# 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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### 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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### 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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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 $(\... 0answers 196 views ### Joint continuity of tensor product Let$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$... 0answers 123 views ### Two notions of vector distributions and differential operators in$\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)^{*})... 0answers 58 views ### 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 ... 0answers 30 views ### 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 ... 1answer 79 views ### 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 ... 1answer 26 views ### 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)... 2answers 42 views ### 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, ... 0answers 22 views ### 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) 1answer 108 views ### 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 ... 1answer 26 views ### 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 ... 1answer 32 views ### 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 ... 0answers 51 views ### 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... 2answers 33 views ### 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 ... 1answer 26 views ### 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.$$Assume x= \sum_{i \... 2answers 48 views ### Show that algebraic direct sum is \sigma-weakly dense. Consider the abstract von Neumann algebra$$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 ... 1answer 35 views ### 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 ... 1answer 33 views ### 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 ... 1answer 21 views ### 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\} ... 1answer 59 views ### 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 \... 0answers 23 views ### 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 ... 1answer 18 views ### 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 \... 1answer 31 views ### 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\... 2answers 34 views ### 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 ... 2answers 20 views ### 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 ... 1answer 67 views ### 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^...
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 ...
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 \$\{...