Questions tagged [topological-vector-spaces]

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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Properties of the neighborhoods of $0$ in the cofinite topology

Let $E$ be a vector space with a non countable number of elements. Define $$\mathscr{F}=\{U \subset E:0 \in U \hbox{ and } \#E\setminus U<\infty\}.$$ It's easy to see that $\mathscr{F}$ defines a ...
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A few questions about topologies on $\mathcal{C}^\infty_0 (\Omega)$

Reading about the space $\mathcal{C}^\infty_0(\Omega)$ of all compactly supported functions, I've came across a claim that this space is not complete with respect to the family of seminorms $$ \|\...
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Prove that the product of a family of topological vector space is Hausdorff space

I want to prove that: let $\Lambda \neq \varnothing$ and $E_{\alpha}$ topological vector space, for all $\alpha \in \Lambda$. Then $$E:=\prod_{\alpha \in \Lambda} E_{\alpha}$$ is a Hausdorff space if, ...
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A filter that has no countable basis

Conside a straight line $L$ in the plane $\mathbb{R}^{2}$. The filter of neighborhods of $L$ in $\mathbb{R}^{2}$ is the filter formed by the sets which contain an open set containing $L$. Prove that ...
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Krein-Milman in non locally convex space

In the book "Functional analysis and semi-groups" by Hille and Phillips, the Krein-Milman theorem (p. 28, Theorem 2.6.4) is stated as follows: Let $X$ be a TVS such that the topological dual $X'$ ...
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In a topological vector space $E$ a set different from $\emptyset$ and from $E$ cannot be both open and closed.

I want to prove that: In a topological vector space $E$ a set different from $\emptyset$ and from $E$ cannot be both open and closed, in the other words, if a subset $M\subset E$ is open and closed, ...
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Interesting non locally convex topological vector spaces

In mathematics, different variants of weak topologies on certain vector spaces or their subsets are ubiquitous (e.g. the weak$^*$ topology on the convex set of probability measures $\mathcal{P}(X)$ of ...
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31 views

Question about the proof of Theorem 3.7 from Rudin's Functional Analysis

Below is Theorem 3.7 from Rudin's Functional Analysis. In the proof, why does $\Lambda_1 x_0 = re^{i\theta}$ lie outside $\overline{\Lambda_1(B)}=K$ from Theorem 3.4 (b)? 3.4(b) shows that the real ...
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Fixed point theorem of Markov-Kakutani

Let $V$ be a Hausdorff topological vector space and $C \subseteq V$ a non-empty, compact, convex subset. Let $\mathcal{T}$ be a collection of continuous affine maps $C \to C$ such that every two maps ...
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On Köthe sequence spaces

Hans Jarchow in his "Locally convex spaces" defines the Köthe sequence space $\Lambda(P)$ by the condition $$ \lambda\in\Lambda(P)\quad\Leftrightarrow\quad \forall \alpha\in P\quad \sum_{n=1}^\infty \...
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Does equivalent norm imply the same order?

Consider two equivalent norms $|.|_{a}$ and $|.|_{b}$ on some vector space $V$. A general result then states that the norm induces the same topology on $V$. Does that imply that $$|x|_{a} \leq |y|_{a}...
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Finite-dimensional subspaces of TVSes

The following result is presented in many sources on topological vector spaces (TVSes): Any finite-dimensional subspace of a Hausdorff TVS is closed. However, having had a look at various sources,...
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Definition - filter basis

In the Trèves' book, Topological Vector Spaces, Distributions and Kernels he defines: A family $\mathscr{B}$ of subset of $E$ is a basis of a filter $\mathscr{F}$ on $E$ if the following two ...
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Example of a non compactly generated complete locally convex topological vector space over $ \mathbb{R}$

I am looking for an example of a non compactly generated complete locally convex topological vector space over $\mathbb{R}$. Being familiar with the fact that every complete locally convex topological ...
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A finite dimensional topological vector space $V$ can be equipped with a norm $\|\cdot\|$ which gives the same topology.

Definition A topological vector space $V$ is a space equipped with a topology $\mathcal{T}$ for which the vector sum $+:V\times V\rightarrow V$ and scalar multiplication $*:\Bbb{K}\times V\...
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John Horváth - Topological Vector Spaces and Distributions Volume 2

In at least three places in the book Topological Vector Spaces and Distributions, written by John Horváth, the author comments on a second volume of his book that has never been published. In ...
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Question on nowhere denseness: Why do we include the closure in the definition?

$(X,\tau)$ be a topological space and $A\subseteq X$. \begin{equation}A\text{ nowhere dense in }X \iff \overline{A}\text{ has empty interior.}\; \; \; \;\;(1)\end{equation} I know that this means ...
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Understanding a proof about the form of continuous linear functionals on L(X,Y)

I found a proof in a book and can't see how an inequality holds. From (J. Lindenstrauss, L. Tzafriri; Classical Banach spaces 1: sequence spaces; page 31 proposition 1.e.3): Let $X,Y$ be Banach ...
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Is the space of simple predictable processes topologized by uniform convergence metrizable?

Im following the book by Protter on Stochastic Integration and he is currently introducing Semimartingales. In doing so he first introduces the class of simple predictable processes $S$ and then ...
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norm form closed maps on a topoloical vector space

I'm now studying topological vector spaces. There is this notion of a closed map, which I am still learning, I have not found much on the subject other than the wikipedia page. I wanted to (dis)...
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Suggest a textbook PDF with definition of all metric spacea

Suggest a textbook PDF with definition of all metric space like ultra, cone, partial metric spaces etc
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Constructing metric on $l^2$ that agrees with the usual topology with respect to which the usual unit ball is unbounded

This problem is from a textbook on topological vector spaces by Bogachev. Let $f_n:l^2\to[0,n]$ be continuous functions such that the support of $f_n$ is the open ball with respect to the usual norm ...
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Conditions for extending a restricted vector bundle of an embedded submanifold to the whole manifold

Suppose $(\pi,E,M)$ is a vector bundle (total space $E$, base space $M$) and that $S \subset M$ is an embedded submanifold. If $E$ has positive rank, show that every smooth section of $E|_S$ (the ...
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Is a topological vector space that is $T_0$ already $T_2$?

If $A$ is a topological vector space, then I know that being $T_1$ is equivalent to beeing $T_2$. Now I was wondering if $A$ is a $T_0$-space, is it then automatically $T_1$ (and therefore $T_2$). I ...
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Rudin's Functional Analysis Theorem 1.41

While I was reading the proof of Theorem 1.41 in Rudin's Functional Analysis, I was stuck in the equation \begin{equation*} \pi(\{x:d(x,0)<r\}) = \{u:\rho(u,0)<r\} \end{equation*} where $N$ is a ...
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Definition of Local Convex Spaces [duplicate]

Let $(X,\tau)$ be a topological linear space. Is it true that the following equivalence hold? $(X,\tau)$ is a locally convex space $\Longleftrightarrow$ There is a familly of seminorms $\mathcal{P}=(...
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Which topological properties are inherited from a TVS? Like locally convexity [closed]

Let $X$ be a TVS and $Y$ be a subspace of $X$ with relative topology. If $X$ is locally convex or locally bounded then $Y$ is the same?
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Kernel of a continuous linear transformation $T$ on a topological vector space $X$

$Z$ is a closed subspace of a topological vector space $X$. Is it possible to find a continuous linear transformation $T$ from $X$ into itself, such that the $kernel(T)$ is $Z$? In the case of finite ...
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Is a Banach space a complete topological vector space?

A net $(v_\alpha)_{\alpha\in I}$ in a topological vector space (=TVR) $(V, \mathcal{T})$ is called Cauchy if $$\forall U \in \mathcal{V}_V(0): \exists \alpha_0 \in I : \forall \alpha, \beta \geq \...
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On a result of Mazur about convergence in locally convex spaces

My question is about the following result from Simon (2011) (Theorem 5.3). Let $X$ be a locally convex space and $Y$ its space of continuous [linear] functionals. Let $\{x_n\}$ be a sequence in $X$ ...
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Can any two points in an open and connected subspace of a locally convex t.v.s. be connected by a continuously differentiable path?

Can any two points in an open and connected subspace of a Hausdorff locally convex space be connected by a continuously differentiable path? Some known related facts: An open and connected subspace ...
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Existence of a continuously differentiable function with prescribed properties

Given a sequence $t_n \rightarrow 0$ of pairwise distinct positive real numbers and a sequence $x_n \rightarrow x$ in a Hausdorff locally convex space $X$, is there a continuously differentiable ...
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Conclude that the following topological vector space is not first-countable

Suppose that we have a normed vector space $(X,\|\cdot\|)$. We endow $X$ with a (locally convex) topology $\tau$ such that any $\tau$-convergent is norm bounded. Suppose that there exists a countably ...
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How do I show that for an open, nonempty subset $A$ of a topological vector space $V$ is convex?

I'm trying to prove the following: $V$ topological vector space. $A,B\subseteq V$ nonempty, open subsets. Suppose that a hyperplane $H\subseteq V$ separates $A$ and $B$. Show $H$ strictly separates ...
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Topological Vectorial Space, Existence of a neighborhood

Suppose $\mathscr{P}$ is a separating family of seminorms on a vector space $X$. Associate to each $p\in\mathscr{P}$ and each positive integer $n$ the set $$V(p,n)=\left\{x;p(x)<\frac{1}{n}\right\...
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Show that : $ \|A\|=\sup_{x^*\in B^*}{s(x^*|A)} $

Let $X$ be a Banach space and : $$ \mathcal{P}_{wkc}(X)=\{A\subset X\, |\, A\,\text{is nonempty, weakly compact, convex\}}. $$ Let $A\in \mathcal{P}_{wkc}(X)$, we define the radius of $A$ by: $$ \|A\|...
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Closedness of the sum of two cones

Consider two closed convex cones $K_1$, $K_2$ in a topological vector space. It is known that, in general, the Minkowski sum $K_1 + K_2$ (which is the convex hull of the union of the cones) need not ...
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Is it possible to change the order of limits in $\lim_\alpha \lim_\beta x_\alpha^*x_\beta$

I came across with this problem, Let $X$ be a topological vector space with $X^*$ its dual. Let $B$ be weak* dense in $X^*$. Is it true that $x_\alpha\to x$ (weakly) if $x^*x_\alpha\to x^*x$ for each $...
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Is there a complete non barrelled locally convex space?

These answers provides two examples of non barrelled locally convex spaces, but they are both incomplete. Furthermore, their completion is a Banach space and so necessarily barrelled. This answer ...
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Why is a subrepresentation a *closed* invariant subspace?

If $(\rho,V)$ is a topological representation of a topological group $G$, we usually say that a subrepresentation consists of a closed subspace $U\subset V$ which is $G$-invariant. I don't understand ...
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Does $A_{n+1} + A_{n+1} \subset A_n$ for $n = 0,1,2,\dots$ imply $A_1 + A_2 + \cdots \subset A_0$?

Let $\{A_n\}$ be a local base of a topological vector space satisfying $A_{n+1} + A_{n+1} \subset A_n$ for $n = 0,1,2,\dots$ where the sum is defined as usual: $A + B = \{a + b\mid a\in A,\,b\in B\}$. ...
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Open mapping theorem proof from Rudin. Why is the limit going to $0$?

Below is the proof of the Open Mapping Theorem from Rudin's Functional Analysis. Near the end of the proof, I cannot figure out why $y_{m+1} \to 0$ as $m \to \infty$ by the continuity of $\Lambda$. I ...
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There is a sequence converging to $0$ in the vector space $X$ of all functions $f:[0,1]\rightarrow \mathbb{C}$ with topology of pointwise convergence

The problem is from Rudin's Functional Analysis, first part of exercise 7, chapter 1. Here is what it says: Let $X$ be the vector space of all complex functions $f:[0,1]\rightarrow \mathbb{C}$ ...
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algebraic and topological dual (vector space)

If we have a infinite dimensional topological vectorspace $X$, how can one show that the algebraic dual $X^*$ and the topological dual $X'$ are not the same. We thought about using the fact that for $...
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Finding a topology for a vector space $X$, such that $X$ is a topologic vector space and the topologic and algebraic dual spaces are the same

Given is a vector space $X \neq \{0\}$. I should find a topology such that $X$ is a topologic vector space and the topologic $X'$ and algebraic $X^*$ dual spaces are the same. My idea is to use the ...
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Continuity of determinant of vector bundle morphism

Suppose $E$ is a real topological vector space and $B$ is any topological space. We can generate a trivial vector bundle $\xi=(B\times E,B, \pi,\cdot, +)$ where the projection $\pi:B\times E\...
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Are convex sets definable?

We know that the real field is not interpretable in the complex field (with the language of field theory) and the usual way to define convex sets in complex vector spaces is via real coefficients in ...
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Topological Vector Space is separable if its dual space $X^*$ is separable?

Let $(X,\tau)$ be a Topological Vector Space such that the associated dual space $X^*$ is separable. Can we say that $X$ is separable ? I know that this property is valid for Banach spaces but for ...
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the Mackey topology on the dual space $X^*$.

I read an article, I saw a sentence that I did not understand because I do not know the topology of Mackey. This sentence is: "We shall denote by $\tau$ the Mackey topology on the dual space $X^*$. ...
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Hahn Banach. separate $A:=\{f \in L^2(-1,1): f \mathrm{\ is\ continuous}, f(0)=0\}$, $B:=\{f \in L^2(-1,1): f \mathrm{\ is\ continuous}, f(0)=1\}$

Consider $L^2(-1,1)$ and the sets $A, B$ defined as $A:=\{f \in L^2(-1,1): f \mathrm{\ is\ continuous}, f(0)=0\}$, $B:=\{f \in L^2(-1,1): f \mathrm{\ is\ continuous}, f(0)=1\}$ Question: Does ...

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