# Questions tagged [topological-vector-spaces]

A topological vector space is a vector space $V$ over a topological field $\mathbb{K}$ (typically $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$), together with a topology on $V$ such that vector addition and scalar multiplication are both continuous. Hilbert spaces and Banach spaces are ...

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### Mackey topology characterising property

Let $V$ be a topological $k$-vector space.
Let $V^{\star}$ denote the space of all linear functionals $V \rightarrow k$ and $V' \subset V^{\star}$ the subspace of all continuous linear functionals.
...

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69 views

### Openness of invertibility in Fréchet spaces for families parameterized by compact spaces

Consider the following setup. Let $K$ be a compact topological space, $X$ a Fréchet space and $T:K \times X \to X$ a continuous family of linear maps (i.e. $T$ is a continuous map and $T_k \equiv T(k, ...

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14 views

### Compactness if Emery (semi-martingale) topology

The set $\mathscr{S}$, of semi-martingales is a topological vector space under the Emery topology on the space of semi-martingales. There has been some recent research on closures in this topology (...

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85 views

### Gelfand-Pettis integral: what does it mean for a topological vector space to “admit a dual space?”

I am trying to understand more about the Gelfand-Pettis integral. From wikipedia:
What does it mean that $V$ "admits a dual space?" When $V$ is a Banach space, $V^{\ast}$ is taken to be the space ...

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63 views

### Is the compact-open topology on the dual of a separable Frechet space sequential?

Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the ...

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113 views

### The completeness of spaces of continuous functions with the compact-open topology

For a Tychonoff space $X$ let $C_k(X)$ denote the space of continuous real-valued functions on $X$, endowed with the compact-open topology.
Problem. Is the space $C_k(X)$ Polish if it is Polishable ...

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167 views

### On convergent sequences in locally convex topological vector spaces

Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and ...

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89 views

### References on categorical TVS theory

A survey by Castillo lists results on the category of Banach spaces and on Banach space constructions, such as:
Existence of limits in Banach spaces or suitable subcategories
Demonstrations of ...

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214 views

### Is restriction a closed map?

Originally asked on MSE.
Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ ...

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105 views

### Compactness of operators and norming sets

Originally asked on MSE.
Let $T$ be a linear map from a normed space $E$ into a Banach space $F$.
Let $D\subset \overline{B}_{F^{\ast}}$ be norming, i.e., there is $r>0$ such that $\sup\limits_{v\...

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56 views

### Smooth functions with values in bornological vector space

Let $U$ be an open set in $\mathbb{R}^n$ (or more generally, a manifold) and let $V$ be a separated bornological vector space. Do we have
$$C^\infty(U, V) \cong C^\infty(U) \,\hat{\otimes}\, V,$$
as ...

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102 views

### Is each closed subgroups of $\mathbb R^\omega$ isomorphic to a Tychonoff product of locally compact Abelian groups?

It is known that any closed linear subspace of $\mathbb R^\omega$ is topologically isomorphic to $\mathbb R^n$ for some $n\in\omega$.
Problem 1. Is each closed subgroup of $\mathbb Z^\omega$ (or ...

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209 views

### Approximation of the identity by finite range functions in topological vector spaces

Let $X$ be a topological vector space. Assume that there exists a sequence $\phi_n:X\to X$ of finite range measurable functions with $\lim\phi_n(x)=x$ for every $x\in X$. Can we concluded there exists ...

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139 views

### Are second-countable subsets of topological vector spaces metrizable?

Let $X$ be a topological vector space of size $\mathfrak{c}$. Assume that there exists a countable union $X=\cup X_n$ such that all subsets $X_n$'s are relatively second countable.
Q. Does there ...

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136 views

### Boundedness of Dirac deltas

Suppose that $X$ is a metric space and let $C_k(X)$ denote the space of real functions on $X$ with the topology of uniform convergence on compact sets. Then $C_k(X)$ is a topological vector space. Let ...

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195 views

### Separable Lindelöf locally convex spaces that are not second-countable

A Lindelöf space is a topological space in which every open cover has a countable subcover.
Does there exists a Lindelöf locally convex space which is not second countable?
I am also looking for a ...

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**1**answer

90 views

### Closure in the strong dual topology

Originally asked on MSE.
Let $E$ be a metrizable locally convex topological vector space and let $E^{*}$ be its dual space endowed with the strong topology = topology of uniform convergence on (...

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**1**answer

87 views

### Measurability of the product on particular topological vector spaces

Let $X$ be a topological vector space. Let us say that $X$ has property P if there exists a sequence of closed subsets $\{X_n\}$ such that
1- $X=\bigcup X_n$
2- The relative topology is both ...

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**1**answer

95 views

### On sections into Banach bundles over a compact manifold

Let $M$ be a smooth, compact manifold and $\xi: \mathcal B \to M$ a smooth complex Banach bundle over $M$. Here, smooth is understood to be in the Fréchet-sense. Further, let $p: V \to M$ be an ...

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911 views

### Topological vector space textbook with enough applications

(Sorry for my bad English.)
For "applications", I mean applications in math, not real-life.
There are many textbooks about topological vector space, for example, GTM269 by Osborne, Modern Methods in ...

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52 views

### Countable dense subset of functions of exponential type 1 that decay along the positive real axis

I am interested in the space of all holomorphic function of exponential type one, that decay exponentially along the positive real axis. I tried to define it as follows.
Let
$$\|f\|_n = \sup_{z\in\...

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148 views

### Bilinear product of two summable families

Consider the following statement, which I suspect is false as written:
Let $E,F,G$ be (Hausdorff) topological vector spaces (over $\mathbb{R}$), let $\varphi\colon E\times F\to G$ be continuous and ...

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127 views

### Recognizing locally convex spaces on which all bounded linear functionals are continuous

Is it possible to characterize the Hausdorff locally convex spaces on which all bounded linear functionals are continuous?
It is known that a space is bornological if and only if the space is Mackey ...

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88 views

### When is the strict topology bornological?

Let $X$ be a completely regular Hausdorff space. Are there known conditions under which the algebra of bounded continuous functions on $X$, endowed with the strict topology, is bornological?
(Of ...

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**1**answer

285 views

### Closed vector subspaces of large powers of R

By a large power of $\mathbb R$ is meant a topological vector space which is the product of infinitely many copies of the real line.
Is every closed subspace of such a TVS linearly homeomorphic to ...

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368 views

### Bohr compactification as a topological compactification

Let $G$ be a locally compact Hausdorff group. Denote its Bohr compactification by $bG$.
Despite group structure, $G$ has several (Hausdorff) compactifications that, in a sense, the smallest one is ...

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57 views

### An equivalent condition for second countable locally convex spaces

Let $(X,\tau)$ be a locally convex topological vector space.
Assume for any arbitrary topological base $\mathcal{E}$ of $\tau$ we have that: the Borel sigma algebras coming from $\mathcal{E}$ and $\...

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154 views

### Can a topological vector space be probabilistic metric space too? [closed]

Is it possible to have probabilistic metric space (S,F,T) be a topological vector space too?
In specific way, the probabilistic metric space is Menger and does not have a norm, however with Menger ...

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286 views

### References on topological rings

What is a good book on topological rings and modules?
I'm interested in topological rings and modules typically endowed with non-linear topologies, e.g.. non-linearly topologized normed rings.
I ...

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131 views

### Criterion of reflexivity 2

Originally I meant to ask this question here, but got confused and ended up asking another question, which had some mathematical meaning, but was not what I vaguely had in mind.
Let me restate the ...

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103 views

### When are the closed convex subsets countable intersections of halfspaces

For what kind of topological vector spaces (separable maybe?) are the closed convex subsets countable intersections of halfspaces.
I've seen somewhere that it's true for separable Hilbert spaces, ...

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147 views

### Criterion of reflexivity

Let $E$ be a Banach space.
It is known that if for any equivalent norm on $E^*$ the closed unit ball of $E^*$ is weakly* closed, then $E$ is reflexive (a very short proof is in the book by Fabian, ...

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176 views

### Is any dual metrizable locally convex space a Frechet space?

[I have posted this question on MSE some time ago, but received no answer.]
The title basically says all of it.
If a normed space $F$ is a dual of a normed space $E$, then $F$ is a Banach space. I ...

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78 views

### Topologies of pairs and closed bounded convex sets

[I have posted this question on MSE some time ago, but received no answer.]
It is known, that if two locally convex topologies on a vector space determine the same collection of continuous linear ...

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**1**answer

111 views

### Criterion for weak compactness

Let $F$ be a metrizable locally convex space (you may assume it is a Banach space), and let $E$ be a complete locally convex space (you may assume it is a Frechet space). Let $T$ be a continuous ...

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189 views

### A criterion for norming sets

Let $F$ be a Banach space with the closed unit ball $B$. Let $E\subset F^*$ be a total subspace such that $B$ is complete with respect to the norm $|||f|||=\sup \limits_{e\in E,~e\ne 0} \frac{|\left&...

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334 views

### Well definition of a function

I've edited, just skip the first attempt and go to the second one.
THE FRAMEWORK: let us consider a real topological vector space $V$.
We denote with $\mathscr C_k(V)$ the set of all continous ...

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414 views

### Fixed point of a group action

Let $\mathbb{R}^\infty$ be the product of countably many real lines.
Assume that a finitely generated group $\Gamma$ acts on $\mathbb{R}^\infty$ (linearly and continuously) and there is a nonempty ...

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**1**answer

60 views

### Semi-embeddings and weak compactness

Let $F$ and $H$ be normed spaces and let $E$ be a locally convex space.
Let $T:F\to H$ and $S:H\to E$ be linear operators, such that $\|T\|= 1$, $S$ is an injective semi-embedding (i.e. $S\overline{B}...

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**1**answer

203 views

### Why is an inductive limit of bornological spaces bornological?

Let $(E_\alpha,\tau_\alpha,g_\alpha)$ be a family of bornological (locally convex) topological vector spaces $(E_\alpha,\tau_\alpha)$, where a LCTVS $E$ is said to be bornological if every circled, ...

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318 views

### Proof of the Schauder Lemma

Schauder's Lemma in functional analysis states the following:
Let $E$ and $F$ be metrizable locally convex topological vector spaces, and let $E$ be Fréchet. Then if the linear continuous map $A:E\...

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67 views

### DF-algebras and DF-modules

Recall Lemma 0.5.1 from the Helemskii's monograph "The homology of Banach and Topological Algebras":
$\textbf{Lemma}$ Let $\phi\colon X\to Y$ be an injective map between Banach spaces with dense ...

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134 views

### Topology of the Hamel basis in a TVS

Let $V$ be a complex topological vector space, and let $I$ be a Hamel basis of it. Then as a subset $I\subset V$ acquires an induced topology, becoming a topological space. For a topological space $X$ ...

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123 views

### Commutative discrete cyclic operator groups on topological vector spaces

Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...

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120 views

### A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff

We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...

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137 views

### The sheaf of generalized functions on compact subsets

For $K\subseteq \mathbb{R}^d$ compact, let $C_{\mathrm{c}}^{\infty}(K)$ denote the space of smooth functions on (an open neighborhood of) $K$ with compact support contained in $K$ with the usual ...

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88 views

### DF-spaces and F spaces

It is well known that when $E$ is a $DF$-space and $F$ is a Fréchet space, the space $\mathcal{L}_{b} (E,F)$ is Fréchet. The converse, that is the fact that $\mathcal{L}_{b} (F,E)$ would be $DF$, is ...

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87 views

### On the embedding of manifolds into infinite-dimensional spaces

Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...

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340 views

### Closure of the interior of a convex set in a topological vector space

Let $C$ be a convex set with nonempty interior in a topological vector space. Do we always have :
$\overline {C^\circ} = \overline{C}$ ?
If not, what is the "minimal" condition on the space so that ...

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257 views

### When is the sigma-algebra generated by closed convex sets the same as the Borel sigma-algebra

For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ?
More precisely, do we have ...