# Reflexive extended locally convex spaces

Akshay Kumar and Varun Jindal  Akshay Kumar: Department of Mathematics,
Malaviya National Institute of Technology Jaipur, Jaipur-302017, Rajasthan,
India<EMAIL_ADDRESS>Varun Jindal: Department of Mathematics, Malaviya
National Institute of Technology Jaipur, Jaipur-302017, Rajasthan, India
<EMAIL_ADDRESS>
###### Abstract.

For an extended locally convex space (elcs) $(X,\tau)$, the authors in [10]
studied the topology $\tau_{ucb}$ of uniform convergence on bounded subsets of
$(X,\tau)$ on the dual $X^{*}$ of $(X,\tau)$. In the present paper, we use the
topology $\tau_{ucb}$ to explore the reflexive property of extended locally
convex spaces. It is shown that an elcs is (semi) reflexive if and only if any
of its open subspaces is (semi) reflexive. For an extended normed space, we
show that reflexivity is a three-space property.

###### Key words and phrases:

Extended locally convex space, extended normed space, weak topolgy, weak∗
topology, strong topology, reflexive spaces

###### 2010 Mathematics Subject Classification:

Primary 46A20, 46A25; Secondary 46A03, 46A17, 54C40

##  1\. Introduction

In classical functional analysis, our attention is directed towards the study
of locally convex spaces. An important characterization of a locally convex
space is that a collection of seminorms induces its topology. However, in
various problems, we encounter functions that possess all the properties of a
seminorm (or even a norm) but can also assume infinite value.

An extended norm on a vector space $X$ is a function satisfying all the
properties of a norm and, in addition, can also attain infinite value. A
vector space $X$ together with an extended norm is called an extended normed
linear space (enls). These spaces were first formally studied by Beer in [1]
and further developed by Beer and Vanderwerff in [4, 5].

Salas and Tapia-García introduced the concept of an extended locally convex
space in [14], which is a generalization of an extended normed linear space.
These new extended spaces are different from the classical locally convex
spaces as the scalar multiplication in these spaces may not be jointly
continuous. As a result, the conventional theory of locally convex spaces may
not be directly applicable to these new spaces. To address this problem, the
finest locally convex topology (flc topology) for an (elcs) $(X,\tau)$ which
is coarser than $\tau$ was studied in [11]. It was shown in [11] that if
$\tau_{F}$ is the flc topology for an elcs $(X,\tau)$, then both $(X,\tau)$
and $(X,\tau_{F})$ have the same dual $X^{*}$ (the collection of all
continuous linear functionals).

In [10], the authors employed the flc topology to examine the dual of an elcs.
Specifically, they studied the weak topology on an elcs $(X,\tau)$ and the
weak∗ topology on the dual $X^{*}$ of $(X,\tau)$. Besides this, on $X^{*}$,
they also studied the topology $\tau_{ucb}$ of uniform convergence on bounded
subsets of $(X,\tau)$.

In the present paper, we use the topology $\tau_{ucb}$ to study reflexive
extended locally convex spaces.

The paper is organized as follows: the second section presents all essential
preliminary results and definitions. In Section 3, we define and study
reflexive extended locally convex spaces. More specifically, we relate the
reflexivity of an elcs $(X,\tau)$ with the reflexivity of its finest space
$(X,\tau_{F})$, where $\tau_{F}$ is the corresponding flc topology. We also
show that an elcs $(X,\tau)$ is reflexive if and only if any of its open
subspaces is reflexive. Further, in the case of an enls, we prove that the
reflexivity property is a three-space property. As an application of our
results, in the final section, we look at the reflexivity of some well known
function spaces.

##  2\. Preliminaries

The underlying field of a vector space is denoted by $\mathbb{K}$ which is
either $\mathbb{R}$ or $\mathbb{C}$. We adopt the following conventions for
$\infty$: $\infty.0=0.\infty=0$; $\infty+\alpha=\alpha+\infty=\infty$ for
every $\alpha\in\mathbb{R}$; $\infty.\alpha=\alpha.\infty=\infty$ for
$\alpha>0$; $\inf\\{\emptyset\\}=\infty$.

An extended seminorm $\rho:X\to[0,\infty]$ on a vector space $X$ is a function
which satisfies the following properties.

  * (1)

$\rho(\alpha x)=|\alpha|\rho(x)$ for each $x\in X$ and scalar $\alpha$;

  * (2)

$\rho(x+y)\leq\rho(x)+\rho(y)$ for all $x,y\in X$.

An extended norm $\parallel\cdot\parallel:X\to[0,\infty]$ is an extended
seminorm with the property: if $\parallel x\parallel=0$, then $x=0$. A vector
space $X$ endowed with an extended norm $\parallel\cdot\parallel$ is called an
extended normed linear space (or extended normed space) (enls, for short), and
it is denoted by $(X,\parallel\cdot\parallel)$. The finite subspace of an enls
$(X,\parallel\cdot\parallel)$ is defined as

$X_{fin}=\\{x\in X:\parallel x\parallel<\infty\\}.$

Note that the extended norm $\parallel\cdot\parallel$ on $X_{fin}$ is actually
a norm. Therefore $(X_{fin},\parallel\cdot\parallel)$ is a conventional normed
linear space.

We say an enls $(X,\parallel\cdot\parallel)$ is an extended Banach space if it
is complete with respect to the metric $d(x,y)=\min\\{\parallel
x-y\parallel,1\\}$ for all $x,y\in X$. One can prove that an enls
$(X,\parallel\cdot\parallel)$ is an extended Banach space if and only if the
finite space $(X_{fin},\parallel\cdot\parallel)$ is a Banach space. For
details about extended normed linear spaces, we refer to [1, 4, 5].

Suppose $(X,\parallel\cdot\parallel_{1})$ and
$(Y,\parallel\cdot\parallel_{2})$ are extended normed linear spaces. Then for
a continuous linear map $T:X\to Y$, we define

$\parallel T\parallel_{op}=\sup\\{\parallel T(x)\parallel_{2}:\parallel
x\parallel_{1}\leq 1\\}.$

In particular, if $f\in X^{*}$, then $\parallel
f\parallel_{op}=\sup\\{|f(x)|:\parallel x\parallel_{1}\leq 1\\}.$ The
following points about an enls $(X,\parallel\cdot\parallel)$ are given in [1].

  1. (1)

$X_{fin}$ is open in $(X,\parallel\cdot\parallel)$.

  2. (2)

$\parallel f\parallel_{op}=\parallel f|_{X_{fin}}\parallel_{op}$ for every
$f\in X^{*}$, where $f|_{X_{fin}}$ is the restriction of $f$ on the normed
linear space $(X_{fin},\parallel\cdot\parallel)$.

  3. (3)

For any linear functional $f$ on $X$, we have $f\in X^{*}$ if and only if
$f|_{X_{fin}}$ is continuous on $X_{fin}$.

  4. (4)

If $f\in X^{*}$ and $\parallel f\parallel_{op}\neq 0$, then
$|f(x)|\leq\parallel f\parallel_{op}\parallel x\parallel$ for every $x\in X$.

It follows from the point (2) given above that $\parallel\cdot\parallel_{op}$
may not be a norm on the dual $X^{*}$ of an enls
$(X,\parallel\cdot\parallel)$. However, following [1], we call it the operator
norm in the sequel.

A vector space $X$ endowed with a Hausdorff topology $\tau$ is said to be an
extended locally convex space (elcs, for short) if $\tau$ is induced by a
collection $\mathcal{P}=\\{\rho_{i}:i\in\mathcal{I}\\}$ of extended seminorms
on $X$, that is, $\tau$ is the smallest topology on $X$ under which each
$\rho_{i}$ is continuous. We define

$X_{fin}^{\rho}=\\{x\in X:\rho(x)<\infty\\}$

for any extended seminorm $\rho$ on $X$ and the finite subspace $X_{fin}$ of
an elcs $(X,\tau)$ by

$X_{fin}=\bigcap\left\\{X_{fin}^{\rho}:\rho\text{ is continuous on
}(X,\tau)\right\\}.$

Suppose $(X,\tau)$ is an elcs and $\tau$ is induced by a family $\mathcal{P}$
of extended seminorms on $X$. Then the following facts are either easy to
verify or given in [14].

  * (1)

There exists a neighborhood base $\mathcal{B}$ at $0$ in $(X,\tau)$ such that
each element of $\mathcal{B}$ is absolutely convex (balanced and convex);

  * (2)

$X_{fin}$ with the subspace topology is a locally convex space;

  * (3)

$X_{fin}=\bigcap_{\rho\in\mathcal{P}}X_{fin}^{\rho};$

  * (4)

if $\rho$ is any continuous extended seminorm on $(X,\tau)$, then
$X_{fin}^{\rho}$ is a clopen subspace of $(X,\tau)$;

  * (5)

$X_{fin}$ is an open subspace of $(X,\tau)$ if and only if there exists a
continuous extended seminorm $\rho$ on $(X,\tau)$ such that
$X_{fin}=X_{fin}^{\rho}$. In this case, we say $(X,\tau)$ is a fundamental
elcs.

It is shown in Proposition 4.7 of [14] that if $\mathcal{B}$ is a neighborhood
base at $0$ in an elcs $(X,\tau)$ consisting of absolutely convex sets, then
$\tau$ is induced by the collection $\\{\mu_{U}:U\in\mathcal{B}\\}$ of
Minkowski functionals.

######  Definition 2.1.

([14]) Suppose $U$ is any absolutely convex set in an elcs $(X,\tau)$. Then
the Minkowski functional $\mu_{U}:X\rightarrow[0,\infty]$ for $U$ is defined
as

$\mu_{U}(x)=\inf\\{\lambda>0:x\in\lambda U\\}.$

The following facts are immediate from the above definition.

  1. (1)

The Minkowski functional $\mu_{U}$ for the set $U$ is an extended seminorm on
$X$. In addition, if $U$ is absorbing, then $\mu_{U}$ is a seminorm on $X$.

  2. (2)

$\\{x\in X:\mu_{U}(x)<1\\}\subseteq U\subseteq\\{x\in X:\mu_{U}(x)\leq 1\\}$.

  3. (3)

The Minkowski functional $\mu_{U}$ is continuous on $X$ if and only if $U$ is
a neighborhood of $0$ in $(X,\tau)$.

If $A$ is any nonempty set in a topological space $(X,\tau)$, then we denote
the closure and interior of $A$ in $(X,\tau)$ by $\text{Cl}_{\tau}(A)$ and
$\text{int}_{\tau}(A)$, respectively. We also adopt the following terminology
for an elcs $(X,\tau)$.

  1. (1)

If $U$ is any absolutely convex subset of $X$, then $X_{fin}^{U}=\\{x\in
X:\mu_{U}(x)<\infty\\}$.

  2. (2)

If $A\subseteq X$, then ab($B$) is the smallest absolutely convex set in $X$
that contains $A$.

  3. (3)

If $A\subseteq X$, then $A^{\circ}=\\{f\in X^{*}:|f(x)|\leq 1\text{ for every
}x\in A\\}$ is called the polar of $A$ in $X^{*}$.

  4. (4)

If $A\subseteq X^{*}$, then $A_{\circ}=\\{x\in X:|f(x)|\leq 1\text{ for every
}f\in A\\}$ is called the polar of $A$ in $X$.

For other terms and definitions, we refer to [13, 15, 16].

##  3\. Reflexivity

The aim of this section is to explore reflexive extended locally convex
spaces. To define reflexivity property of an elcs $(X,\tau)$, we first need to
define the topology $\tau_{ucb}$, on $X^{*}$, of uniform convergence on
bounded subsets of $(X,\tau)$. For an elcs, the topology $\tau_{ucb}$ has been
studied extensively in [10]. We first give the definition of a bounded set in
an elcs.

######  Definition 3.1.

([11]) Suppose $(X,\tau)$ is an elcs. Then $A\subseteq X$ is said to be
bounded in $(X,\tau)$ if for every neighborhood $U$ of $0$, there exist $r>0$
and a finite set $F\subseteq X$ such that $A\subseteq F+rU$.

The following points about bounded sets in an elcs $(X,\tau)$ are either easy
to prove or given in [11].

  1. (1)

Every finite subset of $X$ is bounded.

  2. (2)

Every subset of a bounded set is bounded.

  3. (3)

When $(X,\tau)$ is a conventional locally convex space, $A\subseteq X$ is
bounded in the sense of Definition 3.1 if and only if it is absorbed by each
neighborhood of $0$ in $(X,\tau)$.

  4. (4)

Suppose $(Y,\sigma)$ is any elcs and $T:X\rightarrow Y$ is a continuous linear
operator. Then for every bounded set $A$ in $(X,\tau)$, $T(A)$ is bounded in
$(Y,\sigma)$. In particular, for every $f\in X^{*}$, $f(A)$ is bounded in
$\mathbb{K}$.

  5. (5)

No subspace $\left(\text{other than the zero subspace}\right)$ is bounded.

  6. (6)

If $x_{n}\to 0$ in $(X,\tau)$, then $\\{x_{n}:n\in\mathbb{N}\\}$ is bounded in
$(X,\tau)$.

######  Definition 3.2.

([10]) Let $(X,\tau)$ be an elcs. Then the topology $\tau_{ucb}$, on $X^{*}$,
of uniform convergence on bounded subsets of $(X,\tau)$ is induced by the
collection $\mathcal{P}=\\{\rho_{B}:B~{}\text{is bounded subset
of}~{}(X,\tau)\\}$ of seminorms on $X^{*}$, where $\rho_{B}(\phi)=\sup_{x\in
B}|\phi(x)|~{}\text{for}~{}\phi\in X^{*}$.

The following points for an elcs $(X,\tau)$ are either easy to verify or given
in [1, 10].

  1. (1)

$(X^{*},\tau_{ucb})$ is a locally convex space and
$\mathcal{B}=\\{B^{\circ}:B~{}\text{is bounded in}~{}(X,\tau)\\}$ is a
neighborhood base at $0$ in $(X^{*},\tau_{ucb})$, where the polar $B^{\circ}$
of $B\subseteq X$ is given by $B^{\circ}=\\{\phi\in X^{*}:|\phi(x)|\leq
1\text{ for all }x\in B\\}$.

  2. (2)

Recall from [10] that the weak∗ topology on $X^{*}$ is induced by the
collection $\\{\rho_{x}:x\in X\\}$ of seminorms, where $\rho_{x}(f)=|f(x)|$
for every $f\in X^{*}$. Clearly, $\tau_{w^{*}}$ is coarser than $\tau_{ucb}$.

  3. (3)

If $(X,\parallel\cdot\parallel)$ is an enls such that $X=X_{fin}\oplus M$,
then $(X^{*},\tau_{ucb})$ is isomorphic (linear homeomorphic) to
$(X_{fin}^{*},\parallel\cdot\parallel_{op})\times(M^{*},\tau_{w^{*}})$, where
$\tau_{w^{*}}$ is the weak∗ topology on the dual $M^{*}$ of the enls
$(M,\parallel\cdot\parallel)$ (see, Theorem 4.11 in [1]).

Recall that when $(X,\tau)$ is a classical locally convex space, the topology
$\tau_{ucb}$ is more popularly known as the strong topology. In particular,
for an elcs $(X,\tau)$ with the flc topology $\tau_{F}$ by the strong topology
$\tau_{s}$ on $X^{*}$, we mean the topology of uniform convergence on bounded
subsets of the locally convex space $(X,\tau_{F})$.

######  Remark 3.3.

It is easy to prove that if $\tau_{F}$ is the flc topology of an elcs
$(X,\tau)$, then every bounded set in $(X,\tau)$ is bounded in $(X,\tau_{F})$.
Converse may not be true (see, Proposition 5.3 in [10]). Therefore
$\tau_{ucb}$ is coarser than $\tau_{s}$.

For an elcs $(X,\tau)$ and $x\in X$, if a net $(f_{\lambda})$ converges to $f$
in $(X^{*},\tau_{ucb})$, then $f_{\lambda}(x)\to f(x)$. Consequently, the map
$J_{x}:(X^{*},\tau_{ucb})\to\mathbb{K}$ defined by $J_{x}(f)=f(x)$ for $f\in
X^{*}$ is a continuous linear functional. Hence the canonical map
$J:X\to(X^{*},\tau_{ucb})^{*}$ defined by $J(x)=J_{x}$ for all $x\in X$ is
well defined.

######  Remark 3.4.

For an enls $(X,\parallel\cdot\parallel)$, the map
$J:(X,\parallel\cdot\parallel)\rightarrow(X^{*},\parallel\cdot\parallel_{op})^{*}$
may not be well defined. Since for every $z\notin X_{fin}$ and
$n\in\mathbb{N}$, we can find a continuous linear functional $f_{n}$ on $X$
such that $f_{n}(z)=n$ and $f(x)=0$ for every $x\in X_{fin}$. Clearly,
$\parallel f_{n}\parallel_{op}=0$ for every $n\in\mathbb{N}$. Consequently,
$f_{n}\to 0$ in $(X^{*},\parallel\cdot\parallel_{op})$ but
$|J_{z}(f_{n})|=n\nrightarrow 0$.

######  Definition 3.5.

Let $(X,\tau)$ be an elcs. Then we say $X$ is semi-reflexive if the canonical
map $J$ is surjective, and we say $X$ is reflexive if $J$ is both surjective
and continuous when $(X^{*},\tau_{ucb})^{*}$ is equipped with the topology of
uniform convergence on bounded subsets of $(X^{*},\tau_{ucb})$.

######  Example 3.6.

Let $X$ be a vector space with the discrete extended norm defined by

$\parallel x\parallel_{0,\infty}=\begin{cases}\text{0;}&\quad\text{if
$x=0$}\\\ \text{$\infty$;}&\quad\text{if $x\neq 0.$ }\\\ \end{cases}$

Then only finite subsets are bounded in
$(X,\parallel\cdot\parallel_{0,\infty})$. Since the given space is a discrete
space, the canonical map $J$ is continuous. Now, if
$\psi\in(X^{*},\tau_{ucb})^{*}$, then there exists a finite set
$A=\\{x_{1},x_{2},...,x_{n}\\}\subseteq X$ such that
$A^{\circ}\subseteq\psi^{-1}(-1,1)$. It is easy to show that $\bigcap_{x\in
A}J_{x}^{-1}(0)\subseteq\psi^{-1}(0)$ (if $\epsilon>0$ and $f(x_{j})=0$ for
$1\leq j\leq n$, then $\frac{1}{\epsilon}f(x_{j})=0$ for $1\leq j\leq n$. So
$|\psi(f)|<\epsilon$). By Lemma 3.9, p. 67 in [8], $\psi$ is a linear
combination of $J_{x_{1}},J_{x_{2}},..,J_{x_{n}}$. Therefore $J$ is
surjective. Hence $(X,\parallel\cdot\parallel_{0,\infty})$ is reflexive.

######  Proposition 3.7.

Let $(X,\parallel\cdot\parallel)$ be an enls. Then
$J:(X,\parallel\cdot\parallel)\rightarrow(X^{*},\tau_{ucb})^{*}$ is always
continuous.

###### Proof.

Let $x_{n}\rightarrow 0$ in $(X,\parallel\cdot\parallel)$ and let $B$ be a
bounded subset of $(X^{*},\tau_{ucb})$. Then there exist $\alpha>0$ and
$n_{0}\in\mathbb{N}$ such that $\parallel f\parallel_{op}\leq\alpha$ and
$x_{n}\in X_{fin}$ for each $f\in B$, $n\geq n_{0}$. Consequently,

$|J_{x_{n}}(f)|=|f(x_{n})|\leq\alpha\parallel x_{n}\parallel~{}\text{for every
}f\in B~{}\text{and}~{}n\geq n_{0}.$

Therefore $J_{x_{n}}\rightarrow 0$ in $(X^{*},\tau_{ucb})^{*}$. Hence $J$ is
continuous.∎

######  Corollary 3.8.

Let $(X,\parallel\cdot\parallel)$ be an enls. Then $X$ is reflexive if and
only if it is semi-reflexive.

We next study the reflexivity of an elcs $(X,\tau)$ in relation to the
properties of the corresponding finest space $(X,\tau_{F})$.

Recall that a locally convex space $(X,\tau)$ is said to be barreled if each
barrel (closed, absolutely convex and absorbing set) is a neighborhood of $0$.

######  Proposition 3.9.

Let $(X,\tau)$ be a reflexive elcs with the flc topology $\tau_{F}$. Then
$(X,\tau_{F})$ is barreled.

###### Proof.

Let $B$ be a barrel in $(X,\tau_{F})$. By Theorem 8.8.3, p. 251 in [12],
$B^{\circ}$ is pointwise bounded. Since $(X,\tau)$ is reflexive, we have
$(X^{*},\tau_{ucb})^{*}=\\{J_{x}:x\in X\\}$. So $B^{\circ}$ is bounded in
$(X^{*},\tau_{ucb})$. Consequently, $(B^{\circ})^{\circ}$ is a neighborhood of
$0$ in $(X^{*},\tau_{ucb})^{*}$. As $J:(X,\tau)\to(X^{*},\tau_{ucb})^{*}$ is
continuous, $J^{-1}\left((B^{\circ})^{\circ}\right)$ is a neighborhood of $0$
in $(X,\tau)$. Note that
$J^{-1}\left((B^{\circ})^{\circ}\right)=(B^{\circ})_{\circ}$. Consequently, by
applying Bipolar theorem in $(X,\tau_{F})$, we have $(B^{\circ})_{\circ}=B$.
Therefore $B$ is a neighborhood of $0$ in $(X,\tau)$. Since $B$ is an
absorbing and absolutely convex neighborhood of $0$ in $(X,\tau)$, the
Minkowski functional $\mu_{B}$ is a continuous seminorm on $(X,\tau)$. By
Theorem 3.5 in [11], $\mu_{B}$ is a continuous seminorm on $(X,\tau_{F})$.
Therefore $B$ is a neighborhood of $0$ in $(X,\tau_{F})$. Hence $(X,\tau_{F})$
is barreled.∎

######  Theorem 3.10.

Let $(X,\tau)$ be an elcs with the flc topology $\tau_{F}$. If $(X,\tau_{F})$
is reflexive, then $(X,\tau)$ is reflexive. Converse holds if $(X,\tau_{F})$
is semi-reflexive.

###### Proof.

Let $(X,\tau_{F})$ be reflexive. Since $\tau_{ucb}\subseteq\tau_{s}$ and
$(X^{*},\tau_{s})^{*}=\\{J_{x}:x\in X\\}$, we have
$(X^{*},\tau_{ucb})^{*}=\\{J_{x}:x\in X\\}$. Therefore $(X,\tau)$ is semi-
reflexive. Now, let $B$ be a bounded set in $(X^{*},\tau_{ucb})$. Then $B$ is
bounded in $(X^{*},\tau_{w^{*}})$. By Theorem 8.8.3, p. 241 in [12],
$B_{\circ}$ is an absorbing subset of $X$. Note that
$J^{-1}(B^{\circ})=\\{x\in X:|J_{x}(f)|\leq 1~{}\text{for}~{}f\in
B\\}=B_{\circ}$ is a barrel in $(X,\tau_{F})$ (see, Theorem 8.3.6, p. 234 in
[12]). Since $(X,\tau_{F})$ is reflexive, by Theorem 15.2.6, p. 490 in [12],
$(X,\tau_{F})$ is barreled. Consequently, $B_{\circ}$ is a neighborhood of $0$
in $(X,\tau_{F})$. So $J^{-1}(B^{\circ})=B_{\circ}$ is a neighborhood of $0$
in $(X,\tau)$. Therefore the canonical map $J$ is continuous on $(X,\tau)$.
Hence $(X,\tau)$ is reflexive.

Conversely, suppose $(X,\tau)$ is reflexive and $(X,\tau_{F})$ is semi-
reflexive. Then by Proposition 3.9, $(X,\tau_{F})$ is barreled. By Theorem
15.2.6, p. 490 in [12], $(X,\tau_{F})$ is reflexive. ∎

######  Corollary 3.11.

Let $(X,\tau)$ be an elcs with the flc topology $\tau_{F}$. Suppose anyone of
the following conditions holds

  * (1)

$(X^{*},\tau_{ucb})$ is barreled;

  * (2)

$(X,\tau)$ is a fundamental elcs.

Then $(X,\tau)$ is reflexive if and only if $(X,\tau_{F})$ is reflexive.

###### Proof.

Suppose anyone of the given assumptions holds. Then by Theorem 5.9 and Theorem
5.12(2) in [10], we have $\tau_{ucb}=\tau_{s}$. Therefore $(X,\tau)$ is semi-
reflexive if and only if $(X,\tau_{F})$ is semi-reflexive. By Theorem 3.10,
$(X,\tau)$ is reflexive if and only if $(X,\tau_{F})$ is reflexive. ∎

######  Corollary 3.12.

Suppose $(X,\parallel\cdot\parallel)$ is an enls. Then
$(X,\parallel\cdot\parallel)$ is reflexive if and only if $(X,\tau_{F})$ is
reflexive.

Our next theorem relates the reflexivity of an elcs $(X,\tau)$ with the
reflexivity of its open subspaces.

######  Proposition 3.13.

Suppose $M$ is an open subspace of an elcs $(X,\tau)$. Then there exists a
continuous extended seminorm $\rho$ on $(X,\tau)$ such that
$X_{fin}^{\rho}=M$.

###### Proof.

Since $M$ is an open subspace of $(X,\tau)$, there exists a continuous
extended seminorm $\mu$ on $(X,\tau)$ such that $X_{fin}^{\mu}\subseteq M$.
Define an extended seminorm $\rho$ on $X$ by

$\rho(x)=\begin{cases}\text{0;}&\quad\text{if $x\in M$}\\\
\text{$\infty$;}&\quad\text{if $x\notin M.$ }\\\ \end{cases}$

Note that $\rho(x)\leq\mu(x)$ for every $x\in X$. Consequently, $\rho$ is
continuous on $(X,\tau)$. It is easy to see that $X_{fin}^{\rho}=M$. ∎

We need the following remark and lemma in the proof of Theorem 3.16.

######  Remark 3.14.

Suppose $\rho$ is a continuous extended seminorm on an elcs $(X,\tau)$ and $M$
is a subspace of $X$ with $X=X_{fin}^{\rho}\oplus M$. Then $(M,\tau|_{M})$ is
a discrete space. Therefore $\tau|_{M}$ is induced by the discrete extended
norm $\parallel\cdot\parallel_{0,\infty}$. If $\tau_{F}$ is the flc topology
for $(X,\tau)$, then by Theorem 4.1 in [11], the flc topology for
$(M,\tau|_{M})$ is $\tau_{F}|_{M}$. By Example 3.6 and Corollary 3.12, both
the spaces $(M,\tau|_{M})$ and $(M,\tau_{F}|_{M})$ are reflexive.

######  Lemma 3.15.

Let $(X,\tau)$ be an elcs and let $\rho$ be a continuous extended seminorm on
$(X,\tau)$. If $M$ is a subspace of $X$ with $X=X_{fin}^{\rho}\oplus M$ and
$\tau_{F}$ is the flc topology for $(X,\tau)$, then $(X,\tau_{F})$ is
isomorphic to the product space
$\left(X_{fin}^{\rho},\tau_{F}|_{X_{fin}^{\rho}}\right)\times\left(M,\tau_{F}|_{M}\right)$.

###### Proof.

Consider the map
$\Psi:(X,\tau_{F})\to\left(X_{fin}^{\rho},\tau_{F}|_{X_{fin}^{\rho}}\right)\times\left(M,\tau_{F}|_{M}\right)$
defined by $\Psi(x=x_{f}+x_{m})=(x_{f},x_{m})$. Then $\Psi$ is linear and
bijective. Note that if $U$ and $V$ are neighborhoods of $0$ in
$\left(X_{fin}^{\rho},\tau_{F}|_{X_{fin}^{\rho}}\right)\text{ and
}\left(M,\tau_{F}|_{M}\right)$, respectively, then there exist continuous
seminorms $\mu$ and $\nu$ on
$\left(X_{fin}^{\rho},\tau_{F}|_{X_{fin}^{\rho}}\right)\text{ and
}\left(M,\tau_{F}|_{M}\right)$ such that $\mu^{-1}([0,1))\subseteq U$ and
$\nu^{-1}([0,1))\subseteq V$. It is easy to see that
$\lambda(x)=\mu(x_{f})+\nu(x_{m})$ for $x\in X$ and $x=x_{f}+x_{m}$ is a
continuous seminorm on $(X,\tau)$ as $X_{fin}^{\rho}$ is an open subspace of
$(X,\tau)$ and $\lambda=\mu$ on $X_{fin}^{\rho}$. By Theorem 3.5 in [11],
$\lambda$ is continuous on $(X,\tau_{F})$. Note that
$\Psi(\lambda^{-1}([0,1)))\subseteq U\times V$. Therefore $\Psi$ is
continuous.

For the continuity of $\Psi^{-1}$, let $\rho$ be a continuous seminorm on
$(X,\tau_{F})$. Then $\rho|_{X_{fin}^{\rho}}$ and $\rho|_{M}$ are continuous
on $(X_{fin}^{\rho},\tau_{F}|_{X_{fin}^{\rho}})$ and $(M,\tau_{F}|_{M})$,
respectively. Note that

$\Psi^{-1}\left(\rho|^{-1}_{X_{fin}^{\rho}}\left(\left[0,\frac{1}{2}\right)\right)\times\rho|^{-1}_{M}\left(\left[0,\frac{1}{2}\right)\right)\right)\subseteq\rho^{-1}([0,1)).$

Therefore $\Psi^{-1}$ is continuous. ∎

######  Theorem 3.16.

Suppose $(X,\tau)$ is an elcs with the flc topology $\tau_{F}$. Then the
following statements are equivalent.

  * (1)

$\left(X,\tau\right)$ is reflexive;

  * (2)

$\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)$ is reflexive, for every
continuous extended seminorm $\rho$ on $X$;

  * (3)

$\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)$ is reflexive, for some
continuous extended seminorm $\rho$ on $X$.

###### Proof.

(1)$\Rightarrow$(2). Suppose $\rho$ is a continuous extended seminorm on
$(X,\tau)$ and $\psi$ is a continuous linear functional on
$\left(\left(X_{fin}^{\rho}\right)^{*},\tau_{ucb}^{\rho}\right)$, where
$\tau_{ucb}^{\rho}$ is the topology of uniform convergence on bounded subsets
of $\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)$. Define a linear
functional $\Psi$ on $X^{*}$ by $\Psi(f)=\psi(f|_{X_{fin}^{\rho}})$ for $f\in
X^{*}$. Suppose $(f_{\alpha})$ is a net in $(X^{*},\tau_{ucb})$ converging to
$0$. Then $f_{\alpha}|_{X_{fin}^{\rho}}\to 0$ in
$\left(\left(X_{fin}^{\rho}\right)^{*},\tau_{ucb}^{\rho}\right)$ as every
bounded subset of $\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)$ is also
bounded in $(X,\tau)$. Consequently,
$\Psi(f_{\alpha})=\psi(f_{\alpha}|_{X_{fin}^{\rho}})\to 0$. Thus
$\Psi\in(X^{*},\tau_{ucb})^{*}$. Since $\left(X,\tau\right)$ is reflexive,
there exists an $x_{0}\in X$ such that $\Psi=J_{x_{0}}$. If $x_{0}\notin
X_{fin}^{\rho}$, then there exists an $f\in X^{*}$ such that $f(x_{0})\neq 0$
and $f(X_{fin}^{\rho})=0$. Then $0\neq
J_{x_{0}}(f)=f(x_{0})=\Psi(f)=\psi(f|_{X_{fin}^{\rho}})=0$. So $x_{0}\in
X_{fin}^{\rho}$. For every
$f\in\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)^{*}$,
$\psi(f)=\Psi(f^{\prime})=J_{x_{0}}(f^{\prime})=f^{\prime}(x_{0})=f(x_{0})$,
where $f^{\prime}$ is a continuous linear extension of $f$ on $X$ which is
possible by Corollary 4.2 in [11]. Hence
$\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)$ is semi-reflexive.

To complete the proof it is enough to show that the canonical map
$J_{\rho}:\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)\to\left(\left(X_{fin}^{\rho}\right)^{*},\tau_{ucb}^{\rho}\right)^{*}$
on $X_{fin}^{\rho}$ is continuous. Let $M$ be a subspace of $X$ such that
$X=X_{fin}^{\rho}\oplus M$. Suppose $D$ is any bounded set in
$\left(\left(X_{fin}^{\rho}\right)^{*},\tau_{ucb}^{\rho}\right)$. Consider
$Z=\\{f^{\prime}:f\in D\\}$, where $f^{\prime}$ is the continuous linear
extension of $f$ on $X$ which is $0$ on $M$. Then $Z$ is pointwise bounded.
Since $\left(X,\tau\right)$ is reflexive, by Proposition 3.9, $(X,\tau_{F})$
is barreled. By Theorem 11.3.4 and Theorem 11.3.5, p. 384 in [12], $Z$ is
bounded in $(X^{*},\tau_{s})$. So $Z$ is bounded in $(X^{*},\tau_{ucb})$.
Consequently, $Z^{\circ}$ is a neighborhood of $0$ in
$(X^{*},\tau_{ucb})^{*}$. Thus $J^{-1}\left(Z^{\circ}\right)$ is a
neighborhood of $0$ in $(X,\tau)$ as $\left(X,\tau\right)$ is reflexive.
Therefore $J^{-1}\left(Z^{\circ}\right)\cap X_{fin}^{\rho}$ is a neighborhood
of $0$ in $\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)$. Note that
$J^{-1}\left(Z^{\circ}\right)\cap X_{fin}^{\rho}=J_{\rho}^{-1}(D^{\circ})$ (if
$x\in X_{fin}^{\rho}$, then $|f(x)|\leq 1~{}\text{for every}~{}f\in
D\iff|g(x)|\leq 1~{}\text{for every}~{}g\in Z$). Which implies that
$J_{\rho}^{-1}(D^{\circ})$ is a neighborhood of $0$ in
$\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)$. Therefore $J_{\rho}$ is
continuous. Hence $\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)$ is
reflexive.

The implication (2)$\Rightarrow$(3) is obvious.

(3)$\Rightarrow$(1). Let $(3)$ hold for some continuous extended seminorm
$\rho$ on $(X,\tau)$ and let $M$ be a subspace of $X$ such that
$X=X_{fin}^{\rho}\oplus M$. Suppose $\Psi$ is a continuous linear functional
on $(X^{*},\tau_{ucb})$. Define linear functionals $\Psi_{1}$ and $\Psi_{2}$
on $\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)^{*}$ and
$(M,\tau|_{M})^{*}$, respectively, by $\Psi_{1}(f)=\Psi(\hat{f})$ for
$f\in\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)^{*}$ and
$\Psi_{2}(g)=\Psi(g^{\prime})$ for $g\in(M,\tau|_{M})^{*}$, where

$\displaystyle\hat{f}(x)$
$\displaystyle=\left\\{\begin{array}[]{lll}f(x)&\textnormal{if}&x\in
X_{fin}^{\rho}\\\ 0&\textnormal{if}&x\in M\end{array}\right.$
$\displaystyle\qquad g^{\prime}(x)$
$\displaystyle=\left\\{\begin{array}[]{lll}0&\textnormal{if}&x\in
X_{fin}^{\rho}\\\ g(x)&\textnormal{if}&x\in M.\end{array}\right.$

It is easy to see that if nets $f_{\alpha}\to 0$ in
$\left(\left(X_{fin}^{\rho}\right)^{*},\tau_{ucb}^{\rho}\right)$ and
$g_{\beta}\to 0$ in $\left(M^{*},\tau_{ucb}^{m}\right)$, then
$\hat{f}_{\alpha}\to 0$, $g_{\beta}^{\prime}\to 0$ in $(X^{*},\tau_{ucb})$,
where $\tau_{ucb}^{\rho}$ and $\tau_{ucb}^{m}$ are the topologies of the
uniform convergence on bounded subsets of
$(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}})$ and $(M,\tau|_{M})$, respectively.
Since $\Psi\in(X^{*},\tau_{ucb})^{*}$, we have
$\Psi_{1}\in\left(\left(X_{fin}^{\rho}\right)^{*},\tau_{ucb}^{\rho}\right)^{*}$
and $\Psi_{2}\in\left(M^{*},\tau_{ucb}^{m}\right)^{*}$. So there exists
$x_{f}\in X_{fin}^{\rho}$ such that $\Psi_{1}(f)=f(x_{f})$ for all
$f\in(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}})^{*}$ as
$\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)$ is semi-reflexive. By
Remark 3.14, $M$ is also semi-reflexive. There exists $x_{m}\in M$ such that
$\Psi_{2}(g)=g(x_{m})$ for all $g\in(M,\tau|_{M})^{*}$. Note that for every
$f\in X^{*}$,
$\Psi(f=\widehat{f|}_{X_{fin}^{\rho}}+f|_{M}^{\prime})=\Psi(\widehat{f|}_{X_{fin}^{\rho}})+\Psi(f|_{M}^{\prime})=\Psi_{1}(f|_{X_{fin}^{\rho}})+\Psi_{2}(f|_{M})=f|_{X_{fin}^{\rho}}(x_{f})+f|_{M}(x_{m})=f(x_{f}+x_{m})=J_{x_{f}+x_{m}}(f)$.
Therefore $(X,\tau)$ is semi-reflexive.

Let $D$ be a bounded set in $(X^{*},\tau_{ucb})$. Then it is pointwise
bounded. Since both $\left(X_{fin}^{\rho},\tau|_{X_{fin}^{\rho}}\right)$ and
$(M,\tau|_{M})$ are reflexive, by Theorem 3.10, both the spaces
$\left(X_{fin}^{\rho},\tau_{F}|_{X_{fin}^{\rho}}\right)$ and
$(M,\tau_{F}|_{M})$ are barreled. By Lemma 3.15 and Theorem 11.12.4, p. 409 in
[12], $(X,\tau_{F})$ is barreled. Therefore by Theorem 11.3.4, p. 384 in [12],
$D$ is equicontinuous on $(X,\tau_{F})$. Consequently, $D_{\circ}$ is a
neighborhood of $0$ in $(X,\tau_{F})$. Note that $J^{-1}(D^{\circ})=\\{x\in
X:|f(x)|\leq 1~{}\text{for}~{}f\in D\\}=D_{\circ}$. So $J^{-1}(D^{\circ})$ is
a neighborhood of $0$ in $(X,\tau)$. Hence $J$ is continuous and $(X,\tau)$ is
reflexive. ∎

######  Remark 3.17.

A similar result holds if we replace reflexive by semi-reflexive in the
statement of Theorem 3.16.

######  Corollary 3.18.

Let $(X,\parallel\cdot\parallel)$ be an enls. Then
$(X,\parallel\cdot\parallel)$ is reflexive if and only if
$(X_{fin},\parallel\cdot\parallel)$ is reflexive.

######  Corollary 3.19.

Let $(X,\parallel\cdot\parallel)$ be a reflexive enls. Then
$(X,\parallel\cdot\parallel)$ is an extended Banach space.

###### Proof.

If $(X,\parallel\cdot\parallel)$ is reflexive, then
$(X_{fin},\parallel\cdot\parallel)$ is reflexive. By classical theory for
normed space, $(X_{fin},\parallel\cdot\parallel)$ is a Banach space. Hence by
Proposition 3.11 in [1], $(X,\parallel\cdot\parallel)$ is an extended Banach
space.∎

Recall that for an elcs $(X,\tau)$ with the dual $X^{*}$, the weak topology
$\tau_{w}$ on $X$ is the locally convex space topology induced by the
collection $\mathcal{P}_{w}=\\{\rho_{f}:f\in X^{*}\\}$ of seminorms on $X$,
where $\rho_{f}(x)=|f(x)|$ for every $x\in X$. It is easy to prove that the
weak topologies corresponding to $(X,\tau)$ and $(X,\tau_{F})$ are same.

######  Theorem 3.20.

Let $(X,\parallel\cdot\parallel)$ be an extended Banach space with the flc
topology $\tau_{F}$. Then the following assertions are equivalent:

  * (1)

$(X,\parallel\cdot\parallel)$ is reflexive;

  * (2)

$(X,\tau_{F})$ is reflexive;

  * (3)

$(X_{fin},\parallel\cdot\parallel)$ is reflexive;

  * (4)

the closed unit ball $B_{X}$ is weakly compact;

  * (5)

the weak topology has the Heine-Borel property.

###### Proof.

(2)$\Leftrightarrow$(5). It follows from the fact that a locally convex space
is semi-reflexive if and only if its weak topology has the Heine-Borel
property (see, Theorem 15.2.4, p. 489 in [12]).

(4)$\Leftrightarrow$(3). Note that if $\tau_{w}$ is the weak topology
corresponding to $(X,\parallel\cdot\parallel)$ (or $(X,\tau_{F})$), then
$\tau_{w}|_{X_{fin}}$ is the weak topology of the Banach space
$(X_{fin},\parallel\cdot\parallel)$. Consequently, the equivalence follows
from the fact that a Banach space is reflexive if and only if its closed unit
ball is weakly compact (see, Exercise 15.101, p. 516 in [12]).

The equivalences (1)$\Leftrightarrow$(2)$\Leftrightarrow$(3) follow from
Corollaries 3.12, 3.18. ∎

Recall that a normed linear space $X$ is reflexive if and only if
$J(B_{X})\supseteq B_{X^{**}}$, where $J$ is the canonical map on $X$, and
$B_{X}$ and $B_{X^{**}}$ are the closed unit balls in $X$ and
$(X^{*},\parallel\cdot\parallel_{op})^{*}$, respectively. We next prove an
analogous result for an enls.

######  Theorem 3.21.

Suppose $(X,\parallel\cdot\parallel)$ is an extended normed space and $J$ is
the corresponding canonical map on $X$. Then $X$ is reflexive if and only if
$J(B_{X})=\left(B_{X^{*}}\right)^{\circ}$, where $B_{X^{*}}$ is the closed
unit ball in $(X^{*},\parallel\cdot\parallel_{op})$ and
$\left(B_{X^{*}}\right)^{\circ}$ is the polar of $B_{X^{*}}$ in
$(X^{*},\tau_{ucb})^{*}$.

###### Proof.

It is easy to see that $J(B_{X})\subseteq\left(B_{X^{*}}\right)^{\circ}$. For
the reverse inclusion consider $\psi\in\left(B_{X^{*}}\right)^{\circ}$. Since
$X$ is reflexive, there is an $x_{0}\in X$ such that $\psi=J_{x_{0}}$.
Therefore for any $\phi\in B_{X^{*}}$, we have
$|\phi(x_{0})|=|J_{x_{0}}(\phi)|=|\psi(\phi)|\leq 1$. So by Proposition 4.9 in
[1], $x_{0}\in B_{X}$.

Conversely, suppose $J(B_{X})=\left(B_{X^{*}}\right)^{\circ}$. By Corollary
3.18, it is enough to show that $(X_{fin},\parallel\cdot\parallel)$ is
reflexive. Let $B_{X_{fin}}$ and $B_{X_{fin}^{**}}$ be the closed unit balls
in $(X_{fin},\parallel\cdot\parallel)$ and
$(X_{fin}^{*},\parallel\cdot\parallel_{op})^{*}$, respectively. To show
$(X_{fin},\parallel\cdot\parallel)$ is reflexive, it is enough to show that
$J_{f}(B_{X_{fin}})\supseteq B_{X_{fin}^{**}}$, where $J_{f}$ is the canonical
map on $X_{fin}$. Suppose $\Psi\in B_{X_{fin}^{**}}$. Define a linear
functional $\hat{\Psi}$ on $X^{*}$ by $\hat{\Psi}(f)=\Psi(f|_{X_{fin}})$ for
every $f\in X^{*}$. It is easy to see that
$|\hat{\Psi}(f)|=|\Psi(f|_{X_{fin}})|\leq\parallel\Psi\parallel_{op}\parallel
f\parallel_{op}\leq 1$ for every $f\in B_{X^{*}}$. Then $\hat{\Psi}$ is
continuous on $(X^{*},\parallel\cdot\parallel_{op})$. Consequently,
$\hat{\Psi}\in(X^{*},\tau_{ucb})^{*}$. Also observe that
$\hat{\Psi}\in\left(B_{X^{*}}\right)^{\circ}$. So there exists an $x\in
B_{X}=B_{X_{fin}}$ such that $J_{x}=\hat{\Psi}$ as
$J(B_{X})=(B_{X^{*}})^{\circ}$. For any
$\phi\in(X_{fin},\parallel\cdot\parallel_{op})^{*}$, we have
$\Psi(\phi)=\hat{\Psi}(\phi^{\prime})=J_{x}(\phi^{\prime})=\phi^{\prime}(x)=\phi(x)$,
where $\phi^{\prime}$ is any continuous linear extension of $\phi$ on $X$.
Therefore $\Psi=J_{f}(x)\in J_{f}(B_{X_{fin}})$.∎

In the next result, we show that the reflexive property in an extended Banach
space is a three-space property, that is, if $Y$ is a closed subspace of an
enls $(X,\parallel\cdot\parallel)$ and any two of the spaces
$(Y,\parallel\cdot\parallel)$, $(X,\parallel\cdot\parallel)$ and
$\left(X/Y,\parallel\cdot\parallel_{q}\right)$ are reflexive, then the third
one is also reflexive, where $X/Y=\\{x+Y:x\in X\\}$ and $\parallel
x+Y\parallel_{q}=\inf\\{\parallel x-y\parallel:y\in Y\\}$.

######  Theorem 3.22.

Let $(X,\parallel\cdot\parallel)$ be an enls and let $Y$ be a closed subspace
of $X$. Then $(X,\parallel\cdot\parallel)$ is reflexive if and only if both
$(Y,\parallel\cdot\parallel)$ and the quotient space
$\left(X/Y,\parallel\cdot\parallel_{q}\right)$ are reflexive.

###### Proof.

Suppose $(X,\parallel\cdot\parallel)$ is reflexive. Then $Y_{fin}=\\{y\in
Y:\parallel y\parallel<\infty\\}$ is a closed subspace of a reflexive space
$(X_{fin},\parallel\cdot\parallel)$. By Theorem 15.2.7, p. 490 in [12],
$(Y_{fin},\parallel\cdot\parallel)$ is a reflexive space. Then by Corollary
3.18, $(Y,\parallel\cdot\parallel)$ is reflexive. Note that the finite
subspace $\left(X/Y\right)_{fin}$ of the quotient space
$\left(X/Y,\parallel\cdot\parallel_{q}\right)$ is equal to $\\{x+Y:x\in
X_{fin}\\}$ (see, Theorem 3.21 in [1]). It is easy to see that if $x\in
X\setminus X_{fin}$ and $\parallel y\parallel=\infty$ for some $y\in Y$, then
$\parallel x-y\parallel=\infty$. So for every $x\in X_{fin}$, we have

$\parallel x+Y\parallel_{q}=\inf\\{\parallel x-y\parallel:y\in
Y\\}=\inf\\{\parallel x-y\parallel:y\in Y_{fin}\\}=\parallel
x+Y_{fin}\parallel_{q}.$

Therefore $\left(\left(X/Y\right)_{fin},\parallel\cdot\parallel_{q}\right)$ is
isometrically isomorphic to
$\left(X_{fin}/Y_{fin},\parallel\cdot\parallel_{q}\right)$. Since reflexivity
in a normed linear space is a three-space property (see, p. 491 in [12]), we
have $\left(X_{fin}/Y_{fin},\parallel\cdot\parallel_{q}\right)$ is reflexive.
So $\left(\left(X/Y\right)_{fin},\parallel\cdot\parallel_{q}\right)$ is
reflexive. By Corollary 3.18, $\left(X/Y,\parallel\cdot\parallel_{q}\right)$
is reflexive.

Conversely, suppose both $(Y,\parallel\cdot\parallel)$ and
$\left(X/Y,\parallel\cdot\parallel_{q}\right)$ are reflexive. By Corollary
3.18, both $(Y_{fin},\parallel\cdot\parallel)$ and
$\left((X/Y)_{fin},\parallel\cdot\parallel_{q}\right)$ are reflexive. So
$(Y_{fin},\parallel\cdot\parallel)$ and
$\left(X_{fin}/Y_{fin},\parallel\cdot\parallel_{q}\right)$ are reflexive.
Therefore $(X_{fin},\parallel\cdot\parallel)$ is reflexive. Hence by Corollary
3.18, $(X,\parallel\cdot\parallel)$ is reflexive. ∎

In general, if $\mu$ is a finitely compatible norm for an enls
$(X,\parallel\cdot\parallel)$, that is, both $\mu$ and
$\parallel\cdot\parallel$ induce the same topology on $X_{fin}$, then the
reflexivity of $(X,\mu)$ may not have any relation with the reflexivity of
$(X,\parallel\cdot\parallel)$ (see, Examples 3.24, 3.25). But, there exists a
finitely compatible norm $\nu$ such that $(X,\parallel\cdot\parallel)$ is
reflexive whenever $(X,\nu)$ is reflexive.

######  Theorem 3.23.

Suppose $(X,\parallel\cdot\parallel)$ is an extended normed space with
$X=X_{fin}\oplus M$. Then there exists a finitely compatible norm $\nu$ for
$(X,\parallel\cdot\parallel)$ with the following property: If $(X,\nu)$ is
reflexive, then $(X,\parallel\cdot\parallel)$ is reflexive.

###### Proof.

Let $\phi\in X^{*}$ such that $\phi^{-1}(0)=X_{fin}$. Consider the norm $\nu$
on $X$ by $\nu(x)=\parallel x_{f}\parallel+|\phi(x_{M})|$, where
$x=x_{f}+x_{M}$ with $x_{f}\in X_{fin}$ and $x_{M}\in M$. Then $\nu$ is a
finitely compatible norm for $(X,\parallel\cdot\parallel)$ and
$\phi\in(X,\nu)^{*}.$ As $(X,\nu)$ is reflexive and $\phi\in(X,\nu)^{*}$ with
$\phi^{-1}(0)=X_{fin}$, $(X_{fin},\parallel\cdot\parallel)$ is reflexive. By
Corollary 3.18, $(X,\parallel\cdot\parallel)$ is reflexive.∎

######  Example 3.24.

Let $c_{00}$ be the collection of all eventually zero sequences with the
discrete extended norm $\parallel\cdot\parallel_{0,\infty}$. Then
$(c_{00},\parallel\cdot\parallel_{0,\infty})$ is a reflexive space but the
finitely compatible normed space $(c_{00},\parallel\cdot\parallel_{\infty})$
is not reflexive.

######  Example 3.25.

Let $l_{p}~{}(1<p<\infty)$ be the space of all $p$-summable real sequences.
Suppose $M$ is a subspace of $l_{p}$ with $l_{p}=c_{00}\oplus M$. Define an
extended norm on $l_{p}$ by

$\parallel x\parallel=\begin{cases}\text{$\parallel
x\parallel_{p},$}&\quad\text{if $x\in c_{00}$}\\\
\text{$\infty$,}&\quad\text{otherwise. }\\\ \end{cases}$

Then $X_{fin}=c_{00}$ is not reflexive. Consequently,
$(l_{p},\parallel\cdot\parallel)$ is a non-reflexive extended normed linear
space. Note that the classical $p$-norm $\parallel\cdot\parallel_{p}$ is a
finitely compatible norm on $(X,\parallel\cdot\parallel)$. Also, the normed
space $(l_{p},\parallel\cdot\parallel_{p})$ is a reflexive space.

##  4\. Applications to Function spaces

Let $(X,d)$ be a metric space and let $C(X)$ be the set of all real-valued
continuous functions on $(X,d)$. By a bornology $\mathcal{B}$ on $(X,d)$, we
mean a collection of nonempty subsets of $X$ that covers $X$ and is closed
under finite union and taking subsets of its members. A subfamily
$\mathcal{B}_{0}$ of $\mathcal{B}$ is a base for $\mathcal{B}$ if it is
cofinal in $\mathcal{B}$ under the set inclusion. In addition, if every member
of $\mathcal{B}_{0}$ is closed in $(X,d)$, then we say $\mathcal{B}$ has a
closed base. For more details about metric bornologies, we refer to [2].

In this section, we study the reflexivity of the function spaces
$(C(X),\tau_{\mathcal{B}}^{s})$ and $(C(X),\tau_{\mathcal{B}})$, where
$\tau_{\mathcal{B}}^{s}(\tau_{\mathcal{B}})$ is the topology of strong uniform
convergence (uniform convergence) on the elements of $\mathcal{B}$. We first
define these topologies.

######  Definition 4.1.

([3]) Let $\mathcal{B}$ be a bornology on a metric space $(X,d)$. Then the
topology $\tau_{\mathcal{B}}$ of uniform convergence on $\mathcal{B}$ is
determined by a uniformity on $C(X)$ having base consisting of sets of the
form

$[B,\epsilon]=\left\\{(f,g):\forall x\in
B,~{}|f(x)-g(x)|<\epsilon\right\\}~{}(B\in\mathcal{B},~{}\epsilon>0).$

######  Definition 4.2.

([3]) Let $\mathcal{B}$ be a bornology on a metric space $(X,d)$. Then the
topology $\tau^{s}_{\mathcal{B}}$ of strong uniform convergence on
$\mathcal{B}$ is determined by a uniformity on $C(X)$ having base consisting
of sets of the form

$[B,\epsilon]^{s}=\left\\{(f,g):\exists~{}\delta>0~{}\forall x\in
B^{\delta},~{}|f(x)-g(x)|<\epsilon\right\\}~{}(B\in\mathcal{B},~{}\epsilon>0),$

where for $B\subseteq X$, $B^{\delta}=\displaystyle{\bigcup_{y\in B}}\\{x\in
X:d(x,y)<\delta\\}$.

Suppose $\mathcal{B}$ is a bornology on a metric space $(X,d)$. Then the
topology $\tau_{\mathcal{B}}$ on $C(X)$ is induced by the collection
$\mathcal{P}=\\{\rho_{B}:B\in\mathcal{B}\\}$ of extended seminorms, where
$\rho_{B}(f)=\sup_{x\in B}|f(x)|$ for $f\in C(X)$. Similarly, the topology
$\tau_{\mathcal{B}}^{s}$ on $C(X)$ is induced by the collection
$\mathcal{P}=\\{\rho_{B}^{s}:B\in\mathcal{B}\\}$ of extended seminorms, where
$\rho_{B}^{s}(f)=\inf_{\delta>0}\left\\{\sup_{x\in B^{\delta}}|f(x)|\right\\}$
for $f\in C(X)$. Hence both the function spaces $(C(X),\tau_{\mathcal{B}})$
and $(C(X),\tau_{\mathcal{B}}^{s})$ are actually extended locally convex
spaces. For more details related to $\tau_{\mathcal{B}}$ and
$\tau_{\mathcal{B}}^{s}$, we refer to [3, 6, 7].

Recall that if $(X,\tau)$ is a locally convex space with the dual $X^{*}$,
then the Mackey topology $\tau_{M}$ is a locally convex space topology on $X$
whose neighborhood base at $0$ is given by

$\mathcal{B}_{M}=\\{B_{\circ}:B\text{ is an absolutely convex and weak${}^{*}$
compact subset of }X^{*}\\}.$

We say $(X,\tau)$ is a Mackey space if $\tau=\tau_{M}$. Note that $\tau_{M}$
is the largest locally convex topology on $X$ such that
$(X,\tau_{M})^{*}=X^{*}$. For more details about Mackey topology, we refer to
[12, 13].

######  Proposition 4.3.

Suppose $\tau_{w^{*}}$ and $\tau_{w}$ are the weak∗ and weak topologies for an
elcs $(X,\tau)$, respectively. Then

$\mathcal{B}=\\{B^{\circ}:B\text{ is absolutely convex and compact in
}(X,\tau_{w})\\}$

is a neighborhood base at $0$ in $(X^{*},\tau_{M})$, where $\tau_{M}$ is the
Mackey topology for the locally convex space $(X^{*},\tau_{w^{*}})$.

###### Proof.

Let $D$ be an absolutely convex and weak∗ compact subset of
$(X^{*},\tau_{w^{*}})^{*}$. Then $D=\\{J_{x}:x\in A\\}$ for some $A\subseteq
X$ as $(X^{*},\tau_{w^{*}})^{*}=\\{J_{x}:x\in X\\}$. Since $D$ is absolutely
convex, $A$ is absolutely convex. Note that a net $(x_{\lambda})$ in $X$
converges weakly to $x$ if and only if $f(x_{\lambda})\to f(x)$ for every
$f\in X^{*}$ if and only if $J_{x_{\lambda}}(f)\to J_{x}(f)$ for all $f\in
X^{*}$. Therefore $A$ is weakly compact. It is easy to prove that
$A^{\circ}=D_{\circ}$. Which completes the proof. ∎

######  Theorem 4.4.

Suppose $(X,\tau)$ is an elcs. Then $(X,\tau)$ is semi-reflexive if and only
if every bounded subsets of $(X,\tau)$ is relatively weakly compact.

###### Proof.

Suppose $(X,\tau)$ is semi-reflexive. Then
$(X^{*},\tau_{ucb})^{*}=(X^{*},\tau_{w^{*}})^{*}=\\{J_{x}:x\in X\\}$.
Therefore $\tau_{ucb}\subseteq\tau_{M}$, where $\tau_{M}$ is the Mackey
topology for $(X^{*},\tau_{w^{*}})$. Now, let $A$ be any bounded set in
$(X,\tau)$. Then $A^{\circ}$ is a neighborhood of $0$ in $(X^{*},\tau_{ucb})$.
Consequently, $A^{\circ}$ is a neighborhood of $0$ in $(X^{*},\tau_{M})$. By
Proposition 4.3, there exists an absolutely convex and weakly compact subset
$B$ of $X$ such that $B^{\circ}\subseteq A^{\circ}$. Therefore
$A\subseteq(A^{\circ})_{\circ}\subseteq(B^{\circ})_{\circ}$. By applying
Bipolar Theorem on $B$ in $(X,\tau_{w})$, we obtain $(B^{\circ})_{\circ}=B$.
Hence $A$ is relatively weakly compact.

Conversely, suppose every bounded set is weakly compact in $(X,\tau)$. Then
$\tau_{ucb}\subseteq\tau_{M}$. Therefore
$(X^{*},\tau_{ucb})^{*}=(X^{*},\tau_{M})^{*}=(X^{*},\tau_{w^{*}})^{*}=\\{J_{x}:x\in
X\\}$. Hence $(X,\tau)$ is semi-reflexive. ∎

######  Theorem 4.5.

Let $\mathcal{B}$ be a bornology with a closed base on a metric space $(X,d)$.
If $(C(X),\tau_{\mathcal{B}}^{s})$ $(\text{or }(C(X),\tau_{\mathcal{B}}))$ is
reflexive, then $\mathcal{K}\subseteq\mathcal{B}$.

###### Proof.

It follows from Theorems 3.15, 4.3 in [9] and Proposition 3.9. ∎

######  Theorem 4.6.

Let $\mathcal{B}$ be a bornology with a closed base on a metric space $(X,d)$.
If $(C(X),\tau_{\mathcal{B}}^{s})$ $(\text{or }(C(X),\tau_{\mathcal{B}}))$ is
reflexive, then $X$ is a discrete space.

###### Proof.

Suppose $K$ is any compact set in $(X,d)$. Then by Theorem 4.5,
$K\in\mathcal{B}$. We show that $K$ is a finite set. Let $(x_{n})$ be any
sequence in $K$ converging to $x_{0}$. For every $n\in\mathbb{N}$, there
exists an $f_{n}\in C(X,[0,1])$ such that $f_{n}(x_{j})=0$ for $1\leq j\leq n$
and $f_{n}(x_{0})=1$. Consider $A=\\{f_{n}:n\in\mathbb{N}\\}$. Clearly, $A$ is
bounded in $(C(X),\tau_{\mathcal{B}}^{s})$. By Theorem 4.4, $A$ is relatively
weakly compact in $(C(X),\tau_{\mathcal{B}}^{s})$. Therefore there exists a
cluster point $f$ of $A$. It is easy to prove that the topology of pointwise
convergence is coarser than the weak topology for
$(C(X),\tau_{\mathcal{B}}^{s})$ as the maps
$\Psi_{x}:\left(C(X),\tau_{\mathcal{B}}^{s}\right)\to\mathbb{R}$ defined by
$\Psi_{x}(f)=f(x)$ is continuous for every $x\in X$. For every
$m\in\mathbb{N}$ and $\epsilon>0$, there exists an $n>m$ such that
$|f(x_{m})-f_{n}(x_{m})|<\epsilon$. Therefore $f(x_{m})=0$ for every
$m\in\mathbb{N}$. Consequently, $f(x_{m})\to f(x_{0})=0$. But, there also
exists an $n\in\mathbb{N}$ such that $|f(x_{0})-f_{n}(x_{0})|<\frac{1}{2}$. We
arrive at a contradiction. Hence $X$ is a discrete space. ∎

Suppose $(X,d)$ is a metric space. Then the topology $\tau_{u}$ of uniform
convergence on $C(X)$ is induced by the extended norm $\parallel
f\parallel_{\infty}=\sup_{x\in X}|f(x)|$. It is known that for a compact space
$(X,d)$, the normed space $(C(X),\parallel\cdot\parallel_{\infty})$ is
reflexive if and only if $X$ is finite (Example 15.5.2, p. 502 in [12]). We
now prove a similar result without assuming $(X,d)$ to be compact. The next
theorem also shows that the converse of Theorem 4.6 may not be true.

######  Theorem 4.7.

Suppose $(X,d)$ is a metric space. Then the uniform space
$(C(X),\parallel\cdot\parallel_{\infty})$ is reflexive if and only if $X$ is
finite.

###### Proof.

If $X$ is finite, then $C(X)$ is finite dimensional. Therefore
$(C(X),\parallel\cdot\parallel_{\infty})$ is reflexive. Conversely, suppose
$(C(X),\parallel\cdot\parallel_{\infty})$ is reflexive and $X$ is infinite.
Then by Example 15.5.2, p. 502 in [12], $X$ cannot be compact. So there exists
a countable, closed and discrete subset $T=\\{t_{n}:n\in\mathbb{N}\\}$ of $X$.
If $Y=\\{f\in C(X):f=0~{}\text{on}~{}T\\}$, then $Y$ is a closed subspace of
$C(X)$. By Theorem 3.22, $C(X)/Y$ is reflexive. Now, we show that the space
$l_{\infty}$ of all bounded real sequences is isometrically isomorphic to a
subspace of $C(X)/Y$. Let $z=(z_{n})\in l_{\infty}$ and
$m=\displaystyle{\inf_{n\in\mathbb{N}}}z_{n}$ and
$M=\displaystyle{\sup_{n\in\mathbb{N}}}z_{n}$. Since $T$ is discrete and
closed, by Tietze extension theorem, there exists a $f_{z}\in C(X)$ such that
$f_{z}(t_{n})=z_{n}$ for $n\in\mathbb{N}$. Define
$F_{z}(x)=\max\\{m,\min\\{M,f_{z}(x)\\}\\}$ for $x\in X$. Then $F_{x}\in C(X)$
with $F_{x}(t_{n})=x_{n}$ for $n\in\mathbb{N}$. Consider $\Psi:l_{\infty}\to
C(X)/Y$ by $\Psi(z)=F_{z}+Y$. Then $\Psi$ is linear as $F_{\alpha x+y}=\alpha
F_{x}+F_{y}~{}\text{on}~{}Y$ for $x,y\in l_{\infty}$ and
$\alpha\in\mathbb{R}$. Note that if $z=(z_{n})\in l_{\infty}$ and $f\in Y$,
then $\parallel F_{z}+Y\parallel\leq\parallel
F_{z}\parallel_{\infty}\leq\max\\{|m|,|M|\\}=\parallel z\parallel_{\infty}$
and $\parallel
z\parallel_{\infty}=\displaystyle{\sup_{n\in\mathbb{N}}}|z_{n}|=\displaystyle{\sup_{n\in\mathbb{N}}}|F_{z}(t_{n})|=\displaystyle{\sup_{n\in\mathbb{N}}}|F_{z}(t_{n})-f(t_{n})|\leq\parallel
F_{z}-f\parallel_{\infty}$. Therefore $\Psi$ is an isometry. By Theorem 3.22,
$\Psi(l_{\infty})$ is reflexive. Consequently, by Exercise 3.61, p. 97 in [8],
$l_{\infty}$ is reflexive. Which is not true. ∎

## References

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  * [7] T. K. Chauhan and V. Jindal.  Strong whitney and strong uniform convergences on a bornology.  Journal of Mathematical Analysis and Applications, 505(1):125634, 2022.
  * [8] M. Fabian, P. Habala, P. Hájek, V. M. Santalucía, J. Pelant, and V. Zizler.  Functional analysis and infinite-dimensional geometry.  Springer, 2001.
  * [9] A. Kumar and V. Jindal.  Barreled extended locally convex spaces and uniform boundedness principle.  arXiv preprint arXiv:2305.13816, 2023.
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  * [12] L. Narici and E. Beckenstein.  Topological vector spaces.  Second Edition, CRC Press, 2011.
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# Berezinskii-Kosterlitz-Thouless transitions in an easy-plane ferromagnetic
superfluid

Andrew P. C. Underwood  Department of Physics, Centre for Quantum Science, and
Dodd-Walls Centre for Photonic and Quantum Technologies, University of Otago,
Dunedin, New Zealand  Andrew J. Groszek  ARC Centre of Excellence for
Engineered Quantum Systems, School of Mathematics and Physics, University of
Queensland, Saint Lucia QLD 4072, Australia  ARC Centre of Excellence in
Future Low-Energy Electronics Technologies, School of Mathematics and Physics,
University of Queensland, Saint Lucia QLD 4072, Australia  Xiaoquan Yu
Graduate School of China Academy of Engineering Physics, Beijing 100193, China
Department of Physics, Centre for Quantum Science, and Dodd-Walls Centre for
Photonic and Quantum Technologies, University of Otago, Dunedin, New Zealand
P. B. Blakie  Department of Physics, Centre for Quantum Science, and Dodd-
Walls Centre for Photonic and Quantum Technologies, University of Otago,
Dunedin, New Zealand  L. A. Williamson  ARC Centre of Excellence for
Engineered Quantum Systems, School of Mathematics and Physics, University of
Queensland, Saint Lucia QLD 4072, Australia

###### Abstract

A two-dimensional (2D) spin-1 Bose gas exhibits two Berezenskii-Kosterlitz-
Thouless (BKT) transitions in the easy-plane ferromagnetic phase. The higher
temperature transition is associated with superfluidity of the mass current
determined predominantly by a single spin component. The lower temperature
transition is associated with superfluidity of the axial spin current, quasi-
long range order of the transverse spin density and binding of polar-core spin
vortices (PCVs). Above the spin BKT temperature, the component circulations
that make up each PCV spatially separate, suggesting possible deconfinement
analogous to quark deconfinement in high energy physics. Intercomponent
interactions give rise to superfluid drag between the spin components, which
we calculate analytically at zero temperature. We present the mass/spin
superfluid phase diagram as a function of quadratic Zeeman energy $q$. At
$q=0$ the system is in an isotropic spin phase with $\mathrm{SO}(3)$ symmetry.
Here the fluid response exhibits a system size dependence, suggesting the
absence of a BKT transition. Despite this, for finite systems the decay of
spin correlations changes from exponential to algebraic as the temperature is
decreased.

Spinor Bose gases boast a plethora of spin phases, providing an ideal system
for studying equilibrium and non-equilibrium properties of phase transitions
[1, 2]. At zero temperature a ferromagnetic spin-1 condensate with quadratic
Zeeman energy $0<q<q_{0}$ is in the easy-plane phase, with $q_{0}$ a quantum
critical point [3, 4]. This phase exhibits two broken continuous symmetries: a
$\mathrm{U}(1)$ symmetry associated with global phase coherence and an
$\mathrm{SO}(2)$ symmetry associated with transverse spin coherence [5].
Quenching the system from the polar ($q>q_{0}$) to the easy-plane phase has
revealed rich non-equilibrium dynamics as the system orders to the new ground
state [6, 7, 8, 9, 10, 11, 12, 9, 13, 14, 15, 16, 17].

An even richer phase structure is possible at finite temperature due to the
multitude of possible defects and superfluid currents [18, 19, 20, 21, 22, 23,
24, 25, 26]. In 2D, it is well known that true long-range order is prohibited
[27, 28] and that the onset of superfluidity is instead associated with a BKT
transition [29, 30, 31]. In an easy-plane ferromagnetic spin-1 Bose gas, the
two continuous symmetries give rise to two distinct BKT temperatures [25].
Beyond the existence of these transitions, however, little is known about the
finite temperature phases of this system. At $q=0$, the order parameter
manifold changes from $\mathrm{U}(1)\times\mathrm{SO}(2)$ to $\mathrm{SO}(3)$
[32]. In general, the nature of superfluidity in $\mathrm{SO}(3)$ systems is
not well understood [33, 34, 35, 36, 37, 25].

In this work we explore the finite temperature behaviour of a 2D spin-1
ferromagnetic Bose gas in the easy-plane phase. Results are obtained via
sampling of the dynamical evolution of the system (c.f. Ref. [25], which
employs a Monte Carlo Wolff algorithm). We observe that the system first
transitions to a mass-superfluid state and then, at a lower temperature,
transitions to a spin-superfluid state, in agreement with Ref. [25]. The mass
transition is predominantly determined by the superfluidity of the $m=0$ spin
component (with $m\in\\{-1,0,1\\}$ the spin-1 magnetic sublevels). The spin
transition is driven by the binding/unbinding of PCVs, which consist of
spatially confined equal and opposite circulations in the $m=\pm 1$
components. Above the spin BKT temperature, the $m=\pm 1$ component
circulations spatially separate, suggesting possible deconfinement analogous
to a color plasma. We identify superfluid drag between spin components,
arising from spin interactions [38, 39, 40], which we calculate analytically
at zero temperature. We determine the $(T,q)$ superfluid phase diagram for
$0<q<q_{0}$. At $q=0$, the fluid response exhibits a system size dependence,
suggesting the absence of a BKT transition [25]. Despite this, for finite-
sized systems the decay of correlations of total spin still change from
exponential to algebraic as the temperature is decreased. The decay exponent
is close to $1/2$ at the cross-over, twice that of $U(1)$ systems. The recent
observation of thermalization of a quasi-1D easy-plane spin-1 Bose gas [41]
demonstrates that our results could be observed in current experiments.

## Formalism.

We use a simple-growth stochastic Gross-Pitaevskii model that couples a three-
component spinor field $\Psi=(\psi_{1},\psi_{0},\psi_{-1})^{\mathrm{T}}$ to a
grand canonical reservoir at chemical potential $\mu$ and temperature $T$ [42,
43, 44, 45, 46],

$i\hbar\differential{\Psi}=\left(1-i\gamma\right)\left(\mathcal{L}\left\\{\Psi\right\\}-\mu\Psi\right)\differential{t}+i\hbar\differential{W}.$
(1)

The nonlinear operator $\mathcal{L}\left\\{\Psi\right\\}$ reads

$\mathcal{L}\left\\{\Psi\right\\}\\!=\\!\left[-\frac{\hbar^{2}\nabla^{2}}{2M}\mathbb{1}+qf_{z}^{2}+g_{n}n\mathbb{1}+g_{s}\sum_{\alpha}F_{\alpha}f_{\alpha}\right]\Psi,$
(2)

and describes time evolution arising from the kinetic energy, quadratic Zeeman
shift, density-dependent interactions (coupling constant $g_{n}>0$), and spin-
dependent interactions (coupling constant $g_{s}<0$). Here
$n=\Psi^{\dagger}\mathbb{1}\Psi$ and $F_{\alpha}=\Psi^{\dagger}f_{\alpha}\Psi$
are the density and spin density respectively, with $\mathbb{1}$ the identity
matrix in spin space, and $f_{\alpha}$ the spin-1 matrices
($\alpha\in\\{x,y,z\\}$). The dimensionless rate $\gamma$ determines how
strongly the system couples to the reservoir and
$\differential{W}=(\differential{w}_{1},\differential{w}_{0},\differential{w}_{-1})$
are Gaussian distributed complex noise terms with correlations
$\expectationvalue{\differential{w_{m}^{*}(\mathbf{r})}\differential{w_{m^{\prime}}(\mathbf{r}^{\prime})}}=2\gamma
k_{B}T\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta_{m,m^{\prime}}\differential{t}/\hbar$.
Stationary solutions to Eq. (1) sample the grand canonical ensemble and are
independent of $\gamma$ [43]. In a uniform system, $q_{0}=2|g_{s}|n\approx
2|g_{s}|\mu/g_{n}$.

We simulate a condensate with weak spin interactions $|g_{s}|=0.1g_{n}$ on an
$N\times N$ grid with periodic boundary conditions. We use a plane wave basis
cut off at the thermal energy $k_{B}T$ [47, 48]. With an adjustment to account
for our use of a square grid, this gives a grid spacing $\Delta
x=\sqrt{2\pi\hbar^{2}/Mk_{B}T}$. We take $\mu\approx 5q_{0}$ as a convenient
energy scale with associated length unit $x_{\mu}=\hbar/\sqrt{M\mu}$. We
express the temperature in terms of the dimensionless quantity
$\mathcal{T}=Mg_{n}k_{B}T/\hbar^{2}\mu$, which captures the dependence of
thermodynamic properties on both temperature and chemical potential [47, 48].
Equilibrium states are obtained by evolving Eq. (1) until $t\approx
10^{5}\hbar/\mu$. We then evolve the equilibrium state, sampling at intervals
of $10\hbar/\mu$, to build up an ensemble of $\sim 10^{4}$ states from which
thermal averages are calculated. The system size and hence thermalization time
diverge as $\mathcal{T}$ decreases, therefore we restrict our analysis to
$\mathcal{T}\geq 0.05$.

## Spin and mass BKT transitions.

The two continuous symmetries in the easy-plane phase give rise to two
superfluid currents at low temperatures [25]. The first is a mass superfluid
current arising from the global phase coherence. The second is an $F_{z}$ spin
superfluid current arising from coherence of the transverse spin
$\mathbf{F}_{\perp}=(F_{x},F_{y})$ 111The correspondence between $F_{z}$ spin
current and transverse spin coherence follows from noting that gradients of
transverse spin angle gives rise to an $F_{z}$ current [77], or more formally
since $f_{z}$ is the generator of transverse spin rotations.. The mass
superfluid density can be determined from the response of the system to slowly
moving boundary walls, via the current-current response tensor,

$\chi(\mathbf{k})=\frac{M}{k_{B}T}\int
d^{2}\mathbf{r}e^{-i\mathbf{k}\cdot\mathbf{r}}\langle\mathbf{J}(\mathbf{0})\mathbf{J}(\mathbf{r})\rangle.$
(3)

Here
$\mathbf{J}=(\hbar/M)\operatorname{Im}(\sum_{m}\psi_{m}^{\dagger}\nabla\psi_{m})$
is the total mass current, and angle brackets denote a thermal average. In the
long-wavelength limit, the longitudinal component $\chi^{L}(\mathbf{k})$ of
the tensor $\chi(\mathbf{k})$ is affected by the total fluid response, while
the transverse component $\chi^{T}(\mathbf{k})$ of $\chi(\mathbf{k})$ is only
affected by the normal fluid response [50, 51, 52]. The mass superfluid
density is then $\rho_{n}=\lim_{k\rightarrow
0}[\chi^{L}(\mathbf{k})-\chi^{T}(\mathbf{k})]$ (see Appendix D of [50] for
details 222We absorb a factor of $k_{B}T$ into the definition of $\chi$). We
extend this procedure to determine the $F_{z}$ spin superfluid density
$\rho_{s}$ by considering the response to a “spin dependent” moving boundary,
where the $m=\pm 1$ boundaries move in opposite directions and the $m=0$
boundary is stationary (in experiments, this could be engineered via spin-
dependent light fields [54]). We then replace $\mathbf{J}$ by
$\mathbf{J}_{z}=(\hbar/M)\operatorname{Im}(\sum_{m,m^{\prime}}\psi_{m}^{\dagger}(f_{z})_{mm^{\prime}}\nabla\psi_{m^{\prime}})$
in Eq. (3).

Figure 1: (a) Mass (black circles) and $F_{z}$ spin (red circles) superfluid
densities, determined from Eq. (3), as a function of dimensionless temperature
$\mathcal{T}$. With reducing $\mathcal{T}$ the system first exhibits mass
superfluidity and subsequently $F_{z}$ spin superfluidity. The superfluid
densities obtained from fitted $\eta_{n,s}$, via
$\rho_{n,s}=Mk_{B}T/2\pi\hbar^{2}\eta_{n,s}$, coincide with those determined
from Eq. (3) (matching black and red squares). We identify the precise BKT
temperatures $\mathcal{T}_{n,s}$ to be the temperatures where
$\eta_{n,s}=1/4$. Diamonds are $\rho_{0,0}$ (see text). Inset: correlations of
$\psi_{0}$ decay algebraically below the mass superfluid transition
($G_{0}\sim r^{-\eta_{n}}$), while correlations of $\mathbf{F}_{\perp}$ decay
algebraically below the spin superfluid transition ($G_{\perp}\sim
r^{-\eta_{s}}$). Red lines are fits. (b), (c) Transverse (circles) and
longitudinal (unfilled squares) components of $\lim_{k\rightarrow
0}\chi_{m,m^{\prime}}(\mathbf{k})$. Superfluid drag $\rho_{0,1}$
$(\rho_{-1,1})$ appears below $\mathcal{T}_{n}$ $(\mathcal{T}_{s})$. The
analytic zero-temperature $\chi_{m,m^{\prime}}^{L}(0)$ is indicated by filled
squares. [Results for $q=0.03\mu\approx 0.15q_{0}$, $N=256$.]

In Fig. 1 we plot the mass and spin superfluid densities determined from Eq.
(3). Clearly evident are the two distinct BKT temperatures $\mathcal{T}_{n}$
and $\mathcal{T}_{s}$ associated with the onset of mass and spin
superfluidity, respectively. Below the mass (spin) BKT temperature, two-point
correlations of $\psi_{0}$ ($\mathbf{F}_{\perp}$) change from decaying
exponentially to decaying algebraically,

$\begin{split}G_{0}(r)=\;&\frac{\langle\psi_{0}(\mathbf{0})^{\dagger}\psi_{0}(\mathbf{r})\rangle}{\langle\psi_{0}(\mathbf{0})^{\dagger}\psi_{0}(\mathbf{0})\rangle}\sim
r^{-\eta_{n}}\hskip 14.22636pt(\mathcal{T}\leq\mathcal{T}_{n}),\\\
G_{\perp}(r)=\;&\frac{\langle\mathbf{F}_{\perp}(\mathbf{0})\cdot\mathbf{F}_{\perp}(\mathbf{r})\rangle}{\langle\mathbf{F}_{\perp}(\mathbf{0})\cdot\mathbf{F}_{\perp}(\mathbf{0})\rangle}\sim
r^{-\eta_{s}}\hskip 14.22636pt(\mathcal{T}\leq\mathcal{T}_{s}),\end{split}$
(4)

see inset to Fig. 1(a). The mass (spin) superfluid densities estimated from
the decay exponents $\eta_{n}$ ($\eta_{s}$), via
$\rho_{n,s}=Mk_{B}T/2\pi\hbar^{2}\eta_{n,s}$ [30], show excellent agreement
with those determined from Eq. (3), see Fig. 1. Precisely at the respective
BKT temperatures the exponents take on a universal value of $1/4$ [31]; hence
we identify $\mathcal{T}_{n}$ and $\mathcal{T}_{s}$ in Fig. 1 as the
temperatures where $\eta_{n,s}=1/4$.

_Superfluid drag._ We can also examine the more general response functions

$\chi_{m,m^{\prime}}(\mathbf{k})=\frac{M}{k_{B}T}\int
d^{2}\mathbf{r}e^{-i\mathbf{k}\cdot\mathbf{r}}\langle\mathbf{J}_{m}(\mathbf{0})\mathbf{J}_{m^{\prime}}(\mathbf{r})\rangle,$
(5)

with
$\mathbf{J}_{m}=(\hbar/M)\operatorname{Im}(\psi_{m}^{\dagger}\nabla\psi_{m})$.
Note $\chi_{m,m^{\prime}}=\chi_{m^{\prime},m}$ and, due to symmetry under
$m=1\leftrightarrow m=-1$, $\chi_{m,m^{\prime}}=\chi_{-m,-m^{\prime}}$.
Defining $\rho_{m,m^{\prime}}\equiv\lim_{k\rightarrow
0}[\chi_{m,m^{\prime}}^{L}(\mathbf{k})-\chi_{m,m^{\prime}}^{T}(\mathbf{k})]$,
with $\chi_{m,m^{\prime}}^{T}$ ($\chi_{m,m^{\prime}}^{L}$) the transverse
(longitudinal) components of $\chi_{m,m^{\prime}}$, the mass and spin
superfluid densities can be decomposed as
$\rho_{n}=\sum_{m,m^{\prime}}\rho_{m,m^{\prime}}$ and
$\rho_{s}=\sum_{m,m^{\prime}}mm^{\prime}\rho_{m,m^{\prime}}$. We find that
mass superfluidity is primarily determined from $\rho_{0,0}$ at the mass
transition, see Fig. 1(a). The transverse and longitudinal components of the
four independent $\chi_{m,m^{\prime}}$ are shown in Fig. 1(b) and Fig. 1(c).
Off-diagonal contributions $\rho_{m,m^{\prime}\neq m}$ indicate “superfluid
drag” between components $m$ and $m^{\prime}$ [55], see Fig. 1(c). In other
multicomponent systems, superfluid drag occurs via current-current coupling
and is known as the Andreev-Bashkin effect [56, 55], whereas in the spin-1
system it occurs due to inherent intercomponent interactions [38, 39, 40].

The zero-temperature $\chi_{m,m^{\prime}}^{L}(0)$ can be obtained analytically
as follows. Beginning with a stationary, uniform system, we impart current
$(n_{m}\hbar k_{m}/M)\hat{\mathbf{x}}$ into spin component $m$ (with
$n_{m}=|\psi_{m}|^{2}$). Due to intercomponent interactions, this will induce
currents $(n_{m^{\prime}}\hbar k_{m^{\prime}}/M)\hat{\mathbf{x}}$ in
components $m^{\prime}\neq m$. We obtain the $k_{m^{\prime}\neq m}/k_{m}$ by
minimizing the additional kinetic energy $\delta
E=(\hbar^{2}/2M)\sum_{m^{\prime}\neq m}n_{m^{\prime}}k_{m^{\prime}}^{2}$
subject to the constraint $k_{1}+k_{-1}-2k_{0}=0$ imposed by minimization of
the spin-interaction energy. Defining a wavenumber
$n_{m}k\equiv\sum_{m^{\prime}}n_{m^{\prime}}k_{m^{\prime}}$ and comparing with
the relation $n_{m}=\sum_{m^{\prime}}\chi_{m,m^{\prime}}^{L}(0)$ we surmise
$\chi_{m,m^{\prime}}^{L}(0)=n_{m^{\prime}}k_{m^{\prime}}/k$, where
$k_{m^{\prime}}/k$ has an implicit dependence on $m$. This is confirmed in
Fig. 1(b) and Fig. 1(c).

## Topological properties and confinement.

Figure 2: (a) Right axis: Free $\mathbf{F}_{\perp}$ vortices proliferate for
$\mathcal{T}>\mathcal{T}_{s}$ (red circles), which can be identified as PCVs
for $\mathcal{T}<\mathcal{T}_{n}$. Free $\psi_{0}$ vortices proliferate for
$\mathcal{T}>\mathcal{T}_{n}$ (black circles). Free
$\psi_{1}\psi_{-1}\psi_{0}^{*}\psi_{0}^{*}$ vortices proliferate for
$\mathcal{T}\gtrsim 0.3$ (blue circles), suggesting deconfinement of PCVs,
analogous to a color plasma. Left axis: coherence between
$\psi_{1}\psi_{0}^{*}$ and $\psi_{-1}^{*}\psi_{0}$, measured by
$\langle\cos(\theta_{1}+\theta_{-1}-2\theta_{0})\rangle$ (gray dashed line),
decreases with increasing temperature, along with a decrease of the “coupling
strength” $\langle|\psi_{1}\psi_{-1}||\psi_{0}|^{2}\rangle$ (gray dotted
line). (b) Profiles of $\theta_{1}+\theta_{-1}-2\theta_{0}$ at two
temperatures (inverted triangles in (a)). Unfilled (filled) circles denote
positive (negative) free $\psi_{1}\psi_{-1}\psi_{0}^{*}\psi_{0}^{*}$
circulations. [Results for $q=0.03\mu$, $N=256$.]

In the easy-plane phase the system supports two topologically distinct
vortices, associated with the $\mathrm{U}(1)$ and $\mathrm{SO}(2)$ symmetries
respectively. The destruction of mass superfluidity coincides with a
proliferation of free $\psi_{0}$ vortices, see Fig. 2(a), consistent with the
finding that $\rho_{n}\approx\rho_{0,0}$ close to the mass BKT transition
(Fig. 1). The destruction of spin superfluidity coincides with a proliferation
of free $\mathbf{F}_{\perp}$ vortices, see Fig. 2(a). Since $\psi_{0}$ is
coherent at this temperature, these circulations can be identified as PCVs,
rather than Mermin-Ho vortices [57, 58, 59]. [Free vortices are identified by
convolving the relevant field with a Gaussian filter before detecting
divergences in the vorticity field. We choose a filter of width
$5\sqrt{5}x_{\mu}$, which is on the order of the core size of a cold PCV [60],
with $\sqrt{5}x_{\mu}\approx\hbar/\sqrt{2M|g_{s}|n}$ the approximate spin
healing length.]

A single $\mathbf{F}_{\perp}$ vortex consists of equal and opposite phase
windings of $\psi_{1}\psi_{0}^{*}$ and $\psi_{-1}\psi_{0}^{*}$ bound by the
spin-exchange energy
$2g_{s}\operatorname{Re}\psi_{1}\psi_{-1}\psi_{0}^{*}\psi_{0}^{*}$ [57, 61].
This spin-exchange energy increases linearly with separation between the
$\psi_{\pm 1}\psi_{0}^{*}$ vortices [62], analogous to confinement in quantum
chromodynamics [63, 64, 65]. We find that the coherence between
$\psi_{1}\psi_{0}^{*}$ and $\psi_{-1}^{*}\psi_{0}$, measured by
$\langle\cos(\theta_{1}+\theta_{-1}-2\theta_{0})\rangle$ (with $\theta_{m}$
the phase of $\psi_{m}$), decreases with increasing temperature, see Fig.
2(a). Furthermore, there is a proliferation of free vortices in the quantity
$\psi_{1}\psi_{-1}\psi_{0}^{*}\psi_{0}^{*}$ for $\mathcal{T}\gtrsim 0.3$, see
Fig. 2(a),(b), indicating spatial separation of $\psi_{1}\psi_{0}^{*}$ and
$\psi_{-1}\psi_{0}^{*}$ vortices. This suggests possible PCV deconfinement,
analogous to deconfinement in a color plasma. In such a plasma, this is
enabled by a decrease in the strong-force coupling strength with increasing
temperature [66, 67]. Analogously, the spin-exchange “coupling strength”
$\langle|\psi_{1}\psi_{-1}||\psi_{0}|^{2}\rangle$ decreases with increasing
temperature, see Fig. 2(a). Ignoring fluctuations in $|\psi_{0}|$ and $n$,
this coupling strength is $\propto\sqrt{n^{2}-F_{z}^{2}}$ and hence diminishes
with increasing fluctuations of $F_{z}$ 333Note that in the quadratic regime,
fluctuations of $F_{z}$ increase with increasing transverse spin disorder,
since both are affected by the same Bogoliubov mode [78]..

## Spin/mass superfluid phase diagram.

Figure 3: (a) $(\mathcal{T},q)$ phase diagram showing mass and spin superfluid
transitions (solid black lines). The colormap gives the mass and spin
superfluid densities. The spin BKT temperature decreases linearly with
increasing $q$, approaching zero as $q\rightarrow q_{0}$ (grey dashed line is
$\propto 1-q/q_{0}$ with $q_{0}=2|g_{s}|\mu/g_{n}=0.2\mu$). Results for
$N=256$. (b) At $q=0$, the mass fluid response $\chi^{T}(k)$ (solid lines)
exhibits a system size dependence. For comparison, $\chi^{T}(k)$ at
$q=0.01\mu$ (dashed lines) converge for increasing $N$. (c) Correlations of
total spin, showing a cross-over from exponential to algebraic decay at
$\mathcal{T}\approx 0.3$ (system sizes as in (b)). Correlations decay close to
$r^{-1/2}$ at the cross-over temperature. A sampling time of $2.5\times
10^{6}\hbar/\mu$ is used for the $N=512$ results in (b) and (c) to avoid
autocorrelation.

The dependence of $\mathcal{T}_{n}$ and $\mathcal{T}_{s}$ on $q/\mu$ is shown
in Fig. 3(a). The linear dependence of $\mathcal{T}_{s}$ on $q$ follows from
evaluating the zero-point of the free energy $F=E-TS$ of a single PCV, which
gives $\mathcal{T}_{s}\propto 1-q/q_{0}$ [69, 16]. Here $S=2k_{B}\ln L$ and
$E=\frac{K}{2}\int_{\xi_{s}}^{L}r^{-2}\,d^{2}\mathbf{r}=\pi K\ln(L/\xi_{s})$
are the entropy and energy respectively of a single free (but confined) PCV,
with $L$ the system size and $K\approx\hbar^{2}(1-q/q_{0})\mu/2g_{n}M$ the
spin-wave stiffness. Applying the same free-energy argument to vortices in the
$m=0$ component would give $\mathcal{T}_{n}\propto 1+q/q_{0}$. While this
qualitatively captures the increase in $\mathcal{T}_{n}$ with $q/q_{0}$, the
linear behaviour holds only for large $q/q_{0}$. The spin superfluid
transition extrapolates to zero at $q=q_{0}$. At these low temperatures we
expect ordering behaviour to be affected by both quantum and thermal
fluctuations [70, 71, 72].

At $q=0$ the order parameter manifold is $\mathrm{SO}(3)$ [32, 2], combining
the symmetry of the full spin vector $\mathbf{F}=(F_{x},F_{y},F_{z})$ and
gauge symmetry into a single manifold, with vortices fundamentally distinct
from $U(1)$ systems [32]. The nature of superfluidity and the potential for
BKT transitions in $\mathrm{SO}(3)$ systems is not well understood [33, 34,
35, 36, 37, 25]. We find that the mass fluid response $\chi^{T}(k)$ exhibits a
system size dependence for long wavelengths, see Fig. 3(b). This suggests the
absence of a $q=0$ mass BKT transition in the thermodynamic limit, consistent
with the conclusion of [25]. Despite this, we still see a cross-over from
exponential to algebraic decay in correlations of total spin,

$G_{\mathbf{F}}(r)=\frac{\langle\mathbf{F}(\mathbf{0})\cdot\mathbf{F}(\mathbf{r})\rangle}{\langle\mathbf{F}(\mathbf{0})\cdot\mathbf{F}(\mathbf{0})\rangle},$
(6)

see Fig. 3(c). The cross-over temperature $\mathcal{T}\approx 0.3$ slowly
decreases with increasing system-size, and hence may go to zero in the
thermodynamic limit. In the finite-sized systems explored here,
$G_{\mathbf{F}}(r)$ decays close to $r^{-1/2}$ at the cross-over temperature,
rather than as $r^{-1/4}$ typical for $\mathrm{U}(1)$ systems. This is similar
to the scaling of low-temperature correlations in a finite-sized ferromagnetic
Heisenberg model in 2D [73]. Importantly, though, the order parameter manifold
of the Heisenberg model is $S^{2}$, not $\mathrm{SO}(3)$, and hence does not
support vortices [74]. A cross-over in the behaviour of correlations around a
well-defined temperature has been identified in a Heisenberg antiferromagnet
on a triangular lattice, which also has an $\mathrm{SO}(3)$ order parameter
manifold [36, 37, 75].

## Discussion.

In this work we identified two BKT transitions in the easy-plane phase of a
ferromagnetic spin-1 Bose gas. We characterised the transitions in terms of
relevant vortices and inter- and intra-component fluid responses, and
identified possible deconfinement of PCVs above the spin-superfluid
transition. At $q=0$ the fluid response exhibits a system size dependence,
however correlations of total spin still exhibit a cross-over from exponential
to algebraic decay. It would be interesting to analyse this behaviour for
vanishingly small $q$ and how this depends on system size. Extending our work,
one could explore the role of non-zero axial magnetization, which adjusts the
relative density of the $m=\pm 1$ spin components and modifies the nature of
the PCVs [60]. One could also explore the role of transverse trapping [76],
which may affect the mass and spin superfluidities differently.

Our work opens up the possibility to explore temperature quenches across the
spin BKT transition and to study nonequilibrium processes such as Kibble-Zurek
scaling and coarsening dynamics. Comparisons with the extensive work on zero-
temperature quenches to the easy-plane phase [6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17], particularly close to $q_{0}$, could illuminate the role of
quantum versus thermal fluctuations in the symmetry breaking and
nonequilibrium dynamics [70, 71, 72]. We have shown that the $m=\pm 1$ spin
components decohere for increasing temperature, indicating states beyond the
low-temperature $\mathrm{U}(1)\times\mathrm{SO}(2)$ manifold. For very high
temperatures we expect restoration to the full $\mathrm{SU}(3)$ manifold; our
work paves the way to explore how such a phase emerges.

## Acknowledgements.

A.J.G. acknowledges Tom Billam for useful discussions. This research was
supported by the Australian Research Council Centre of Excellence for
Engineered Quantum Systems (Project No. CE170100009) and the Australian
Research Council Centre of Excellence in Future Low-Energy Electronics
Technologies (Project No. CE170100039). P.B.B acknowledges support from the
Marsden Fund of the Royal Society of New Zealand. X.Y. acknowledges support
from the National Natural Science Foundation of China (Grant No. 12175215),
the National Key Research and Development Program of China (Grant No. 2022YFA
1405300) and NSAF (Grant No. U1930403). We acknowledge the use of New Zealand
eScience Infrastructure (NeSI) high performance computing facilities.

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