# Tagged Questions

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### Sum of Neighborhoods of Zero

When do two neighborhoods of zero over a topological vector space add up as: $$aN+bN=(a+b)N\quad a,b\geq 0$$ I could imagine something like balanced might suffice... The problem is that I'd like to ...
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### Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
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### Completation of an n.v.s. and dimensions of subspaces.

I don't know if the following statement is true: Let $X$ be an n.v.s. with $\text{dim}(X)=\infty$ and not Banach; and $\bar X$ its completation in the bidual space. Let $Y$ be a closed subspace ...
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### Linear bijection non-preserving Hausdorff propery

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more ...
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### Almost Everywhere Function Space

Problem Let $\Omega$ be a measure space with measure $\mu$ and $V$ a topological vector space not necessarily Hausdorff as well as the function space $\mathcal{F}:=\{f:\Omega\to V\}$ topologized by ...
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### Topology Book including specific aspects

I am looking for a basic book about Topology (maybe also a bit of Functional analysis but basically Topology) including the following points (in addition to the basic points): $\bullet$ Seminorms ...
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### Topological Tensor Product is a Topological Ring Independent of the Choice of Basis

Let $A, B$ be commutative rings containing a field $k$, with $B$ a finite dimensional $k$-module, $w_1, ... , w_N$ a basis. If $w_iw_j = \sum\limits_{n=1}^N c_{ijn}w_n$, then we can define $C$ to be ...
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### Balanced Core: $U\text{ open }\implies U^*\text{ open}$

I need one last lemma for the proof of finite dimensional subspaces are closed: Is it true that if a subset is open so is its balanced core??
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### Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
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### Topological Vector Space: Uniform Structure

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! Prove that any topological vector space gives rise ...
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### Infinite Dimensional Topological Vector Space

Let $V$ is a finite-dimensional vector space over $\mathbb{R}$ (or $\mathbb{C}$). To make $V$ a topological space, we may choose the sets $f^{-1}(U)$ as a sub-basis, where $f$ ranges over all linear ...
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### Show that no topological vector space is bounded.

I am studying the concept of topological vector spaces in Grubb's Distributions and Operators. A vector space $X$ (over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$) is called a topological vector ...
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### When a sort of weak topology is enough to generate vector space topology

Consider a vector space $V$, and some functions $f_\alpha: V \rightarrow \mathbb{C}$ where $\alpha$ ranges over some index set $A$. We can think about the coarsest topology which: a) makes the ...
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### Properties of compact set: non-empty intersection of any system of closed subsets with finite intersection property

Let $X$ be a Hausdorff topological vector space. Let $C$ be a nonempty compact subset of $X$ and $\{C_\alpha\}_{\alpha \in I}$ be a collection of closed subsets such that $C_\alpha \subset C$ for each ...
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### strong topology = inductive limit topology on duals of projective limits

I've been bothering with this for some time now, and can't find any source with an actual proof, the statement simply appears to be "well-known". If you know (a source with) a proof, I'd be happy :) ...
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### Are the axioms of a topological space superfluous?

A topology on a set $X$ is a family $\mathcal{T}$ of subsets of $X$, which are open sets and satisfy: (1) $\emptyset, X \in \mathcal{T}$. (2) Any union of elements of $\mathcal{T}$ belongs to ...
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### Every Bounded set contained in a Compact set

In a general metric space, is every bounded set contained in a compact set?
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### Prove that the weak$^*$ topology on the space of tempered distributions is not 1st countable

Please, help me with a proof of this (apparently) known fact whose proof is out of my reach, even though I spent a considerable amount of time looking it up: The weak$^*$ topology on the space of ...
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### When does a dense subspace destine the weak topology?

Let $E$ be a locally convex space, let $E^{\prime}$ be its continuous dual space and let $F$ be a subspace of $E^{\prime}$ which is dense with respect to the strong topology on $E^{\prime}$ (i.e. the ...
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### Prove that a set is dense and of the first category in $L^2(T)$

Define the Fourier coefficients $\hat{f}(n)$ of a function $f\in L^2(T)$, $T$ is a unit circle, by: $\hat{f}(n)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(e^{i\theta})e^{-in\theta}d\theta$ Let ...
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### How to prove in a topological vector space in R^n: cl(A) + cl(B) =cl(clA+clB), where cl denotes closure and A, B compact convex sets?

Let A,B be compact convex sets, then Is equal to cl(A+B)=cl(cl(A)+cl(B))? why? Thank you.
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### About the closedness of Banach space

I'm reading a proof in trying to prove that a Banach space $X$ is reflexive if and only if $X^{*}$ is reflexive. There's a point in the proof saying that $X$ is a closed subspace of $X^{**}$, but I ...
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### About open mapping and closed range theorem

I'm self-learning Functional Analysis in Rudin's book and found some following statement hard to understand. Hope someone can help me clarify this. 1) $X, Y$ are Banach spaces, $T \in B(X,Y)$, let ...
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### No infinite-dimensional $F$-space has a countable Hamel basis

If $X$ is an infinite-dimensional topological vector space which is the union of countably many finite-dimensional subspaces, prove that $X$ is of the first category in itself. Prove that therefore ...
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### Subspaces of a Topological Vector Spaces

I have a few questions about topological spaces which I am currently studying. First some definitions that I am using: Definition of subspace topology: Given a topological space $(X,\tau)$ and a ...
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### In the proof the proving duality of topological vector space

I find hard to understand the proof for this theorem. Hope that some one can help me to clarify this. Thanks Suppose $X$ is a vector space and $X'$ is a separating space of linear functionals on ...
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### About the continuity of a function in the closed graph theorem proof

I'm reading Functional Analysis book of Rudin, and in the proof of the closed graph theorem, there's one point that I don't understand. Can someone please explain it to me? I really appreciate this. ...
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### Question about the continuity property of a function in topological vector space

I'm reading Functional Analysis book of Walter Rudin, and there's one point in this book that I don't know why he states that. Here is the statement: $f$ is a linear mapping from F-space $X$ into ...
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### There exists function sequence $\{f_{n}\}$ converges to $0$ such that $\{a_{n}f_{n}\}$ not converges to $0$

Let $X$ be the vector space of all complex functions on the unit interval $[0,1]$, topologized by the family of seminorms $$p_{x}(f) = |f(x)|, \quad (0 \le x \le 1).$$ Show that there exists a ...
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### In a topological vector space, show if $A$ and $B$ are bounded, then $A + B$ is bounded?

I get as far as this before I am stuck: Pick any neighbourhood of $0$ and call it $U$. Then there exists $a, b$ such that $A \subseteq aU$ and $B \subseteq bU$. So hence $A + B \subseteq aU+bU$. ...
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### Compactness in a new convexity

Let $V$ be a topological vector space. For any $m,n\in\mathbb{N}$, denote by $M_{m,n}(V)$ the vector space of all $m\times n$ matrices with entries in $V$. In particular, we denote ...
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### Countable subsets of TVSs

This is something which is not clear to me. Take any countable subset $C$ of a compact set $K$ in a locally convex topological vector space $X$. Can we conclude that there is a point $x\in X$ such ...
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### A separable, regular space which has cardinality of the continuum but is not first countable?

Actually the title says it all. Is there such a topological space which is separable, regular, has cardinality of the continuum but is not first countable? If so, is there also an example of a ...
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### Homeomorphism via Minkowski functional?

Suppose $E$ is an infinite dimensional topological vector space and $\Omega\subset E$ is open, convex and $0\in \Omega$. The Minkowski-functional of $\Omega$ is defined by:  p_\Omega:E\to ...
Suppose that $\Omega$ is a convex open subset of an infinite dimensional vector space $E$ such that $\Omega$ is not contained in any finite dimensional subspace of $E$. Let $Q_m\subset \Omega$ denote ...
Let $\mathcal{H}_n \mathbb{C}$ be the set of hermitian $n \times n$ complex matrices. This set carries the structure of a vector space over $\mathbb{R}$ under usual addition. It also inherits the ...