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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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Counterexample in topological vector space.

Let's $X$ be a topological vector space. So it has two operations $+$ and $\times$. As we know these operations could be continous. But can it be that one of this operation is continous, but the ...
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44 views

Problem 3, chapter 2 Rudin functional analysis

Put $K = [-1,1]$; define $\mathcal{D}_K$ as in section 1.46 with ($\mathbb{R}$ in place of $\mathbb{R}^n$). Suppose $\left\{ f_n > \right\}$ is a sequence of Lebesgue integrable functions such ...
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Is every probability measure a Radon measure? [closed]

Let $\mu$ be a probability measure defined on a compact convex subset $K$ of a locally convex Hausdorff space $X$. Is $\mu$ a Radon measure?
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1answer
47 views

Closed subspaces of weak star topology

Let $ V $ be a vector space over a field $ k$. Give $ k $ the discrete topology and give $ V^\ast $ the coarsest topology for which the maps $ j_v: V^\ast \to k$ for all $ v \in V $, defined by $ j_v(...
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1answer
43 views

Link between topological dimension and Hamel (algebraic) dimension of a vector space

I was wondering if there is a link between this two dimension definitions in the case of a Topological Vector Space in fact I know that sometimes topological dimension coincides with other notions of ...
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2answers
33 views

In a Topological Vector Space T0 implies T3½ (completely regular)? And other separation properties.

I will describe my doubt. I know that in a TVS T1 implies T2. Now since a TVS admits a uniformisable topology, we have that T2 implies the uniform structure is separating. Now a separating uniform ...
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0answers
10 views

$A$ bounded $\Leftrightarrow A$ bounded for all $P_\alpha \in P$. [closed]

Problem: Let $E$ be a locally convex space with topology defined by a seminorm family $P=\{P_{\alpha}\}_{\alpha \in I}$ and $A \in E$. Prove that $A$ bounded $\Leftrightarrow$ $A$ bounded for all $P_\...
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1answer
20 views

Polar set of the open ball in the dual space $X'$

Assume $(X,\Vert\cdot\Vert)$ is a normed space with the dual space $X'$. I want to show that the polar set of the open disk $U_{r}^{X'}(0)$ is equal to the closed disk $K_{\frac{1}{r}}^{X}(0)$, e.i.: $...
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1answer
42 views

Topological vector spaces with a unique trivial nonempty convex set which is algebraically open

Let $X$ be a topological vector space. We recall that a set $A\subseteq X$ is algebraically if, for each $x,y \in X$, the set $\{t \in \mathbf{R}: x+ty \in A\}$ is open. Moreover, it is well known (...
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1answer
34 views

Open connected which does not contain open convex subsets

It is well known that convex implies connected, and it is clear that if $X$ is a locally convex topological vector space and $\emptyset \neq A\subseteq X$ is open then $A$ contains a nonempty open ...
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1answer
138 views

Existence of a symmetric subset $B\subseteq A$ such that $2A-A\subseteq 8A$

Let $A$ be a nonempty open connected subset of a (real) topological vector space $X$ such that $$2A-A \subseteq 8A$$ (for instance one could take $A=(-1,2)$). Question. Is it true that there exists ...
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1answer
14 views

Existence of symmetric subsets

Let $A$ be a nonempty open connected subset of a (real) topological vector space $X$. Question. Is it true that there exists a nonempty open connected set $B\subseteq A$ such that $B$, in addition,...
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31 views

Projection is continuous

Let $V$ be a Banach space and let $W,Z$ be closed subspaces such that $V= W \oplus Z$. I have to show that the natural projection $P:V \to W$ is continuous. I want to use the generalization of the ...
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1answer
27 views

Infinite intersection of topological subspaces. What is its topoloy?

If I have a decreasing sequence (infinite, at least countable) of subspaces of topological vector spaces: $$\ldots E_{n+1} \subseteq E_n \subseteq \ldots \subseteq E_2 \subseteq E_1$$ What is the ...
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1answer
21 views

There is no $0$ neighbourhood in the infinite product of topological vector spaces with $X_i \neq \{0\}$

Assume $(X_i,\mathcal{T}_i)$, $i \in I$ is a family of topological vector spaces and $X:=\prod\limits_{i\in I} X_i$ with the product topology $\mathcal{T}$. $I$ is infinite and for all $i\in I: X_i\...
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1answer
30 views

Representation of the elements of the dual space of the product of topological vector spaces

Assume $(X_i,\mathcal{T}_i)$, $i \in I$ is a family of topological vector spaces and $X:=\prod\limits_{i\in I} X_i$ with the product topology $\mathcal{T}$. Let $\pi_i: X \rightarrow X_i$ be the ...
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1answer
44 views

$B \subseteq X$ is bounded $\Leftrightarrow$ $\forall U \in \mathcal{U}(0) ~ \exists \lambda_U: ~B \subseteq \lambda_u U $

I have some troubles with the following problem and hope some of you can help me. Let $X$ be a vector space, equipped with the $\sigma$-weak topology $\sigma(X,Y)$, where $Y$ is a subspace of the ...
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0answers
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direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps $$ 0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0. $$ We can take inductive limit (...
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1answer
25 views

Sobolev space identification

Let $\Omega$ be a open subset of $\mathbb{R}^{d}$. It is well known that $L^2(\Omega\times (0,T))$ can be identified with $L^2(0,T;L^2(\Omega))$. Now, let us consider $$H^{1}(0,T;H_{0}^{1}(\Omega))=\{...
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Are normed spaces equipped with the weak topology sequential [duplicate]

Let $X$ be a normed space equipped with the weak topology. Is $X$ a sequential space (using this definition)? That is can we test closedness in the weak topology using weakly convergent sequences? I ...
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1answer
42 views

Theorem 1.27 in Rudin's Functional Analysis

1.27 Theorem: Suppose $Y$ is a subspace of a topological vector space $X$, and $Y$ is an $F$-space (in the topology inherited from $X$). Then $Y$ is a closed subspace of $X$. Here is Rudin's proof: ...
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1answer
37 views

Every continuous linear functional in $(\mathbb{R}^\infty)^*$ is of the form $\sum_n a_n x(n)$

I'm reading lecture notes about analysis on infinite dimensional spaces and I ran into this exercise: Every continuous linear functional $f\in (\mathbb{R}^\infty)^*$ is of the form $$f(x)=\...
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1answer
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Is the space of real sequences normable

Intuition says the vector space of real sequences $R^N$ ($N$ the natural numbers, pointwise addition of real coordinates) is not normable. I have found this surprisingly hard to prove. I am aware ...
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1answer
47 views

Continuity of representation of topological group

First, We set notations as follows. $G$ : topological group , $k$ : field , $V$ : linear topological space over $k$ , $\mathrm{Map}(V,V)$ : Set of all continuous maps from $V$ to $V$ $\mathrm{Aut}_k (...
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1answer
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A linear topological space over real number field is locally compact ???

Let $V$ be a topological $\mathbb{R}$-vector space with ${\rm dim}(V) < \infty$ Then, $V$ is locally compact $??$
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1answer
55 views

About the locally convex topology

I know that if a locally convex space Hausdorff $(X,S)$ is first numerable then for the $\hat{0}\in X$ exists a countable local base $\{V_n, n \in \mathbb{N}\}$ and to each $V_n$ corresponds a ...
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0answers
37 views

Properties of topological vector spaces

I'd like to understand better the significance of certain properties of topological vector spaces. It would be great if someone could give me an intuitive picture for what makes them "special", and/or ...
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9 views

Cover set of open ball topology

Let $Q_k(x_k, r_k) $ be a cube of center $x_k$ and side length $\frac{r_k} {2} $ suppose that $1<a<a*$ such that $Q_k^{*} =a^{*}Q_k$ so $Q_k^{*}=Q_k(x_k, a^{*}r_k) $ Show that if $$O=\...
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1answer
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Density of test functions in the space of distributions — a clarification

Let $U \subseteq \mathbb{R}^n$ be open and denote by $\mathcal{D}(U)$ the space of all compactly supported smooth functions $U \to \mathbb{R}$. Let $\mathcal{D}^\prime(U)$ be the space of all ...
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1answer
49 views

A subspace of a dual Banach space $X^*$ is norming if and only if its weak$^*$ closure contains a multiple of the unit ball of $X^*$

Definition A subspace $Z$ of a dual Banach space $X^*$ is said to be norming if there exists $c>0$ such that $$\sup_{f\in B_Z}|f(x)|\geq c||x||$$ for every $x\in X$ (where $B_Z$ is $B_{X^*}\cap ...
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1answer
48 views

Weak-* continuous functionals with bounded level sets

Let $X$ be an infinite-dimensional Banach space, $X^*$ its topological dual and $f:X^*\to\mathbb{R}$ some weak-* continuous functional (not necessarily linear). Is it possible for $f^{-1}(a)$ to ...
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1answer
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Rudin functional analysis theorem 4.13, (a) and (b)

Let $U$ and $V$ be the open unit balls in the Banach spaces $X$ and $Y$, respectively. If $T \in \mathcal{B}(X,Y)$ and $\delta > 0$, the the implications $$ (a)\to(b)\to(c)\to(d) $$ hold among ...
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1answer
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Rudin Functional analysis, theorem 4.12 corollary (b)

Suppose $X$ and $Y$ are Banach spaces, and $T \in \mathcal{B}(X,Y)$ Then $$ \mathcal{N}(T^*) = \mathcal{R}(T)^{\perp} \;\;\text{and}\;\; \mathcal{N}(T) = ^{\perp}\mathcal{R}(T^*) $$ The corollary ...
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2answers
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How is Schwartz space different from Hilbert space?

I know that Schwartz space can be considered a dense subset of the Hilbert space isomorphic to $\ell^2$. What I wish to understand is, how really different Schwartz space is from the Hilbert space. ...
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1answer
40 views

Rudin functional analysis, theorem 4.9 part (b)

Let $M$ a closed subspace of a Banach space $X$. b) Let $\pi:X \to X/M$ be the quotient map. Put $Y = X/M$, For each $y^* \in Y^*$, define $$ \tau y^* = y^* \pi $$ Then $\tau$ is an isometric ...
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2answers
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$+:X\times X\to X,(x,y)\mapsto +(x,y)=x+y$ and $\cdot:\Bbb{R}\times X\to X,(\lambda,x)\mapsto \cdot(\lambda,y)=\lambda\cdot x$ are weakly continuous

$$+:X\times X\to X,\\(x,y)\mapsto +(x,y)=x+y$$ and $$\cdot:\Bbb{R}\times X\to X,\\(x,y)\mapsto \cdot(\lambda,y)=\lambda\cdot x$$ are weakly continuous, where $X$ is an infinite dimensional normed ...
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1answer
43 views

This convex hull is balanced?

Let be (X,S) a locally convex space, and $B \subset X$ a nonempty sequentially closed bounded and convex set such that $\hat{0} \notin clB$,(the closure of B). Define the set T:=s-clco$\{B \cup-B \}$ ...
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1answer
33 views

Prove that $\|f\|_{X^*} = \sup_{\|x\| \leq 1} |f(x)|.$

Let $X$ be a normed vector space. Let $f: X \rightarrow \mathbb{R}$ be a bounded linear functional. We define: $$\|f\|_{X^*} = \sup_{x \in X \\ x \neq 0} \dfrac{|f(x)|}{\|x\|_X}$$ Prove that: $$\|f\|...
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1answer
28 views

Rudin's functional analysis, theorem 4.1.(completeness of $\mathcal{B}(X,Y)$)

Suppose $X$ and $Y$ are normed space. Associate to each $\Lambda \in \mathcal{B}(X,Y)$ the number $$\lVert \Lambda \rVert = \sup \left\{ \lVert \Lambda x \rVert : x \in X, \lVert x \rVert \leq 1 \...
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1answer
37 views

Completeness condition for Fréchet space

In Wikipedia's definition of Fréchet space, it is stated that a Fréchet space is a topological vector space that satisfies the following: It is locally convex Its topology can be induced by a ...
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6answers
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why a subspace is closed?

Let $E$ be $K-$vector space with a norm $\|\cdot\|$, and $F$ a subspace with dimension $n$. Show that $F$ is a closed set . I am trying to show that any convergent sequence of elements $(x_n)_{n \in ...
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1answer
27 views

Rudin functional analysis theorem 3.28, $P$ is weak*-compact in $C(Q)^*$.

Follow up question to this one. It is later proved that defining $\phi : C(Q)^* \to X$ as $$ \phi(\mu) = \int_Q x d \mu (x) $$ if $P$ is the set of all regular probability measures on $Q$ we have $ ...
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1answer
29 views

Rudin functional analysis theorem 3.28, application of Reisz representation theorem.

Suppose (a) $X$ is a topological vector space on which $X^*$ separates points, (b) $Q$ is a compact subset of $X$, and (c) the closed convex hull $\overline{H}$ of $Q$ is compact ...
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0answers
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Is there a characterization of smooth $G$-equivariant functions $\mathbb R^n \to \mathbb R^m$?

If $G$ is a compact Lie group acting smoothly on $\mathbb R^n$ by orthogonal matrices, a result of G. Schwartz characteriezs the smooth $G$-invariant functions $f : \mathbb R^n \to \mathbb R$ as ...
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Rudin functional analysis, theorem 3.27

Few bits a bit unclear of a) $X$ is a topological vector space on which $X^*$ separates points, and b) $\mu$ is a Borel probability measure on a compact Hausdorff space $Q$. If $f : Q \to X$ ...
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Rudin functional analysis, Definition 3.26

Straight to the point Suppose $\mu$ is a measure on a measure space $Q$,$X$ is a topological vector space on which $X^*$ separates points, and $f$ is a function from $Q$ into $X$ such that the ...
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1answer
33 views

Closed sets with filters

I am studying topological vector spaces, and, being at the begin of the textbook (Sevres) I frequently encounter rephrasing and generalisations of known results about metric spaces in terms of filters....
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1answer
42 views

Rudin Functional Analysis Theorem 3.25

This is probably the first question regarding the theorem in the title... If $K$ is a compact set in a locally convex space $X$, and if $\bar{co}(K)$ is also compact, then every extreme point of $\...
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70 views

Hahn-Banach in dual space

Let $X$ be a Banach space. Let us take a subspace $V \subset X^*$ of the topological dual space and a linear functional $\varphi \colon V \to \mathbb R$. We assume that $\varphi$ is continuous w.r.t. ...
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If the dual of a topological vector space separates points, does it separate a point and a closed subspace?

The Hahn-Banach Theorem implies that if $X$ is a normed vector space, then the dual space $X^*$, consisting of continuous linear functionals on $X$, has the following two properties: $X^*$ separates ...