Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 ...

1
vote
0answers
92 views

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. ...
3
votes
1answer
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, ...
0
votes
0answers
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 (...
0
votes
0answers
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 ...
3
votes
1answer
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 ...
1
vote
1answer
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 ...
3
votes
2answers
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 ...
5
votes
0answers
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 ...
6
votes
1answer
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 $X$ ...
1
vote
1answer
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\...
2
votes
0answers
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 ...
3
votes
1answer
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 ...
4
votes
1answer
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 ...
3
votes
1answer
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 ...
2
votes
1answer
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 ...
4
votes
1answer
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 ...
2
votes
1answer
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 (...
2
votes
1answer
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 ...
2
votes
1answer
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 ...
6
votes
5answers
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 ...
0
votes
0answers
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\...
11
votes
1answer
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 ...
3
votes
1answer
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 ...
1
vote
1answer
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 ...
10
votes
1answer
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 ...
4
votes
2answers
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 ...
2
votes
0answers
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 $\...
1
vote
0answers
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 ...
3
votes
1answer
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 ...
1
vote
1answer
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 ...
3
votes
1answer
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, ...
1
vote
1answer
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, ...
6
votes
1answer
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 ...
1
vote
2answers
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 ...
0
votes
1answer
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 ...
2
votes
2answers
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&...
6
votes
1answer
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 ...
4
votes
1answer
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 ...
0
votes
1answer
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}...
2
votes
1answer
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, ...
5
votes
3answers
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\...
3
votes
1answer
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 ...
3
votes
0answers
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 $X$ ...
3
votes
0answers
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 ...
4
votes
0answers
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 ...
3
votes
1answer
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 ...
3
votes
1answer
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 ...
4
votes
0answers
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 ...
1
vote
1answer
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 ...
7
votes
0answers
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 ...