Questions tagged [locally-convex-spaces]
For questions about topological vector spaces whose topology is locally convex, that is, there is a basis of neighborhoods of the origin which consists of convex open sets. This tag has to be used with (topological-vector-spaces) and often with (functional-analysis).
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Why is a certain projective limit of weighted symmetric Fock space, namely $\bigcap\limits_{\tau \in T, p\ge 1 } \mathcal{F}(H_\tau,p)$, separable
I have a question regarding separability of a certain locally convex space.
Let $H_{\tau}:=H^{\tau_1}(\mathbb{R}^n,\tau_2(x)dx)$ the weighted Sobolev Hilbert space with $\tau_1 \in \mathbb{N}, \tau_2(...
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Convexity of a connected, compact, and locally convex set in $\mathbb{R}^n$
Is a compact, connected, and "locally convex" set in $\mathbb{R}^n$ convex?
Here I mean a space $A$ locally convex as: For any point $x\in A$, there exists a neighborhood $U$ of $x$ s.t. $U$...
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Understanding the locally convex topology induced by a family of seminorms intuitively
In our functional analysis course, we defined the locally convex topology induced by a family of seminorms as follows:
Let $X$ be a vector space, $I \neq \emptyset$ an index set and $\{p_{\alpha}\}_{\...
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1
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Critical point for composite function implies $f'(x_{0}) = 0$ in locally convex spaces
Let $X, Y$ be locally convex spaces over $\mathbb{R}$ and $f: X \to Y$. Suppose $f$ is Fréchet differentiable at a point $x_{0} \in X$. Given an interval $0 \in I \subset \mathbb{R}$, we call a smooth ...
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Equality of Seminorms in vector bundles
In this notes Geometric Wave Equations by Stefan Waldmann at page 7 he has
Let $E \longrightarrow M$ be a vector bundle of rank $N$. For a chart $(U, \psi)$ we consider a compact subset $K \subseteq ...
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Regarding a non-standard definition of tempered distributions in Schuller's lecture
Is the below definition of tempered distributions correct?
$\newcommand{\rr}{\mathbb{R}}
\newcommand{\cc}{\mathbb{C}}
\newcommand{\nn}{\mathbb{N}_0}
\newcommand{\schwartz}{\mathcal{S}}
\newcommand{\...
- 140
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Textbook on TVS that contains Theorem 32.2 about the completeness of the space of continuous linear maps
I have come across this thread about TVS. The OP took below screenshot of THEOREM 32.2.
Could you please elaborate on the book from which the screenshot was taken?
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Existence of measure with given Fourier transform
Bogachev's "Gaussian measures" contains the following theorem:
Theorem 3.3.1:
Let $\mathcal{X}$ be a locally convex space and $G\subset \mathcal{X}^\ast$ a linear subspace separating the ...
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1
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Weak and strong relative topologies coincide on compact sets
Let $\left(\mathcal{X}, \tau\right)$ be a locally convex topological space and $K$ a compact subset. Is there a reference for the proof that on $K$ the relative topology coincides with the relative ...
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Equivalent definitions of bounded sets in topological vector spaces
I want to show equivalence of following two definitions:
Definition 1: A subset $U$ of a topological vector space is called bounded, if for every neighborhood of $0$ $V$, there is a scalar $s\in\...
- 2,143
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Understanding equivalent definitions of locally convex spaces
As stated in the title, I am trying to make sense of the equivalent definitions of locally convex spaces. Especially, I am confused about the part, where I try to prove, that the topologies coincide.
...
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Is the weak operator topology equal to $\sigma(L(X,Y), X\times Y^*)$?
I'm asking myself if the weak operator topology is equal to the weak topology $\sigma(L(X,Y), X \times Y^{*},b)$ with
$$
\begin{align}
b:L(X,Y)\times (X \times Y^{*})\to& [0, \infty)\\
(T,(x,y') \...
- 278
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$\varphi\mapsto x^\alpha \varphi$ is a continuous endomorphism in Schwartz space
I'm following some notes on functional analysis where it's shown the result presented on the title.
The proofs in the notes is not clear to me.
Before showing the proof I give some context.
The ...
- 278
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On a class of compacts broader then Eberlein's
The class of Eberlein compacts (those compacts spaces homeomorphic to a weakly compact subset of a Banach space) is well known and well studied; one of the many properties it enjoys is that every set ...
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$x_\beta \rightarrow x$ in a locally convex space if and only if $\rho_\alpha(x_\beta, x) \rightarrow 0$ for every seminorm $\rho_\alpha$
Let $X$ be a locally convex space with the family of seminorms $\{p_\alpha\}$. I am trying to get a feel for convergence of nets (or sequences) in these spaces aside from the general topological ...
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Unit semiball of the supremum of seminorms is closed
I don't know how to prove the second part of this exercise (ex. 7.2 in F. Treves, "Topological vector spaces, distributions and kernels").
Let $\mathcal{P}$ be a family of continuous ...
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0
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Confusion about locally convex (topological vector) spaces
The author in the book I am reading defines a locally convex space as follows.
Definition: A topological vector space $(X,\tau)$ is called a locally convex space if there is an index set $A$ and a ...
- 2,143
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Subbase for initial topology of quotient mappings (locally convex spaces)
Consider a family $\{p_i\}_{i\in I}$ of seminorms defined on $X$ and consider for each $i\in I$ the quotient mapping $q_i:X\rightarrow X/\ker(p_i)$. I know, that each quotient $X/\ker(p_i)$ is a ...
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On an infinite dimensional locally convex space, no weakly-continuous semi-norm is a norm
Let $X$ be an infinite dimensional locally convex space that separates points and $X^*$ its dual. I would like to prove that no $\sigma(X, X^*)$-continuous semi-norm is actually a norm.
What I have ...
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Proving a map $\mathbb{R}^n \rightarrow \mathscr{S}(\mathbb{R}^3)$ is $C^\infty$
I am working through Reed and Simon's book on functional analysis. This is one of their exercises:
Let $\mathscr{S}$ denote the space of Schwartz functions, $\mathscr{S}'$ the space of tempered ...
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The Mackey topology $\tau(X^*, X)$
Let $X$ be a Banach space and $X^*$ its dual. I would like to better understand the Mackey topology $\tau(X^*, X)$ on $X^*$.
The Mackey topology on $X$, $\tau(X,X^*)$ is defined as the topology of ...
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1
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Are the weakly bounded subsets of the double dual bounded for the natural topology?
I'm trying to get a better understanding of the topologies on the double dual of a Hausdorff locally convex space (l.s.c.). The following question has then come up, and I'm unable to find an answer.
...
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A Hausdorff topological vector space is locally convex if and only if $0$ has a neighborhood base of balanced, convex, absorbing space
The following is Theorem V.1 in Reed & Simon's book on functional analysis.
Let $V$ be a vector space with a Hausdorff topology in which addition and scalar multiplication are separately ...
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Containment of kernels of continuous seminorms
Let $X$ be a locally convex space, whose topology is defined by a family of seminorms $\mathcal{P}$.
Fact. For every continuous seminorm $q: X\to \mathbb{F}$ and positive number $\epsilon>0$, the ...
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Is Schwartz space on $\mathbb{R}^n$ the closure of the span of tensor products of functions in Schwartz space on $\mathbb{R}$?
I have spent my spare time picking up Fourier analysis on $\mathcal{S}(\mathbb{R})$. Instead of rewriting everything to accommodate notation and proofs on $\mathcal{S}(\mathbb{R}^n)$, I think it would ...
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If $x_n \to x$ weakly then $\exists y_n \in \operatorname{co}\{x_1, \dots, x_n\}$ s.t. $y_n \to x$.
Let $X$ be a normed space and $x_n \to x$ weakly. I can show that there exists $\{y_n\} \subseteq \operatorname{co}\{x_1, x_2, \dots \}$ s.t. $\|y_n -x \| \to 0$ where co is convex hull. To show the ...
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Extreme points in the space of ucp maps
Suppose $M$ and $N$ are $\mathrm{II}_1$ factors. Let $\tau\mathrm{UCP}(M,N)$ be the convex space of trace-preserving UCP maps from $M$ to $N$, equipped with the topology of pointwise weak* convergence....
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How to check if a locally convex space is complete?
I am given the definition that a locally convex space $V$ is complete if and only if it is isomorphic to its completion $\tilde{V}$.
The completion is constructed in the following way: We have a ...
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The algebra of smooth functions as a projective tensor product
In this quesiton it is mentioned that the algebra of smooth functions from a compact Manifold $M$ to a Frechet space $F$ is isomorphic to the projective tensor product $C^{\infty}(M) \otimes_\pi F$, i....
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Pettis integral on locally convex space and seminorms
Let $E$ be a locally convex Hausdorff space, and $X$ be a locally compact Hausdorff space which we fix a positive Radon measure $\mu$. Assume that $f: X \to E$ is a function such that the Pettis-...
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Derivative of vector-valued functions using duality
Given a function $f \colon \mathbb{R^n} \to E$, where $E$ is a locally convex Hausdorff TVS, we say that $f$ is differentiable at $x \in \mathbb{R^n}$ if there exists some $Df(x) \in \mathcal{L}(\...
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Why Gateaux derivative is a distribuition?
Let $E$ be a vector bundle , $E^*$ the dual bundle and $D$ a density bundle. Denote by $\Gamma(E)$ the space of section of the bundle $E$.
By definition the a distribution $\omega$ in a vector ...
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Hausdorff property of locally convex spaces
In the book "Funktionalanalysis" by Werner were locally convex spaces defined by seminorms. One statement is that a locally convex space is Hausdorff, if there exist a neighborhoodbasis of ...
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Uniform continuity and seminorms
I recall that a seminorm is basically a norm that is not necessarily positive definite.
Let $(E,(p_n)_{n\in\mathbb{N}})$ be a Fréchet space, meaning each $p_n$ is a seminorm and if we equip $E$ with ...
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Is infrabarrelled and quasibarrelled the same?
Am I right in saying that infrabarrelled and quasibarrelled is defined exactly the same way.
If so, which terminology is used most nowadays?
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Weak quasi-completion of a locally convex space
Let $X$ be a locally convex space.
Then as far as I understand the bidual of $X$ with the weak topology, $(X_\beta')_\sigma'$, is like a quasi-completion of $X_\sigma$.
Namely, if $B \subseteq X$ is ...
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$A$ closed and $B$ compact $\implies \exists U$ open convex and balanced with $(A + U) \cap (B + U) = \emptyset$
Let $X$ be a topological vector space, a set $U \subset X$ is said to be balanced if $\lambda u \in U$ for any $u \in U$ and $\lambda \in [-1,1]$
I would like to prove the following theorem
Theorem
...
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why $q\left(T\left(x-x_j\right)\right) \rightarrow 0$ implies $T\left(x_j\right) \rightarrow T(x)$?
Let $\mathscr{X}$ and $\mathscr{Y}$ be locally convex vector spaces, let $\mathscr{P}$ and $\mathscr{Q}$ be inducing collections of seminorms for $\mathscr{X}$, respectively $\mathscr{Y}$, and let $T: ...
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Intersection of the convex hull of a compact set with a finite dimensional subspace
Let $H$ be a real Hilbert space, $K\subset H$ be a compact subset, $E\subset H$ be a finite dimensional linear subspace, then is the intersection $\mathrm{co}(K)\cap E$ closed in $E$? (Here $\mathrm{...
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Closed convex subset of $l^1$ has infinitely many elements of minimal norm
I need help with the following:
Closed convex subset $M=\{x \in l^1: \sum_{n=1}^{\infty}x_n=1\}$ of space $l^1$ containes infinitely many elements of minimal norm. Prove this statement.
I wanted to ...
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A Schauder basis of Schwartz space. [closed]
It is well-known that hermite functions $\{h_n(x)\}_n$ form a Schauder basis of the Schwartz space $\mathcal{S}(\mathbb{R})$.
Let $\alpha, \beta \in \mathbb{R}^*$. Does the 'modified' family of ...
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Semi-norms generating the usual topology of Schwartz space
Consider the following family of semi-norms on the Schwartz space
$$\|f\|_{m,n}=\sup_{x\in \mathbb{R}}|(1+|x|)^m f^{(n)}(x)| \;\;\; m,n\in \{0,1,2,...\} $$
It is well known in the litterature that ...
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Convexity notion versus local pathwise connectedness
Let $X$ be a topological vector space. Let us call a set $S\subset X$ almost-convex if, for every $x,y\in S$ and every $V$ a neighborhood of zero, there exists a continuous path from $x$ to $y$ ...
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The closure of a face of a convex set is a face of the closure of the set
Let $C$ be a convex set in a normed space and let $F$ be a face of $C.$ By a "face", it is understood a set such that if the segment $u + (0,1)(v-u)$ with two end points in $C$ intersects $F$...
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Subspaces and Mackey topology
Let $E$ be a locally convex (Hausdorff) topological vector space. It's known that if $G$ is a linear subspace of $E$, then if $E$ has weak topology, then $G$ as a subspace also has weak topology.
We ...
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Convex hull of a set of extreme points
Consider the space $\ell^p$ over the reals. Consider a subset of the unit ball
$$
A = \lbrace (x_n)_n^\infty \in \ell^p; \lVert x \rVert_p \leq 1 \ \& \ \lVert x \rVert_\infty \leq a \rbrace
$$
...
user860263
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Extreme points of a modified unit ball $\ell^p$
Consider the space $\ell^p$ over the reals. Denote
$$
A = \lbrace x \in \ell^p; \lVert x \rVert_p \leq 1 \ \& \ \lVert x \rVert_\infty \leq \alpha \rbrace.
$$
For $p \in (1, \infty)$, $\alpha > ...
user860263
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0
answers
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Compare weak and finest locally convex topology on $\mathbb{R} [x]$
Let $V = \mathbb{R} [x] \cong \bigoplus_{\mathbb{N}} \mathbb{R}$ be the vector space of univariate polynomials, or the space of real sequences that have all but finitely many elements equal to zero. ...
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What's the problem with the evaluation map not being continuous?
When introducing differentiable functions between locally convex spaces, many authors (e.g. Bastiani, Keller, Kriegl-Michor) notice that the evaluation map
$$ E\times E^*\to\mathbb R,\qquad (x,L)\...
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A proof of Krein-Milman theorem
Disclaimer
This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question.
Anyway, it is written as problem. Have fun! :)
Let $X$ be a locally convex Hausdorff ...
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