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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Some properties of Cartesian product space

Does the Cartesian product space of two reflexive (Resp. uniformly convex) space still reflexive(Resp. uniformly convex)? Thanks for any answers!
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Infinite product of infinite sums of formal power series: proof?

Teaching a course on algebraic combinatorics has made me aware of a technical fact about formal power series that is used throughout the subject, but that I have never seen formally stated, let alone ...
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How did nuclear spaces come about?

I researched a lot what the point of nuclear spaces is. From what I understand they were invented by Grothendieck to make a more general statement for the Kernel Theorem by Schwartz. He figured out ...
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Conditions for locally convex space to be normable

Let $X$ be a Hausdorff locally convex space and $P$ is a family of seminorms on $X$. How to show that $X$ is normable iff $P$ is equivalent to a finite subfamily $P_0 \subset P$? One implication is ...
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relative interior in a topological vector space

In $\mathbb{R}^n$, the relative interior of a convex set $C$ is defined as $$relint(C)\doteq\{x\in\mathbb{R}^n: \exists\epsilon>0, N_\epsilon(x)\cap aff(C)\subset C\},$$ where $N_\epsilon(x)$ is an ...
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Discontinuous linear operator.

Does there exists a discontinuous linear operator $T$ between Hausdorff locally convex spaces $X$ and $Y$ such that $T$ is continuous with respect to the weak topologies on $X$ and $Y$?
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When topological vector space metrizable

This question comes from Chapter 5 of Folland's Real Analysis.It's a proposition (5.16)without proof. Let $X$ be a vector space equipped with the topology defined by a family $\left\{ p_{\alpha} \...
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Proof of Hahn Banach theorem for locally convex vector space

Let $X$ be a locally convex topological vector space, let $L$ be a linear subspace of $X$ with the corresponding induced topology, and let $l$ be a continuous linear functional defined over $L$. Then $...
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linear functional is continuous if and only if it is locally bounded at orgin

Given a topological vector space $X$, a functional $f:X\rightarrow\mathbb{R}$ is continuous if and only if there exists an open neighbourhood $N$ of $0$ s.t. $|f(x)|\leq 1$ for all $x\in N$. It easy ...
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Hamel Basis for Homeo$^+(\mathbb{R}^2)$

In the 50's through 70's there was a lot of research into the group of orientation-preserving homeomorphisms of the plane, denoted as Homeo$^+(\mathbb{R}^2)$ (in the compact-open topology, which in ...
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Why is the following set open in the dual space?

Let $E$ be a normed vector space over $\mathbb{C}$, and consider $E^{\ast}$ the space of continuous linear functionals on $E$ with the topology induced by the norm $\|f\| = \sup_{\|x\| \leq 1 } |f(x)|$...
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Heine-Borel theorem for $\mathbb{C}^2$

Lang's Complex Analysis contains a proof of the Heine-Borel theorem for $\mathbb{C}^1$, discussed previously on StackExchange. In the reals, we know that the Heine-Borel theorem holds for $\mathbb{R}^...
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A doubt about the weak topology, $(E,\, \sigma(E,\, E'))'$ is Banach for all $E$ normed space?

We have a result that says that the space formed by the linear continuous functionals in $E$ with the weak topology coincides with the topological dual of $E$, that is, $E' = (E,\, \sigma(E,\, E'))'$ ...
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Is there at least one continuous norm on $\mathbb{K}^{X}$?

Let's denote by $\mathbb{K}^X$ the topological vector space of all functions from the set $X$ to base field $\mathbb{K}$ with topology generated by family of seminorms $$\{\|f\|_x = |f(x)|: x \in X\}$$...
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Which topological vector spaces have uncountable unordered sums?

If $P$ is an uncountable locally finite poset, then the incidence algebra $I(P)$ is a topological vector space (in fact a topological algebra) with the interesting property that every element $f$ can ...
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Is it true that $A=int(A)$ or $A=int(A) \cup fr(A)$ and true that $A^C=ext(A)$ or $A^C=ext(A) \cup fr(A)$?

I'm trying to understang some topology of $\mathbb{R}^n$ space. I want to know if it's true that $A=int(A)$ or $A = \text{int}(A)\cup \partial A$ and true that $A^{c} = \text{ext}(A)$ or $A^{c} = \...
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Shape of a unit Ball and metric

Playing around with metrics defined in $\Bbb{R}^2$, such as $d_p$, where $p\ge1$, or even such as $σ=\sqrt{d_p}, τ=\frac{d_p}{1+d_p},π=\min \{1,d_p\}$, I realised that all produce sets in $\Bbb{R}^2$ (...
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Locally convex spaces are topological vector spaces?

I've come across the below definition of a 'locally convex space' and am trying to prove that addition and multiplication are continuous with respect to the locally convex topology generated by the ...
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$x\in U\implies x\in (1-\delta)U$

Let $X$ be a topological $\mathbb{R}$-vector space. I want to show that if $U$ is open and $x\in U$ then there exists a $\delta>0$ such that $x\in (1-\delta)U$. I think it has something to do with ...
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Topology of space of test functions and convergence

Let $D$ denote the space of all compactly supported infinitely differentiable functions on $\mathbb{R}$, and suppose $D_m$ is the subspace of all infinitely differentiable functions on $\mathbb{R}$ ...
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Defining a topology in terms of filter neighbourhoods

I am a little confused about the following: if $X$ is a set for which there is a filter $\mathcal{F}(x)$ of sets containing $x\in X$ assigned to every point $x\in X$, and these filters are such that ...
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The set of continuous linear operators is a vector subspace of the space of all linear operators.

I am trying to proof that the set $L(X,Y)$ of continuous linear operators between to topological vector spaces is a vector subspace of the space of all linear operators $Hom(X,Y)$. I proved that ...
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Which Schauder bases of $F[[x]]$ have the multiplication property?

Both $\{\frac{x^n}{n!}:n\in\mathbb{N}\}$ and $\{\frac{x^n}{n^2+1}:n\in\mathbb{N}\}$ are Schauder bases for the ring of formal power series $\mathbb{R}[[x]]$ as a topological vector space over $(\...
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Joint continuity of tensor product

Let $X, Y$ be locally compact Hausdorff spaces and consider the spaces $\mathcal{K}_{\mathbb{C}}(X), \mathcal{K}_{\mathbb{C}}(Y)$ of continuous functions with compact support on $X$ and $Y$ ...
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Two notions of vector distributions and differential operators in $\mathbb{R}^3$

Two possible ways of defining vector/vector valued distributions in $\mathbb{R}^3$ are: $$ X := [\mathcal{D}(\mathbb{R}^3; (\mathbb{R}^3)^{*})]^{*} = \{ T: \mathcal{D}(\mathbb{R}^3; (\mathbb{R}^3)^{*})...
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Let X be a metrizable TVS of infinite dimension. Show that there exists a discontinuous linear functional on X.

The exercise had one hint: Use an algebraic basis of $X$ and Problem 22. And the Problem 22: Let $X$ be a metrizable TVS and let $(x_n)$ be a sequence of elements of $X$. Show that there exists a ...
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reference request: Borel measure is Gaussian iff Fourier transform involves positive definite bilinear form

I am looking for a reference of the following theorem A Borel probability measure $\mu$ on a topological vector space $X$ is centered Gaussian if and only if there exists a positive semidefinite ...
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Do falling factorials form a Schauder basis for formal power series in some topology?

We usually talk about $F[[x]]$, the set of formal power series with coefficients in $F$, as a topological ring. But we can also view it as a topological vector space over $F$ where $F$ is endowed ...
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Show two topologies coincide on the unit ball.

Consider the following lemma from "Lectures on von Neumann algebras": I understand the proof of $(i)$ and $(ii)$. However, the proof says that $(iii)$ and $(iv)$ follow immediately from $(i)...
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Question about initial topology and dual vector space

Consider the following fragment from the book "Lectures on von Neumann algebras". Why is the line $\varphi$ is $\sigma(\mathcal{E}, \mathcal{F})$-continuous $\implies$ there exist $\psi_1, ...
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Can any one help me with a good book or ressource talking about the next concepts

Topological vector spaces, seminorms, the topology comming from seminorms. (Language:english or french)
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Examples of the difference between Topological Spaces and Condensed Sets

There is apparently cutting-edge research by Dustin Clausen & Peter Scholze (and probably others) under the name Condensed Mathematics, which is meant to show that the notion of Topological Space ...
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Alternative characterisation of weakly complete Banach spaces

Let $V$ be a topological vector space over some locally compact field $\Bbb K$. Let $V'$ denote all continuous functionals on $V$, the weak topology is defined to be the coarsest topology on $V$ ...
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Weak$^*$ continous functionals

Let $X$ be a Banach space (not necessarily reflexive) and let $X^{*}$ denote the continuous dual of $X$. Let $\psi:X \to X^{**}$ denote the canonical embedding of $X$ into its double dual $X^{**}$. We ...
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Is this a topology? (induced by weak convergence)

Let $\mu,\nu$ probability measures on two compact sets respectively $X,Y\subseteq\mathbb{R}^n$. Let $\Pi(\mu,\nu)$ be the space of measures on $X\times Y$ whose first and second marginals are $\mu,\nu$...
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Does there exist a topological space Y containing more than one point such that any function $f : X\to Y$ is continuous?

I found this question in one of my past question papers of the college. Let $X$ be any topological space. Does there exist a topological space $Y$ containing more than one point such that any function ...
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Infinite convex combination is in the closure of the convex hull.

Let $V$ be a topological vector space and $\{x_i\}_{i \in I}$ be a net in $V$. Further, let $\{\lambda_i\}_{i\in I}$ be a net in $[0,1]$ such that $$\sum_{i \in I}\lambda_i =1.$$ Assume $x= \sum_{i \...
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Show that algebraic direct sum is $\sigma$-weakly dense.

Consider the abstract von Neumann algebra $$M:= \ell^\infty-\bigoplus_{i \in I} B(H_i)$$ which consists of elements $(x_i)_i$ with $\sup_i \|x_i\| < \infty$ and $x_i \in B(H_i)$. Let $N$ be the ...
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Smallest vector topology

This is with regard to this question: Topology induced by seminorms and initial topology I saw somwehere that topology $\mathcal{S}$ is the smallest topology with respect to which all the seminorms ...
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Equivalence of two seminorms

Let us consider the space $C^0([0,1])$ of real continuous functions over $[0,1]$. Let $A = \{a_n : n \in \mathbb{N}\}$ be a countable set of $[0,1]$ and $\alpha_n$ be strictly positive real numbers ...
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If $\tau$ is a vector topology on a normed space s.t. a closed ball is $\tau$-compact, is a continuous function on a closed ball even uniformly cont.?

Let $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$, $r>0$, $X$ be a normed $\mathbb K$-vector space, $\tau$ be a vector topology on $X$ s.t. $\overline B_r^X(0):=\{x\in X:\left\|x\right\|_X\le r\}$ ...
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If $E$ is a normed vector space and $\tau$ is a vector topology on $E$ s.t. $f$ is $\tau$-continuous on every closed ball, is $f$ $\tau$-continuous?

Let $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$, $E$ be a normed $\mathbb K$-vector space, $\tau$ be a vector topology on $E$ such that $\overline B_1(0):=\{x\in E:\left\|x\right\|_E\le1\}$ is $\...
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cofinite subspaces in weak* topology

I am learning hopf algebras by D.E Radford, questions about cofiniteness occur to me but I can't find out the answers. My question is Let $U$ be a vector space over a field $k$. Is there any subspace ...
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Every first contable locally convex space has a countable neighborhood basis of balanced and convex sets

Terminology: By a neighborhood of a point $x$ on a topological space, I mean any subset $V$ which contains an open set containing $x$. A set $B$ in a vector space $X$ is called balanced if $\lambda B \...
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Prove that a real normed space $V$ of dimension $n$ is disconnected in two connected parts by any linear subspace of dimension $n-1$

Theorem All norms on a finite dimensional vector space $V$ are equivalent. Proof. Omitted. So to follow we will use the infty norm $\lVert\cdot\lVert_\infty$ given by the equation $$ \lVert\vec v\...
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Topological linear space

Let $E$ be a topological linear space. A subset $M$ of $E$ is said to be bounded if $\forall U\in I(0)$ there exist a scalar $\lambda$ such that $M\subseteq \lambda U$, where $I(0)$ denotes the family ...
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Topology on $\mathbb{K}\times E$

This is probably a really basic question, but here it goes. Let $E$ be a vector space over $\mathbb{K}$ and let $\tau$ be a topology on $E$. Then $E$ is called a topological vector space if the sum ...
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weak star convergent net is uniformly convergent on weakly compact sets?

Let $X$ be a Banach space and $X^\ast$ its (topological)dual space. Let $x_\alpha^\ast,x^\ast\in B_{X^\ast}$, where $B_{X^\ast}$ denotes the closed unit ball of $X^\ast$, such that $x_\alpha^\ast\to x^...
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The Exponential Group in a Banach Algebra: a proof of Rudin.

Define $G$ to be the group of invertible elements in a Banach algebra $A$ and $G_1$ the component of $G$ that contains $e\in G.$ Now, $\exp x=\sum_{n=0}^\infty \frac{x^n}{n!}$ and if $\Gamma$ is the ...
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Does this proof of the sequential completeness of a reflexive space require the Eberlein–Šmulian theorem?

I'm self-teaching functional analysis through Conway's book at the moment, and can't seem to get one of the steps in this proof: The statement that $\{ x\in X : ||x||<M\}$ implies the sequence $\{...

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