# 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 examples of topological vector spaces.

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### Two notions of boundedness in metrizable topological vector space [closed]

In a metrizable topological vector space X with the metric d, a subset A is said to be bounded if it can be absorbed by any neighbourhood of 0 and a subset A is said to be d-bounded if its diameter ...
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### Is each cometrizable space a subspace of a cometrizable topological group?

Following Gruenhage we call a topological space $X$ cometrizable if $X$ admits a weaker metrizable topology such that every point $x\in X$ has a (not necessarily open) neighborhood base consisting of ...
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### Spectral theory without topology

How much of spectral theory can be developed just working with vector spaces (finite or infinite dimensional) without referring to a choice of topology ? Something along these lines, for example: ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $|||f|||=\sup \limits_{e\in E,~e\ne 0} \frac{|\left&... 1answer 338 views ### Well definition of a function I've edited, just skip the first attempt and go to the second one. THE FRAMEWORK: let us consider a real topological vector space$V$. We denote with$\mathscr C_k(V)$the set of all continous ... 1answer 434 views ### Fixed point of a group action Let$\mathbb{R}^\infty$be the product of countably many real lines. Assume that a finitely generated group$\Gamma$acts on$\mathbb{R}^\infty$(linearly and continuously) and there is a nonempty ... 1answer 60 views ### Semi-embeddings and weak compactness Let$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}...
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, ...