The study of vector space with a topology which makes the maps which sums two vectors and which multiply a vector by a scalar continuous. It's a natural generalization of normed spaces.

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2answers
77 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? ...
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1answer
57 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 ...
0
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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 ...
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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 ...
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1answer
236 views

closed subspace of normed vector space

Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.
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0answers
75 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. ...
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1answer
65 views

Does a barrel contain a neighborhood of $0$? [closed]

Suppose $X$ is topological vector space which is of the second category in itself. Let $K$ be a closed, convex, absorbing subset (a barrel) of $X$. Prove that $K$ contains a neighborhood of $0$.
2
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1answer
67 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 ...
7
votes
1answer
244 views

Strong topology vs Natural topology

Let $X$ be a locally convex space and $\left< X, X^{\prime} \right>$ stands for the dual pair. The bidual of $X$ is denoted by $X^{\prime \prime}$ and this is a dual of $X^{\prime}$ with a ...
3
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1answer
151 views

Locally convex space characterization in terms of duality

Let $(X,\tau)$ be a topological vector space (TVS for short). Denote by $X^*$ the topological dual of $(X,\tau)$. If there exists a locally convex topology $\mu$ on $X$ compatible with the duality ...
2
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0answers
69 views

Pseudo norm-exercice

Let $f$ be a measurable function with finite values almost everywhere. We put $$N_0(f) = \displaystyle\int \dfrac{|f|}{1 + |f|} d \mu.$$ We denoted by $L^0$ the set of measurable functions $f$ such ...
0
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1answer
91 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
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1answer
84 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 ...
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1answer
44 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 ...
1
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1answer
42 views

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. ...
4
votes
1answer
32 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 ...
2
votes
1answer
73 views

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 ...
10
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1answer
1k views

If $A$ and $B$ are compact, then so is $A+B$.

This is an exercise in Chapter 1 from Rudin's Functional Analysis. Prove the following: Let $X$ be a topological vector space. If $A$ and $B$ are compact subsets of $X$, so is $A+B$. My guess: ...
2
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1answer
38 views

Tempered fundamental solutions

According to the Malgrange–Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a ...
2
votes
1answer
61 views

Measure-valued maps equal a.e. if their integrals equal over any set

Let $X,Y$ be two standard Borel spaces and let $p,q:X\to\mathcal P(Y)$ be two stochastic kernels, which can be alternatively seen as measure-valued maps. Suppose that for some measure $\mu\in \mathcal ...
0
votes
2answers
69 views

Example of a sequence of functions

Construct an example of a sequence of functions $(f_n)$ defined on $[0,1]$ such that $f_n$ converges pointwise to $0$ and for every sequence of numbers $(a_n)$ that tends to $\infty$, sequence ...
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1answer
65 views

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 ...
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1answer
95 views

is the vector space $\mathbb{R}^\mathbb{N}$ locally compact?

is the vector space $\mathbb{R}^\mathbb{N}$ locally compact? for example, let $x=(x_1,x_2,....)$ any point of $\mathbb{R}^\mathbb{N}$ and let $V=[x_1-\epsilon,x_1+\epsilon] \times ...
0
votes
1answer
58 views

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 $0+0=0$ ...
2
votes
2answers
249 views

First theorem in Topological vector spaces.

I came across this theorem and I am disappointed not being able to understand or to have intuition to understand it . I would be glad to get help . Theorem : If $K$ and $C$ are subset of topological ...
3
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1answer
70 views

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$. ...
2
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2answers
415 views

projection operators on topological vector spaces

Suppose $A \in \mathbb{R}^{m\times n}$. Then there exists a projection matrix $P$ onto the range of $A$. In other words, there exists a matrix $P \in \mathbb{R}^{m\times m}$ such that $P^2=P$, and ...
3
votes
1answer
59 views

Constructing a closed, convex subset of $X^{\ast}$ that is not weakly-* closed

I'm asked to show that if $X$ is a non-reflexive Banach space, there exists (norm) closed and convex subsets of $X^\ast$ that are not $w^{\ast}$-closed. In other words, there's no analogue of Mazur's ...
2
votes
1answer
152 views

Why are the Differential- and multiplication mapping on $C^{\infty}(\Omega)$ continuous?

Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out ...
2
votes
1answer
484 views

Sum of closed and compact set in a TVS

I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
0
votes
1answer
59 views

How can we ensure that a space is a subset of locally convex topological space?

I am looking for fast ways to ensure that a given set is a subset of topologically locally convex space. I have already read the posts post1:seminorms-in-locally-convex-spaces, ...
3
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0answers
52 views

How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is ...
2
votes
1answer
39 views

Notions of topological spaces and matrices

How can I show that: For all $r>0$, exists $A$ nonsingular matrix, such that $B_r(A)\subseteq GL_n(\mathbb{C})$ For all $A\in GL_n(\mathbb{C})$ and for all $r>0$, there exists $\alpha>0$ ...
3
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0answers
40 views

Every locally connected non separating plane continuum is an AR

A continuum is a compact connected metric space. A plane continuum is a continuum contained in $R^2$. We say that a plane continuum does not separate the plane provided that $R^2\setminus X$ is ...
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2answers
20 views

if $v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is member of $K$. How is this possible?

Let $(V,K)$ and $u,v$ is a member of $V$. Suppose that $M$ is a subset of $V$ is a subspace of $V$ with basis $B_m=\{m_1,...,m_r\}$ with $r$ less than and equal to $n$. Let $H$ be a subspace spanned ...
5
votes
1answer
86 views

Dual of a topological vector space. Is it nontrivial?

In the case of normed spaces we know their duals are nonempty using a quick application of the Hahn Banach Theorem. If we step back to the larger class of locally convex spaces, an enthralling ...
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0answers
14 views

Seminorm topology on $C^{\infty}(\Omega)$ and proof of continuity [duplicate]

Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out ...
0
votes
0answers
25 views

On function of bounded semi-variation

The following ideas were taken from the article of DUCHON, entitled "On Vector Measures and Distributions." Definition. Let $X$ be a locally convex space whose topology is generated by a family $P$ ...
3
votes
1answer
242 views

proving that $SO(n)$ is path connected

Our professor gave us exercise to show that $G=SO(n,\mathbb R)$ is path connected. He gave some hints, using them I have come upto this far: I have shown that $SO(n)$ acts on $S^{n-1}$ transitively ...
0
votes
1answer
96 views

Equicontinuous sequence of linear maps and a closed subspace

I'm having some difficulty with a homework problem regarding Exercise #14 from Chapter 2 of Rudin's $\textit{Functional Analysis}$. (a) Suppose $X,Y$ are topological vector spaces, $\{f_n\}$ is an ...
0
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0answers
51 views

An Extended Definition of Function of Bounded Variation

I am trying to formulate a definition of function of bounded variation in the setting of locally convex spaces. This is what I've tried. Definition. Let $X$ be a locally convex space whose topology ...
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0answers
47 views

Differential Operators over the space of Analytic Functions

Let $\mathcal{A}(-a,a)$ be the vector space of functions that are analytic on the interval $(-a,a)$ Is there a common topology to place on this space, if yes what is the topology and is it induced ...
3
votes
1answer
215 views

About Henstock integrable vector-valued function

In what follows, $X$ is a Hausdorff locally convex topological vector space over the reals whose topology is generated by a family $P$ of all continuous seminorms on $X$. We consider the following ...
3
votes
1answer
69 views

What norm makes $C^\infty[a,b]$ a complete space?

I have been searching for some common norms used on vector spaces of functions but I am not having any luck finding what the most common norm is to use on $C^\infty[a,b]$ More specifically I would ...
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votes
1answer
50 views

Are all metric space as a euclidean space?

I believe that all euclidean space is a metric space? But I need to know about inverse? I mean: are all metric space as a euclidean space? Is there any kind of metric space which is not euclidean ...
2
votes
1answer
93 views

Topological vector space with discrete topology is the zero space

Hello i have a question about topological vector spaces. To remind the definition of such a space: A topological vector space is a pair $(X,\tau)$ with $X$ a vector space and $\tau$ a topology on ...
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1answer
114 views

Exercise involving topological vector spaces, linear maps, and the quotient map

I'm doing a homework problem out of Rudin's $\textit{Functional Analysis}$ which is basically a proof of which I have completed some of it, but I'm not sure about the rest of it. Without further ado, ...
1
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1answer
55 views

Some properties about $L^p$ with $0<p<1$

We are coming across many Banach spaces $L^p$ with $1\leq p\leq\infty$. But how about $0<p<1$? Can it be normed? How about its metric induced by the norm? And how about its ...
4
votes
1answer
218 views

Confused by proof in Rudin Functional Analysis, metrization of topological vector space with countable local base

I'm working through Rudin's Functional Analysis, and I am confused by a step in his proof for Theorem 1.24, which states that if X is a topological vector space with a countable local base, then there ...
1
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1answer
136 views

The topological vector space that is not metrizable.

Let $C_0(\mathbb{R})$ denote the vector space of continuous functions on the real line with compact support. For any positive function $\rho$ let $$||f||_{\rho}:=\sup_x\rho(x)|f(x)| \ \ .$$ 1) I could ...