# Tagged Questions

1answer
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### How to show $\mu_A(x)=\inf\{\alpha>0: \alpha^{-1} x\in A\}$ where $\mu_A(x)$ is the Minkowski functional of $A$?

Let $X$ be a $\mathbb K$-vector space ($\mathbb K=\mathbb R$ or $\mathbb C$) and suppose $A\subset X$ is convex (and absorbing). How to show $$\{x\in X: \mu_A(x)<1\}\subset A?$$ Above ...
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### When a locally convex space is metrizable

Let $X$ be a locally convex Hausdorff topological space, whose topology if generated by the countable family of seminorms $\{p_i:\space i\in\mathbb{N}\}$. I'd like to prove that $X$ is metrizable. So, ...
1answer
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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??
0answers
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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 ...
1answer
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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 ...
2answers
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### Topological Vector Space: $\dim V=n\implies V\cong\mathbb{K}^n$

Let $V$ be a finite dimensional Hausdorff topological vector space. Prove that it is is isomorphic to the Euclidean vector space of the same dimension: $$\dim V=n\implies V\cong\mathbb{K}^n$$
1answer
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### Balanced Core: Explicit Expression?

Denote the collection of all balanced subsets by: $\mathcal{B}:=\{B\subseteq X: B\text{ balanced}\}$ Since the union of arbitrary balanced sets is balanced we can form the balanced core of arbitrary ...
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### A vector space with topology generated by a family of typologies each makes it a topological vector space is a topological vector space

Let $V$ be a vector space, and let $(\mathcal F_ \alpha ) _{ \alpha \in A}$ be a family of topologies on V, each of which turning $V$ into a topological vector space. Let $\mathcal F$ be the vector ...
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### How do they call the topological tensor product that classifies operators from Hilbert space?

Let $V$ and $W$ be topological vector spaces. There are different ways to complete the tensor product $V \otimes W$, and the only ones that are usually discussed in introductory literature are the ...
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### Infinite Dimensional Vector Space: Finite Dim Subspace Closed and Nowhere Dense

Show that any finite-dimensional subspace $(S,\|\cdot\|)$ of an infinite-dimensional normed vector space $(V,\|\cdot\|)$ is closed and nowhere dense. Proof: Let $\{x^{(n)}\}_{n\geq1}$ be a ...
1answer
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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 ...
2answers
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### Why do we need dual space [closed]

In functional analysis there are many places where dual space is mentioned, but I still don't understand the real power of that concept. Why do we need the dual space?
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### Closed Monoidal Structures On The Category Of Complete Topological Vector Spaces

Context: The category of Banach spaces, with the projective tensor product is a closed monoidal category. Question 1: Is there a tensor product on the category of complete topological vector spaces, ...
1answer
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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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### Weak* continuity

Let $B$ be the open unit ball in $\mathbb{R}^2$ and $\mathcal{M}^+$ the set of nonnegative Radon measures on $B$ and $\mathcal{M}^2$ the set of $\mathbb{R}^2 \text{-valued}$ Radon measures on $B.$ I ...
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### Given a Banach space $X$, are weak$^*$ bounded subsets of the dual space $X '$ also strongly bounded (with respect to the usual norm in $X '$)?

Some related facts I already know: 1) In a Banach space $X$, weakly bounded sets are strongly bounded and vice-versa (Thm 3.18 - "Functional Analysis", Rudin); 2) From 1, it follows that my question ...
1answer
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### Is Reflexivity Necessary for the Weak and Weak* Topologies to Coincide?

Let $X$ be a normed vector space, not necessarily Banach. Suppose that $X$ is not reflexive, implying the existence of such $\varphi\in X^{**}$ ($X^{**}$ being the double dual of $X$) of that for any ...
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### Question on the proof that $C(\Omega)$ is a Frechet space

I am using Rudin's book on Functional Analysis. I am studying the proof that the space $C(\Omega)$ of continuous functions on an open set $\Omega \subseteq \mathbb{C}$ is a Frechet space. I ...
1answer
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### Proof details for the first theorem in functional analysis

The Proof details for the first theorem in functional analysis is given in this website: I have one questions for the proof of lemma 5 given here. (1) Why this claim hold? from equality $0+0=0$ ...
2answers
107 views