\ell_{p}$
(or $T:E\to c_{0}$ if $p=\infty$) be an operator. For every $n\in\omega$, we
set $\chi_{n}:=T^{\ast}(e^{\ast}_{n})$. Then the inclusion
$S_{p}\big{(}\mathcal{L}(E,\ell_{p})\big{)}\subseteq\ell_{p}^{w}(E^{\prime}_{\beta})$
implies that the sequence $\\{\chi_{n}\\}_{n}$ is weakly $p$-summable in
$E^{\prime}_{\beta}$. Since $A$ is a $(p,p)$-$(V^{\ast})$ set, it follows that

$\big{(}\sup_{a\in A}|\langle\chi_{n},a\rangle|\big{)}\in\ell_{p}\;\mbox{ (or
$\in c_{0}$ if $p=\infty$). }$ (4.1)

Assume that $p<\infty$. Then (4.1) implies

$\sup\left\\{\sum_{n=m}^{\infty}|\langle e^{\ast}_{n},T(a)\rangle|^{p}:a\in
A\right\\}\leq\sum_{n=m}^{\infty}\Big{(}\sup_{a\in A}|\langle
e^{\ast}_{n},T(a)\rangle|\Big{)}^{p}<\infty.$

Therefore, by (i) of Proposition 2.4, $T(A)$ is relatively compact in
$\ell_{p}$. Thus $A$ is a coarse $p$-limited set.

If $p=\infty$, then (4.1) yields $\lim_{n\to\infty}\sup\big{\\{}|\langle
e^{\ast}_{n},T(a)\rangle|:a\in A\big{\\}}=0$. Hence, by (ii) of Proposition
2.4, $T(A)$ is relatively compact in $c_{0}$. Thus $A$ is a coarse
$\infty$-limited set.

(ii) Assume that $1<p<\infty$. Then, by (iii) of Proposition 4.17 of [16],
$S_{p}\big{(}\mathcal{L}(E,\ell_{p})\big{)}\subseteq\ell_{p}^{w}(E^{\prime}_{\beta})$.
Thus, by (i), $A$ is a coarse $p$-limited set.

(iii) Taking account (i), we have to prove that every coarse $\infty$-limited
subset $A$ of $E$ is an $\infty$-$(V^{\ast})$ set. To this end, let
$\\{\chi_{n}\\}_{n}$ be a weakly $\infty$-summable sequence in
$E^{\prime}_{\beta}$. Then the equality
$S_{\infty}\big{(}\mathcal{L}(E,c_{0})\big{)}=c_{0}^{w}(E^{\prime}_{\beta})$
implies that there is $T\in\mathcal{L}(E,c_{0})$ such that
$\chi_{n}=T^{\ast}(e^{\ast}_{n})$ for every $n\in\omega$. Since $A$ is a
coarse $\infty$-limited set we obtain that $T(A)$ is a relatively compact
subset of $c_{0}$. Therefore, by (ii) of Proposition 2.4, we have

$\sup\big{\\{}|\langle\chi_{n},a\rangle|:a\in A\big{\\}}=\sup\big{\\{}|\langle
e^{\ast}_{n},T(a)\rangle|:a\in A\big{\\}}\to 0\;\mbox{ as $n\to\infty$},$

which means that $A$ is an $\infty$-$(V^{\ast})$ set.

(iv) Let $A$ be a $p$-limited subset of $E$, and let $T:E\to\ell_{p}$ (or
$T:E\to c_{0}$ if $p=\infty$) be an operator. For every $n\in\omega$, we set
$\chi_{n}:=T^{\ast}(e^{\ast}_{n})$. Then, by Proposition 4.17(i) of [16], the
sequence $\\{\chi_{n}\\}_{n}$ is weakly $p$-summable in
$E^{\prime}_{w^{\ast}}$. Now proceeding exactly as in (i) we obtain that $A$
is a coarse $p$-limited subset of $E$.

For the last assertion it suffices to note that, by (xi) of Lemma 3.1, every
finite subset of $E$ is a $p$-limited set.

###### Proposition 4.3

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

  1. (i)

a subset $K$ of $E=\prod_{i\in I}E_{i}$ is a coarse $p$-limited set if and
only if so are all its coordinate projections;

  2. (ii)

a subset $K$ of $E=\bigoplus_{i\in I}E_{i}$ is a coarse $p$-limited set if and
only if so are all its coordinate projections and the support of $K$ is
finite.

###### Proof 21

The necessity follows from (iii) of Lemma 4.1 since $E_{i}$ is a direct
summand of $E$ and, for the case (ii), the well known fact that any bounded
subset of a direct locally convex sum has finite support.

To prove the sufficiency, let $K$ be a subset of $E$ such that each coordinate
projection $K_{i}$ of $K$ is a coarse $p$-limited set in $E_{i}$, and, for the
case (ii), $K_{i}=\\{0\\}$ for all but finitely many indices $i\in I$. By (ii)
of Lemma 4.1, we can assume that $0\in K$. We distinguish between the cases
(i) and (ii).

(i) Let $T:E\to\ell_{p}$ (or $T:E\to c_{0}$ if $p=\infty$) be an operator. It
is easy to show (see for example Lemma 2.6 in [16]) that there is a finite
subset $F$ of $I$ such that $\\{0\\}^{F}\times\prod_{i\in I{\setminus}F}E_{i}$
is in the kernel of $T$. Then, taking into account that $0\in K$, we obtain
$T(K)\subseteq\sum_{i\in F}T(K_{i})$. Since, by assumption, all $T(K_{i})$ are
relatively compact in the Banach space $\ell_{p}$ (or in $c_{0}$) it follows
that $\sum_{i\in F}T(K_{i})$ (we identify $K_{i}$ with
$K_{i}\times\prod_{I{\setminus}\\{i\\}}\\{0_{i}\\}$) and hence also $T(K)$ are
relatively compact in $\ell_{p}$ (or $c_{0}$). Thus $K$ is a coarse
$p$-limited set in $E$.

(ii) Let $F\subseteq I$ be the finite support of $K$. Then
$T(K)\subseteq\sum_{i\in F}T(K_{i})$. As above in (i), it follows that $T(K)$
is relatively compact in $\ell_{p}$ (or $c_{0}$). Thus $K$ is a coarse
$p$-limited set in $E$.

##  5 Limited type sets and $p$-convergent operators

Let $1\leq p\leq q\leq\infty$, and let $E$ be a locally convex space. By Lemma
7.2 of [16] and Lemmas 3.1 and 4.1, the family $\mathsf{V}^{\ast}_{(p,q)}(E)$
of all $(p,q)$-$(V^{\ast})$ sets, the family $\mathsf{L}_{(p,q)}(E)$ of all
$(p,q)$-limited sets and the family $\mathsf{CL}_{p}(E)$ of all coarse
$p$-limited sets in $E$ are saturated bornologies. Therefore one can naturally
define the following polar topologies on the dual space $E^{\prime}$.

###### Definition 5.1

Let $1\leq p\leq q\leq\infty$, and let $E$ be a locally convex space. Denote
by $V^{\ast}_{(p,q)}(E^{\prime},E)$ (resp., $EV^{\ast}_{(p,q)}(E^{\prime},E)$,
$L_{(p,q)}(E^{\prime},E)$, $EL_{(p,q)}(E^{\prime},E)$ and
$CL_{p}(E^{\prime},E)$) the polar topology on $E^{\prime}$ of uniform
convergence on $(p,q)$-$(V^{\ast})$ (resp., $(p,q)$-$(EV^{\ast})$,
$(p,q)$-limited, $(p,q)$-${\mathcal{E}}$-limited or coarse $p$-limited)
subsets of $E$.

Since the families $\mathsf{V}^{\ast}_{(p,q)}(E)$ and $\mathsf{L}_{(p,q)}(E)$
depend only on the duality, the topologies $V^{\ast}_{p}(E^{\prime},E)$ and
$L_{(p,q)}(E^{\prime},E)$ are topologies of the dual pair $(E,E^{\prime})$.
However, (vi) of Lemma 4.1 shows that the topology $CL_{p}(E^{\prime},E)$ is
not a topology of $(E,E^{\prime})$. By this reason in what follows we consider
only the topologies $\mathsf{V}^{\ast}_{(p,q)}(E)$ and
$\mathsf{L}_{(p,q)}(E)$.

For further references we select the next simple lemma.

###### Lemma 5.2

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

  1. (i)

$\sigma(E^{\prime},E)\subseteq L_{(p,q)}(E^{\prime},E)\subseteq
V^{\ast}_{(p,q)}(E^{\prime},E)\subseteq\beta(E^{\prime},E)$,

  2. (ii)

$L_{(p,q)}(E^{\prime},E)\subseteq\mu(E^{\prime},E)$ if and only if every
$(p,q)$-limited set $A$ in $E$ is relatively weakly compact;

  3. (iii)

$L_{(p,q)}(E^{\prime},E)=\mu(E^{\prime},E)$ if and only if every
$(p,q)$-limited set $A$ in $E$ is relatively weakly compact and every weakly
compact absolutely convex subset of $E$ is $(p,q)$-limited.

###### Proof 22

(i) follows from (viii) and (xi) of Lemma 3.1.

(ii) and (iii) follow from the Mackey–Arens theorem and the fact that
$\mathsf{L}_{(p,q)}(E)$ is a saturated bornology (see (iii) of Lemma 3.1).

###### Remark 5.3

The inclusion $L_{(p,q)}(E^{\prime},E)\subseteq
V^{\ast}_{(p,q)}(E^{\prime},E)$ can be strict. Indeed, let $X$ be a Tychonoff
space containing an infinite functionally bounded subset. Then, by Corollary
3.9, the space $C_{p}(X)$ contains $(p,q)$-$(V^{\ast})$ sets which are not
$(p,q)$-limited. This fact, the inclusion $L_{(p,q)}(E^{\prime},E)\subseteq
V^{\ast}_{(p,q)}(E^{\prime},E)$ and the fact that
$\mathsf{V}_{(p,q)}^{\ast}\big{(}C_{p}(X)\big{)}$ and
$\mathsf{L}_{(p,q)}^{\ast}\big{(}C_{p}(X)\big{)}$ are saturated bornologies
imply that $L_{(p,q)}(E^{\prime},E)\subsetneq V^{\ast}_{(p,q)}(E^{\prime},E)$.

It is well known that if $T\in\mathcal{L}(E,L)$, then $T^{\ast}$ is weak∗ and
strongly continuous. The following assertion shows that $T^{\ast}$ is also
continuous with respect to the topology $L_{(p,q)}$.

###### Proposition 5.4

Let $1\leq p\leq q\leq\infty$, and let $T:E\to L$ be an operator between
locally convex spaces $E$ and $L$. Then:

  1. (i)

the adjoint map
$T^{\ast}:\big{(}L^{\prime},L_{(p,q)}(L^{\prime},L)\big{)}\to\big{(}E^{\prime},L_{(p,q)}(E^{\prime},E)\big{)}$
is continuous;

  2. (ii)

the adjoint map
$T^{\ast}:L^{\prime}_{w^{\ast}}\to\big{(}E^{\prime},L_{(p,q)}(E^{\prime},E)\big{)}$
is $p$-convergent.

###### Proof 23

(i) To show that $T^{\ast}$ is continuous, let $A^{\circ}$ be a standard
$L_{(p,q)}(E^{\prime},E)$-neighborhood of zero, where $A$ is a $(p,q)$-limited
set in $E$. Then, by Lemma 3.1, $T(A)$ is a $(p,q)$-limited set in $L$. Then
for every $\eta\in T(A)^{\circ}$ and each $a\in A$, we have $|\langle
T^{\ast}(\eta),a\rangle|=|\langle\eta,T(a)\rangle|\leq 1$ and hence
$T^{\ast}\big{(}T(A)^{\circ}\big{)}\subseteq A^{\circ}$. Thus $T^{\ast}$ is
continuous.

(ii) Since $L_{(p,q)}(E^{\prime},E)\subseteq L_{(p,\infty)}(E^{\prime},E)$ by
(vi) of Lemma 3.1, it suffices to consider the case $q=\infty$. Let
$\\{\chi_{n}\\}_{n\in\omega}$ be a weakly $p$-summable sequence in
$L^{\prime}_{w^{\ast}}$. To show that $T^{\ast}(\chi_{n})\to 0$ in
$\big{(}E^{\prime},L_{(p,\infty)}(E^{\prime},E)\big{)}$, fix an arbitrary
$B\in\mathsf{L}_{(p,\infty)}(E)$. By Lemma 3.1, $T(B)$ is a
$(p,\infty)$-limited set in $L$ and hence

$\lim_{n\to\infty}\sup_{b\in B}\big{|}\langle
T^{\ast}(\chi_{n}),b\rangle\big{|}=\lim_{n\to\infty}\sup_{b\in
B}\big{|}\langle\chi_{n},T(b)\rangle\big{|}=0.$

Therefore $T^{\ast}(\chi_{n})\in B^{\circ}$ for all sufficiently large
$n\in\omega$. Since $B$ was arbitrary this means that $T^{\ast}(\chi_{n})\to
0$ in $L_{(p,\infty)}(E^{\prime},E)$, as desired.

Below we give a complete answer to the problem posed in the introduction for
the $(p,\infty)$-case (namely, characterize those operators $T$ which map all
bounded  sets into $(p,q)$-limited sets or into coarse $p$-limited sets). We
are interested in this special case because it is dually connected with
$p$-convergent operators, see in particular Theorem 5.5 and Theorem 5.10 for
the case $q=\infty$.

One can naturally also ask when the topology $L_{(p,q)}(E^{\prime},E)$ in (ii)
of Proposition 5.4 can be replaced by the strong topology
$\beta(E^{\prime},E)$. We answer this question in the next theorem. For $1\leq
p<\infty$, it generalizes a characterization of operators $T$ between Banach
spaces for which $T^{\ast}$ is $p$-convergent, see Ghenciu [23]. Following
Definition 13.8 of [16], if $p\in[1,\infty]$, a locally convex space $E$ is
called weakly sequentially $p$-complete if every weakly $p$-Cauchy sequence is
weakly $p$-convergent.

###### Theorem 5.5

Let $p\in[1,\infty]$, and let $T:E\to L$ be an operator between locally convex
spaces $E$ and $L$. Then the following assertions are equivalent:

  1. (i)

for every $B\in\mathsf{Bo}(E)$, the image $T(B)$ is a $(p,\infty)$-limited set
in $L$;

  2. (ii)

$T^{\ast}:L^{\prime}_{w^{\ast}}\to E^{\prime}_{\beta}$ is $p$-convergent.

If $L^{\prime}_{w^{\ast}}$ is sequentially complete and
$T^{\ast}:L^{\prime}_{w^{\ast}}\to\big{(}E^{\prime}_{\beta}\big{)}_{w}$ is
sequentially continuous, then (i) and (ii) are equivalent to

  1. (iii)

$T^{\ast}\circ S$ is a sequentially precompact operator for any operator
$S:\ell_{p^{\ast}}\to L^{\prime}_{w^{\ast}}$ $($or $S:c_{0}\to
L^{\prime}_{w^{\ast}}$ if $p=1$$)$.

If $1<p<\infty$, $L^{\prime}_{w^{\ast}}$ is sequentially complete and
$T^{\ast}:L^{\prime}_{w^{\ast}}\to\big{(}E^{\prime}_{\beta}\big{)}_{w}$ is
sequentially continuous, then (i)-(iii) are equivalent to the following

  1. (iv)

$T^{\ast}\circ S$ is a sequentially compact operator for any operator
$S:\ell_{p^{\ast}}\to L^{\prime}_{w^{\ast}}$.

If $p=1$, $E^{\prime}_{\beta}$ and $L^{\prime}_{w^{\ast}}$ are sequentially
complete,
$T^{\ast}:L^{\prime}_{w^{\ast}}\to\big{(}E^{\prime}_{\beta}\big{)}_{w}$ is
sequentially continuous and $L^{\prime}_{w^{\ast}}$ is weakly sequentially
$1$-complete, then (i)-(iii) are equivalent to the following

  1. (v)

$T^{\ast}\circ S$ is a sequentially compact operator for any operator
$S:c_{0}\to L^{\prime}_{w^{\ast}}$.

###### Proof 24

(i)$\Rightarrow$(ii) Let $\\{\chi_{n}\\}_{n\in\omega}$ be a weak∗ $p$-summable
sequence in $L^{\prime}$. To show that $T^{\ast}(\chi_{n})\to 0$ in
$E^{\prime}_{\beta}$, fix an arbitrary $B\in\mathsf{Bo}(E)$. Since $T(B)$ is a
$(p,\infty)$-limited set in $L$ we have

$\sup_{b\in B}\big{|}\langle T^{\ast}(\chi_{n}),b\rangle\big{|}=\sup_{b\in
B}\big{|}\langle\chi_{n},T(b)\rangle\big{|}\to 0\;\mbox{ as }\;n\to\infty,$

and hence $T^{\ast}(\chi_{n})\in B^{\circ}$ for all sufficiently large
$n\in\omega$. Since $B$ was arbitrary this means that $T^{\ast}(\chi_{n})\to
0$ in $E^{\prime}_{\beta}$, as desired.

(ii)$\Rightarrow$(i) Let $B\in\mathsf{Bo}(E)$. To show that $T(B)$ is a
$(p,\infty)$-limited set in $L$, take any weakly $p$-summable sequence
$\\{\chi_{n}\\}_{n\in\omega}$ in $L^{\prime}_{w^{\ast}}$. For every
$\varepsilon>0$, the polar $\varepsilon
B^{\circ}=\big{(}\tfrac{1}{\varepsilon}B\big{)}^{\circ}$ is a neighborhood of
zero in $E^{\prime}_{\beta}$. Since $T^{\ast}$ is $p$-convergent, we have
$T^{\ast}(\chi_{n})\to 0$ in $E^{\prime}_{\beta}$ and hence there is
$N_{\varepsilon}\in\omega$ such that $T^{\ast}(\chi_{n})\in\varepsilon
B^{\circ}$ for all $n\geq N_{\varepsilon}$. Therefore

$\sup_{b\in B}\big{|}\langle\chi_{n},T(b)\rangle\big{|}=\sup_{b\in
B}\big{|}\langle T^{\ast}(\chi_{n}),b\rangle\big{|}\leq\varepsilon\;\mbox{ for
all }\;n\geq N_{\varepsilon}.$

As $\varepsilon$ was arbitrary it follows that $\sup_{b\in
B}\big{|}\langle\chi_{n},T(b)\rangle\big{|}\to 0$. Thus $T(B)$ is a
$(p,\infty)$-limited set.

The equivalences (ii)$\Leftrightarrow$(iii) and
(ii)$\Leftrightarrow$(iii)$\Leftrightarrow$(iv)$\Leftrightarrow$(v)
immediately follow from Theorem 13.17 of [16] applied to
$E_{1}=L^{\prime}_{w^{\ast}}$, $L_{1}=E^{\prime}_{\beta}$ and
$T_{1}=T^{\ast}$.

###### Remark 5.6

The condition on $T$ to be such that $T^{\ast}:L^{\prime}_{w^{\ast}}\to
E^{\prime}_{\beta}$ is continuous is sufficiently strong. It is satisfied if
$E$ is a feral space because $E^{\prime}_{\beta}=E^{\prime}_{w^{\ast}}$ and
hence $T^{\ast}$ is automatically continuous by Theorem 8.10.5 of [33]. Recall
that an lcs $E$ is feral if every bounded subset of $E$ is finite-dimensional.

Theorem 5.5 applied to the identity map $T=\mathsf{id}_{E}:E\to E$ immediately
implies the following characterization of spaces for which every bounded
subset is a $(p,\infty)$-limited set. Recall that a locally convex space $E$
is called Grothendieck or has the Grothendieck property if the identity map
$\mathsf{id}_{E^{\prime}}:E^{\prime}_{w^{\ast}}\to\big{(}E^{\prime}_{\beta}\big{)}_{w}$
is sequentially continuous.

###### Corollary 5.7

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

  1. (i)

every bounded subset of $E$ is a $(p,\infty)$-limited set $\big{(}$i.e.,
$\mathsf{Bo}(E)=\mathsf{L}_{(p,\infty)}(E)$$\big{)}$;

  2. (ii)

the identity map $\mathsf{id}_{E^{\prime}}:E^{\prime}_{w^{\ast}}\to
E^{\prime}_{\beta}$ is $p$-convergent.

If $E^{\prime}_{w^{\ast}}$ is sequentially complete and $E$ has the
Grothendieck property, then (i)-(ii) are equivalent to

  1. (iii)

any operator $S:\ell_{p^{\ast}}\to E^{\prime}_{w^{\ast}}$ $($or $S:c_{0}\to
E^{\prime}_{w^{\ast}}$ if $p=1$$)$ is sequentially precompact.

If $1<p<\infty$, $E^{\prime}_{w^{\ast}}$ is sequentially complete and $E$ has
the Grothendieck property, then (i)-(iii) are equivalent to the following

  1. (iv)

any operator $S:\ell_{p^{\ast}}\to E^{\prime}_{w^{\ast}}$ is sequentially
compact.

If $p=1$, $E^{\prime}_{w^{\ast}}$ is a sequentially complete, weakly
sequentially $1$-complete, Grothendieck space and $E^{\prime}_{\beta}$ is
sequentially complete, then (i)-(iii) are equivalent to the following

  1. (v)

any operator $S:c_{0}\to E^{\prime}_{w^{\ast}}$ is sequentially compact.

Applying Corollary 5.7 for $p=\infty$, we obtain the following assertion.

###### Corollary 5.8

Let $E$ be a locally convex space. Then the following conditions are
equivalent:

  1. (i)

every bounded subset of $E$ is limited $\big{(}$i.e.,
$\mathsf{Bo}(E)=\mathsf{L}(E)$$\big{)}$;

  2. (ii)

the identity map $\mathsf{id}_{E^{\prime}}:E^{\prime}_{w^{\ast}}\to
E^{\prime}_{\beta}$ is completely continuous (= $\infty$-convergent); in
particular, $E$ is a Grothendieck space.

Below we give a useful construction of operators from $\ell_{1}^{0}$ into
locally convex spaces whose adjoint is $p$-convergent.

###### Proposition 5.9

Let $\\{x_{n}\\}_{n\in\omega}$ be a bounded sequence in a locally convex space
$(E,\tau)$, and let $T:\ell_{1}^{0}\to E$ be a linear map defined by

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

Then $T$ is continuous. Moreover, if $E$ is locally complete, then $T$ can be
extended to a continuous operator from $\ell_{1}$ to $E$. In any case, if
$\\{x_{n}\\}_{n\in\omega}$ is a $(p,\infty)$-limited set, then
$T^{\ast}:E^{\prime}_{w^{\ast}}\to\ell_{\infty}$ is $p$-convergent.

###### Proof 25

The continuity of $T$ and, in the case $E$ is locally complete, the existence
of the extension of $T$ are proved in Proposition 14.9 of [16]. Assume now
that $A=\\{x_{n}\\}_{n\in\omega}$ is a $(p,\infty)$-limited set. To show that
the adjoint linear map $T^{\ast}$ is $p$-convergent, let
$\\{\chi_{n}\\}_{n\in\omega}$ be a weak∗ $p$-summable sequence in
$E^{\prime}$. Since $A$ is a $(p,\infty)$-limited set, by definition, we have
$\lim_{n\to\infty}\sup_{x\in A}|\langle\chi_{n},x\rangle|=0$. Therefore

$\|T^{\ast}(\chi_{n})\|_{\ell_{\infty}}=\sup_{k\in\omega}\big{|}\langle
T^{\ast}(\chi_{n}),e_{k}\rangle\big{|}=\sup_{k\in\omega}\big{|}\langle\chi_{n},x_{k}\rangle\big{|}\leq\sup_{x\in
A}\big{|}\langle\chi_{n},x\rangle\big{|}\to 0\;\;\mbox{ as }\;n\to\infty.$

Thus $\big{\\{}T^{\ast}(\chi_{n})\big{\\}}_{n\in\omega}$ is a null sequence in
$\ell_{\infty}$, and hence $T^{\ast}$ is $p$-convergent.

We know that $(p,q)$-limited sets are bounded. In the next theorem we give an
operator characterization of those spaces $E$ in which the $(p,q)$-limited
sets have stronger topological properties than just being bounded as, for
example, being weakly sequentially (pre)compact.

###### Theorem 5.10

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 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 weakly sequentially compact (resp., sequentially compact, weakly
sequentially precompact, sequentially precompact, weakly sequentially
$p$-compact or weakly sequentially $p$-precompact);

  2. (ii)

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

  3. (iii)

each $(p,q)$-limited subset of $E$ is relatively weakly sequentially compact
(resp., relatively sequentially compact, weakly sequentially precompact,
sequentially precompact, relatively weakly sequentially $p$-compact or weakly
sequentially $p$-precompact).

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

  1. (iv)

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

###### Proof 26

(i)$\Rightarrow$(ii) and (i)$\Rightarrow$(iv) are clear.

(ii)$\Rightarrow$(iii) and (iv)$\Rightarrow$(iii): 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, 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 (ii) or (iv), the operator $T$ belongs to the corresponding
class described in (i). Therefore $S=\\{T(e_{n})\\}_{n\in\omega}$ has a weakly
convergent (resp., convergent, weakly Cauchy, Cauchy, weakly $p$-convergent,
or weakly $p$-Cauchy) subsequence, as desired.

(iii)$\Rightarrow$(i) Let $T:L\to E$ be an operator from a normed space such
that $T^{\ast}:E^{\prime}_{w^{\ast}}\to L^{\prime}_{\beta}$ is a
$(q,p)$-convergent operator. Then, by Theorem 3.5, $T(B_{L})$ is a
$(p,q)$-limited set and hence it is relatively weakly sequentially compact
(resp., relatively sequentially compact, weakly sequentially precompact,
sequentially precompact, relatively weakly sequentially $p$-compact or weakly
sequentially $p$-precompact). Thus $T$ belongs to the corresponding class
described in (i).

The definition of coarse $p$-limited sets allows to reformulate Theorem 14.16
of [16] as follows.

###### Theorem 5.11

Let $1<p<\infty$, and let $E$ be a quasibarrelled space such that
$E^{\prime}_{\beta}$ is an $\ell_{\infty}$-$V_{p}$-barrelled space. Then the
class of $p$-$(V^{\ast})$ sets in $E$ coincides with the class of coarse
$p$-limited sets.

In Theorem 5.11 the condition on $E$ being a quasibarrelled space is essential
as Example 5.13 below shows. First we prove the next simple lemma.

###### Lemma 5.12

Let $E$ be a locally convex space such that $E=E_{w}$, and let $L$ be a normed
space. Then every $T\in\mathcal{L}(E,L)$ is finite-dimensional.

###### Proof 27

Observe that $T$ can be extended to an operator ${\bar{T}}$ from a completion
${\bar{E}}$ of $E$ to a completion ${\bar{L}}$ of $L$. As $E$ carries its weak
topology, we obtain ${\bar{E}}=\mathbb{F}^{\kappa}$ for some cardinal
$\kappa$. Since ${\bar{T}}$ is continuous, there is a finite subset $\lambda$
of $\kappa$ such that
${\bar{T}}\big{(}\\{0\\}^{\lambda}\times\mathbb{F}^{\kappa{\setminus}\lambda}\big{)}$
is contained in the unit ball $B_{{\bar{L}}}$ of ${\bar{L}}$. Taking into
account that $B_{{\bar{L}}}$ contains no non-trivial linear subspaces we
obtain that $\\{0\\}^{\lambda}\times\mathbb{F}^{\kappa{\setminus}\lambda}$ is
contained in the kernel $\ker({\bar{T}})$ of ${\bar{T}}$. Therefore
${\bar{T}}[\mathbb{F}^{\kappa}]={\bar{T}}[\mathbb{F}^{\lambda}]$ is finite-
dimensional. Thus also $T$ is finite-dimensional.

###### Example 5.13

Let $1<q\leq p<\infty$. Then the space $E:=(\ell_{q})_{w}$ satisfies the
following conditions:

  1. (i)

$E$ is not quasibarrelled, but $E^{\prime}_{\beta}=\ell_{q^{\ast}}$ is a
Banach space;

  2. (ii)

$B_{\ell_{q}}$ is a coarse $p$-limited set in $E$ which is not a
$p$-$(V^{\ast})$ set.

###### Proof 28

The clause (i) is clear, and Lemma 5.12 shows that $B_{\ell_{q}}$ is a coarse
$p$-limited set in $E$. To show that $B_{\ell_{q}}$ is not a $p$-$(V^{\ast})$
set, let $\chi_{n}=e^{\ast}_{n}$ for every $n\in\omega$. Then the sequence
$\\{\chi_{n}\\}_{n\in\omega}\subseteq E^{\prime}_{\beta}=\ell_{q^{\ast}}$ is
weakly $p$-summable (indeed, if
$(x_{k})\in\ell_{q}=\big{(}\ell_{q^{\ast}}\big{)}^{\prime}$, then
$\big{(}\langle(x_{k}),\chi_{n}\rangle\big{)}_{n}=(x_{n})\in\ell_{q}\subseteq\ell_{p}$).
However, $\sup_{(x_{k})\in
B_{\ell_{q}}}|\langle\chi_{n},(x_{k})\rangle|=1\not\to 0$. Thus $B_{\ell_{q}}$
is not a $p$-$(V^{\ast})$ set.

Below for an important case which includes all strict $(LF)$-spaces, we
characterize coarse $1$-limited sets. First we recall some definitions and
results. Following [14], 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 [19], 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 [38].

###### Theorem 5.14

Every locally complete locally convex space $E$ whose every separable bounded
set is metrizable has the Rosenthal property.

Note that for every $\ell_{1}$-sequence $A=\\{a_{n}\\}_{n\in\omega}$ in $E$ in
Theorem 5.14, a topological isomorphism $R$ from $\overline{\mathsf{span}}(A)$
to $\ell_{1}$ is onto. Observe also that strict $(LF)$-spaces satisfy Theorem
5.14.

###### Theorem 5.15

Let $E$ be a locally complete space whose separable bounded sets are
metrizable. Then for a bounded subset $A$ of $E$, the following assertions are
equivalent:

  1. (i)

$A$ is a coarse $1$-limited set;

  2. (ii)

$A$ does not contain an $\ell_{1}$-sequence $\\{x_{n}\\}_{n\in\omega}$ such
that the closed span $\overline{\mathsf{span}}\\{x_{n}\\}_{n\in\omega}$ is
complemented in $E$.

If in addition $E$ is barrelled, then (i) and (ii) are equivalent to

  1. (iii)

$A$ is a $(1,\infty)$-limited set.

###### Proof 29

(i)$\Rightarrow$(ii) Assume that $E$ has only the Rosenthal property, and
suppose for a contradiction that there is an $\ell_{1}$-sequence
$\\{x_{n}\\}_{n\in\omega}$ in $A$ such that
$L:=\overline{\mathsf{span}}\\{x_{n}:n\in\omega\\}$ is complemented in $E$.
Let $S$ be a projection from $E$ onto $L$, and let $R$ be a linear
homeomorphism of $L$ onto a subspace of $\ell_{1}$ such that $R(x_{n})=e_{n}$
for every $n\in\omega$. Then $T:=R\circ S:E\to\ell_{1}$ is an operator such
that $T(A)$ contains $\\{e_{n}:n\in\omega\\}$. Therefore $T(A)$ is not
relatively compact in $\ell_{1}$. Thus $A$ is not coarse $1$-limited, a
contradiction.

(ii)$\Rightarrow$(i) Suppose for a contradiction that $A$ is not a coarse
$1$-limited set. Then there is $T\in\mathcal{L}(E,\ell_{1})$ such that $T(A)$
is not relatively compact in $\ell_{1}$. By Theorem 1.4 of [34], there is a
sequence $\\{x_{n}\\}_{n\in\omega}$ in $A$ such that the sequence
$S_{0}=\\{T(x_{n})\\}_{n\in\omega}$ is equivalent to the standard unit basis
$\\{e_{n}\\}_{n\in\omega}$ of $\ell_{1}$ and such that the subspace
$H_{0}:=\overline{\mathsf{span}}(S_{0})$ is a complemented subspace of
$\ell_{1}$. Let $R_{0}:H_{0}\to\ell_{1}$ be a linear topological isomorphism
such that $R_{0}\big{(}T(x_{n})\big{)}=e_{n}$ for every $n\in\omega$. Since a
continuous image of a weakly Cauchy sequence is weakly Cauchy, the sequence
$\\{x_{n}\\}_{n\in\omega}$ has no weakly Cauchy subsequences, and hence, by
the Rosenthal property of $E$ (see Theorem 5.14), there is a subsequence
$\\{x_{n_{k}}\\}_{k\in\omega}$ of $\\{x_{n}\\}_{n\in\omega}$ which is
equivalent to $\\{e_{k}\\}_{k\in\omega}$. Let
$R:\overline{\mathsf{span}}\\{x_{n_{k}}\\}_{k\in\omega}\to\ell_{1}$ be a
linear topological isomorphism such that $R\big{(}x_{n_{k}}\big{)}=e_{k}$ for
every $k\in\omega$. Observe that the subspace
$H_{1}:=\overline{\mathsf{span}}\\{T(x_{n_{k}})\\}_{k\in\omega}$ of $H_{0}$
satisfies the following two conditions:

  1. (1)

$H_{1}$ is complemented in $H_{0}$ (since $R_{0}$ is a topological
isomorphism, $R_{0}\big{(}T(x_{n_{k}})\big{)}=e_{n_{k}}$ and
$\overline{\mathsf{span}}\\{e_{n_{k}}\\}_{k\in\omega}$ is complemented in
$\ell_{1}$), and hence $H_{1}$ is complemented also in $\ell_{1}$, and

  2. (2)

$H_{1}$ is topologically isomorphic to $\ell_{1}$ (since $R_{0}$ is a
topological isomorphism and
$\overline{\mathsf{span}}\\{e_{n_{k}}\\}_{k\in\omega}$ is topologically
isomorphic to $\ell_{1}$).

Let $Q:\ell_{1}\to H_{1}$ be a projection (so
$Q\big{(}T(x_{n_{k}})\big{)}=T(x_{n_{k}})$ for every $k\in\omega$), and let
$R_{1}:H_{1}\to\ell_{1}$ be a linear topological isomorphism such that
$R_{1}\big{(}T(x_{n_{k}})\big{)}=e_{k}$ for every $k\in\omega$. Since

$R^{-1}\circ R_{1}\circ Q\circ T(x_{n_{k}})=R^{-1}\circ
R_{1}\big{(}T(x_{n_{k}})\big{)}=R^{-1}(e_{k})=x_{n_{k}}\;\;\mbox{ for every
}k\in\omega,$

it follows that $R^{-1}\circ R_{1}\circ Q\circ T$ is a continuous projection
from $E$ onto $\overline{\mathsf{span}}\\{x_{n_{k}}\\}_{k\in\omega}$ and
$\\{x_{n_{k}}\\}_{k\in\omega}$ is equivalent to $\\{e_{k}\\}_{k\in\omega}$.
But this contradicts (ii).

(i)$\Leftrightarrow$(iii) immediately follows from Proposition 3.13.

###### Corollary 5.16

Let $E$ be a strict $(LF)$-space which does not contain an isomorphic copy of
$\ell_{1}$ which is complemented in $E$. Then every bounded subset $A$ of $E$
is a coarse $1$-limited and a $(1,\infty)$-limited set.

By the classical Pitt theorem [10, 4.49], all operators
$\mathcal{L}(\ell_{p},\ell_{1})$ ($1<p<\infty$) and
$\mathcal{L}(c_{0},\ell_{1})$ are compact. Below we generalize this result.

###### Corollary 5.17

If $E$ is a Banach space containing no an isomorphic copy of $\ell_{1}$ which
is complemented in $E$, then the class of all bounded subsets of $E$ coincides
with the class of all coarse $1$-limited sets. Consequently, every
$T\in\mathcal{L}(E,\ell_{1})$ is compact.

The condition of being a barrelled space in (iii) of Theorem 5.15 is essential
as the following example shows.

###### Example 5.18

Let $E=(c_{0})_{p}$ be the Banach space $c_{0}$ endowed with the pointwise
topology induced from $\mathbb{F}^{\omega}$, and let

$B=\big{\\{}(x_{n})_{n\in\omega}\in E:|x_{n}|\leq(n+1)^{2}\mbox{ for every
}n\in\omega\big{\\}}.$

Then $B$ is a coarse $1$-limited set in $E$ which is not $(1,\infty)$-limited.

###### Proof 30

It is clear that $B$ is a bounded subset of $E$. Therefore, by (ii) of Example
5.4 of [16], $B$ is a coarse $p$-limited set for every $p\in[1,\infty]$. To
show that $B$ is not $(1,\infty)$-limited, consider the sequence
$\\{\chi_{n}\\}_{n\in\omega}=\big{\\{}\tfrac{e^{\ast}_{n}}{(n+1)^{2}}\big{\\}}_{n\in\omega}$
in $E^{\prime}$. In the proof of (i) of Example 5.4 of [16], we showed that
$\\{\chi_{n}\\}_{n\in\omega}$ is a weak∗ $1$-summable sequence in
$E^{\prime}$. Since

$\sup_{(x_{k})\in B}|\langle\chi_{n},(x_{k})\rangle|=1\;\;\mbox{ for every
}n\in\omega,$

it follows that $B$ is not a $(1,\infty)$-limited set in $E$.

###### Corollary 5.19

Let $E$ be a Banach space. Then:

  1. (i)

if $p=1$, then the class of coarse $1$-limited sets in $E$ coincides with the
class of $(1,\infty)$-limited sets;

  2. (ii)

if $1<p<\infty$, then the class of coarse $p$-limited sets in $E$ coincides
with the class of $p$-$(V^{\ast})$ sets.

###### Proof 31

(i) follows from Theorem 5.15, and (ii) follows from Theorem 5.11.

To generalize (iii) of Theorem 1.2 and its extension given in Proposition 3 of
[20], first we prove the next lemma. 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 ${\bar{T}}:E\to X$.

###### Lemma 5.20

Every injective Banach space is also an injective locally convex space.

###### Proof 32

Let $X$ be an injective Banach space, $H$ be a subspace of a locally convex
space $E$, and let $T:H\to X$ be an operator. It is well known (see for
example Exercise 5.27 of [10]) that there are a set $\Gamma$ and a closed
subspace $Y$ of $\ell_{\infty}(\Gamma)$ such that $X\oplus
Y=\ell_{\infty}(\Gamma)$. Denote by $\pi_{X}:\ell_{\infty}(\Gamma)\to X$ the
canonical projection, and let $I_{X}:X\to\ell_{\infty}(\Gamma)$,
$I_{X}(x):=(x,0)$, be the canonical embedding. Since, by Proposition 7.4.5 of
[28], $\ell_{\infty}(\Gamma)$ is an injective locally convex space, the
operator $I_{X}\circ T:H\to\ell_{\infty}(\Gamma)$ can be extended to an
operator $\overline{I_{X}\circ T}:E\to\ell_{\infty}(\Gamma)$. Set
${\bar{T}}:=\pi_{X}\circ(\overline{I_{X}\circ T})$. Then ${\bar{T}}$ is an
operator from $E$ to $X$ such that

${\bar{T}}(h)=\pi_{X}\circ\big{(}I_{X}\circ
T\big{)}(h)=\pi_{X}\big{(}(T(h),0)\big{)}=T(h)\;\;\mbox{ for each $h\in H$}.$

Thus ${\bar{T}}$ extends $T$ and hence $X$ is an injective locally convex
space.

###### Theorem 5.21

Let $2\leq p\leq\infty$, and let $E$ be a locally convex space with the
Rosenthal property. Then every coarse $p$-limited subset of $E$ is weakly
sequentially precompact.

###### Proof 33

We consider only the case $2\leq p<\infty$ since the case $p=\infty$ can be
considered analogously replacing $\ell_{p}$ by $c_{0}$. Suppose for a
contradiction that there is a coarse $p$-limited subset $A$ of $E$ which is
not weakly sequentially precompact. So there is a sequence
$S=\\{x_{n}\\}_{n\in\omega}$ in $A$ that 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 $n\in\omega$. 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 [9],
the operator $I_{1}$ is $1$-summing. By the Ideal Property 2.4 of [9], $J$ is
also $1$-summing, and hence, by the Inclusion Property 2.8 of [9], the
operator $J$ is $2$-summing. By the discussion after Corollary 2.16 of [9],
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 [9], the Banach space $L_{\infty}(\mu)$ is injective.
Therefore, by Lemma 5.20, $L_{\infty}(\mu)$ 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 $n\in\omega$}.$

Since $A$ and hence also $S$ are coarse $p$-limited sets, (iii) of Lemma 4.1
implies that the canonical basis $\\{e_{n}\\}_{n\in\omega}$ of $\ell_{p}$ is
also a coarse $p$-limited set. Therefore $\mathsf{id}_{\ell_{p}}\circ
T(S)=\\{e_{n}\\}_{n\in\omega}$ is a relatively compact subset of $\ell_{p}$, a
contradiction.

It is noticed in [20, p. 944] that in general Theorem 5.21 is not true for
$p=1$ even for Banach spaces (in fact, the closed unit ball of $C([0,1])$ is a
coarse $1$-limited set which is not weakly sequentially precompact).

###### Corollary 5.22

  1. (i)

If $2\leq p\leq\infty$ and $E$ is a locally convex space with the Rosenthal
property, then every $p$-limited subsets of $E$ is weakly sequentially
precompact.

  2. (ii)

If $E$ is a Banach space and $1\leq p<\infty$, then every $p$-limited subsets
of $E$ is relatively weakly (sequentially) compact.

###### Proof 34

(i) Since $p$-limited sets are coarse $p$-limited by Proposition 4.2(iv), the
assertion follows from Theorem 5.21.

(ii) immediately follows from Theorem 17.19 of [16] (which states that every
$(p,p)$-$(V^{\ast})$ subset of $E$ is relatively weakly compact) and (viii) of
Lemma 3.1.

Concerning the case $p=\infty$ in (ii) of Corollary 5.22, we note that if a
Banach space $E$ does not contain an isomorphic copy of $\ell_{1}$, then every
limited subset of $E$ is relatively weakly (sequentially) compact, for the
proof see [5] (an alternative proof is given in Theorem 1.9 of [21]).

###### Remark 5.23

By Corollary 5.22, each limited subset of a Banach space is weakly
sequentially precompact. It turns out that for non-Banach spaces, this very
useful assertion is not true in general. Indeed, by Example 7.12 of [16], the
product $\mathbb{R}^{\mathfrak{c}}$ contains a uniformly bounded sequence
$S=\\{f_{n}\\}_{n\in\omega}$ which is a $(p,q)$-$(V^{\ast})$ set for all
$1\leq p\leq q\leq\infty$ but is not (weakly) sequentially precompact. Since
$\mathbb{R}^{\mathfrak{c}}$ is reflexive, by (viii) of Lemma 3.1, $S$ is also
a $(p,q)$-limited set.

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