1
vote
2answers
57 views

Counterexample for Mazur–Ulam theorem

We know, Mazur–Ulam theorem states that if $V$ and $W$ are normed spaces over $\mathbb{R}$ and the mapping $f\colon V\to W$ is a surjective isometry, then $f$ is affine. Can somebody say ...
2
votes
1answer
41 views

For an inductive limit $X = \bigcup X_n$ of vector spaces, show that $X$ is complete if $X_n$ is complete for all $n$

Let $X$ be a vector space. Suppose that $\{X_n\}_{n=1}^\infty$ is a sequence of vector subspaces such that $X_n \subseteq X_{n+1}$ for all $n$, Each $X_n$ is a locally convex topological vector ...
1
vote
1answer
28 views

Question about the basis for the topology on an inductive limit of Frechet spaces

Let $X$ be a complex vector space, and let $\{X_n\}_{n=1}^\infty$ be a sequence of vector subspaces such that $X_n \subseteq X_{n+1}$ for all $n$ and $X = \bigcup_n X_n$. Furthermore, suppose that: ...
1
vote
0answers
51 views

Can complex-valued affine function be approximated?

If $K$ is a compact convex set of a locally convex Hausdorff space $V$ over $\mathbb{R}$ and $A(K)$ is the set of all continuous affine real-valued function on $K$, then the set of all restrictions to ...
3
votes
1answer
59 views

Absorbing at Each point means Open?

In a topological vector space, it seems that open sets have the property that they are absorbing at each of their points, since $(\alpha,x)\to \alpha x$ is continuous. I am wondering if the converse ...
2
votes
1answer
29 views

How to construct a particular convex set (when defining inductive limits of Frechet spaces in Reed and Simon)

Let $X$ be a topological vector space (over $\mathbb{C}$) whose topology is defined by a family of separating seminorms $\{\rho_\alpha\}$. Let $X_1$ be a vector subspace of $X$ whose topology is the ...
1
vote
0answers
25 views

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 ...
1
vote
1answer
15 views

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 ...
2
votes
0answers
17 views

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, ...
0
votes
1answer
53 views

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??
5
votes
0answers
102 views

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 ...
0
votes
1answer
38 views

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 ...
1
vote
2answers
89 views

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: $$\dim V=n\implies V\cong\mathbb{K}^n$$
1
vote
1answer
22 views

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 ...
2
votes
0answers
25 views

Criteria for a sequence in $C[0,1]$ to converge weakly [duplicate]

Is there a criterion for a sequence in $C[0,1]$ to converge weakly? Let $\{f_n\}$ be a sequence in $C[0,1]$, $f\in C[0,1]$. Suppose $f_n\to f$ (weakly). Then for each $x\in [0,1]$ $ev_x(f_n)\to ...
0
votes
1answer
24 views

Various convergences in the space of bounded operators

Could you please help me to find some classical (counter)examples in functional analysis? Let $X$ and $Y$ be some normed spaces over $\mathbb{C}$. By $\mathcal{B}(X,Y)$ we denote the space of bounded ...
2
votes
1answer
22 views

Closure of a von Neumann bounded set

Let $V$ be a topological vector space and $B \subseteq V$ bounded. Then the closure $\overline{B}$ is bounded. This appears on the Wikipedia page ...
2
votes
1answer
42 views

Please verify this definition of locally convex vector space

A locally convex vector space is a vector space $V$ equipped with a family $P$ of separating semi-norms. Is this a correct definiton? So to determine whether a norm induces a locally convex topology ...
0
votes
0answers
20 views

Hausdorff topologies on a finite dimensional vector spaces

Let $X$ be a finite dimensional vector space (over complex numbers), $\mathcal{P}_1$,$\mathcal{P}_2$ - two families of seminorms. There is the following statement: these families are equivalent if the ...
0
votes
1answer
58 views

Does the continuity at $0$ of the addition map in a vector space imply its continuity?

I have a question about the proof of Theorem 1.41 in Rudin, Functional Analysis, 2/e. The theorem states Let $N$ be a closed subspace of a topological vector space (t.v.s.) $X$. Let $\tau$ be the ...
1
vote
0answers
49 views

Open Convex Subsets of Dense Spaces

So I asked this question yesterday, Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric, and it made my start wondering the following... Suppose the following: ...
2
votes
0answers
15 views

Topological modules with enough continuous linear functionals.

Context: I'm trying to find out which topological (unital) modules are "good enough" for generalizing results from real or complex functional analysis. For example, I say that a module, in order to be ...
1
vote
1answer
41 views

Non-locally convex topologies on $\mathbb{R}^{n}$ compatible with the vector space structure

So I know that every locally convex topology on $\mathbb{R}^{n}$ is equivalent to the norm topology. Are there any non-trivial examples of non-locally convex topologies on $\mathbb{R}^{n}$ that still ...
2
votes
1answer
57 views

Counterexample about non Hausdorff topological vector spaces

I have some troubles with Hausdorffness in TVS: Question 1. Is there any topological vector space $X$ which is not Hausdorff? Question 2. Give an explicit example of a topological vector space $X$ ...
0
votes
0answers
48 views

topological vector space of measure functions

Let $(X, \mathcal X, \mu )$ be a measure space, and let $ L(X)$ be the space of measurable functions $f: X \to \mathbb C$. Show that the sets $B(f, \epsilon ,r ): = \{ g \in L(X) : \mu( \{ x : | f(x) ...
0
votes
0answers
35 views

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 ...
4
votes
0answers
38 views

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 ...
0
votes
2answers
162 views

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 ...
4
votes
1answer
112 views

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 ...
6
votes
2answers
152 views

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?
5
votes
0answers
33 views

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, ...
2
votes
1answer
39 views

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 ...
3
votes
1answer
100 views

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 ...
3
votes
1answer
73 views

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 ...
6
votes
1answer
179 views

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 ...
3
votes
1answer
63 views

Extending continuous, densely-defined linear maps between locally convex spaces

Let $X$ and $Y$ be locally convex topological vector spaces, say over $\mathbb{C}$. To set the stage a bit, I'll say that the topology on $X$ is given by a separating family of semi-norms $(p_i)_{i ...
1
vote
1answer
49 views

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 ...
2
votes
0answers
49 views
0
votes
2answers
79 views

Compactness in Infinite Dimensional Vector Spaces

Show that, in an infinite dimensional normed space $(V,\|\cdot\|)$, the closed ball of radius $2$ $$ B_2:=\{x\in V:\ \|x\|\leq2\} $$ is not compact. I suspect I am not understanding what is going ...
1
vote
1answer
50 views

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 ...
2
votes
2answers
87 views

Convex Sets in Functional Analysis?

Why is it that convex sets and convex functions are a) so important & b) so intrinsically related to functional analysis as to deserve an entire chapter in Bourbaki's topological vector spaces? ...
0
votes
0answers
17 views

A question on a function of bounded semivariation(part 2)

Let $(X,\tau)$ be a Hausdorff locally convex topological vector space and let $P(X)$ be a family of continuous seminorms on $X$ that generates the topology $\tau$. I got the following definition from ...
1
vote
1answer
18 views

Is a function of bounded semivariation bounded?

Let $X$ be a topological vector space and let $f:[a,b]\to X$. We say that $f$ is of bounded semi-variation in $[a,b]$ if the set $SV(f,[a,b])$ consisting of all the elements of the form $$\sum_{i=1}^n ...
1
vote
0answers
83 views

Topology of pointwise convergence - open sets

Let $X$ be the vector space of all complex functions on $[0,1]$, topologized by the family of seminorms $p_{x}(f)=|f(x)|$, $0\le x\le1$. This topology is called the topology of pointwise convergence. ...
1
vote
1answer
59 views

A question on a function of bounded semivariation

Let $(X,\tau)$ be a Hausdorff locally convex TVS and let $P(X)$ be a family of continuous seminorms on $X$ that generates the topology $\tau$. I got the following definition from one of the articles ...
2
votes
1answer
99 views

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 ...
0
votes
1answer
138 views

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 ...
3
votes
1answer
96 views

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 ...
1
vote
1answer
47 views

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 ...
4
votes
1answer
35 views

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 ...