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Questions tagged [locally-convex-spaces]

Topological vector space with a locally convex topology, i.e. induced by a system of seminorms.

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Topology on the space of compactly supported functions

Let $X$ be a locally compact Hausdorff topological space, and let $C_c(X)$ be the space of compactly supported $\mathbf{R}$-valued continuous functions. It is a well-known fact that this space is not ...
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3 votes
1 answer
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Is compact-open topology stable with respect to injective limits?

Let $X$ be a locally convex space, and $\{Y_i;\ i\in I\}$ a covariant system of locally convex spaces over a partially ordered set $I$, i.e. a system of linear continuous mappings is given $\sigma^j_i:...
Sergei Akbarov's user avatar
3 votes
1 answer
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Is the exponential map of a locally compact group a local homeomorphism?

We consider a locally compact abelian group $G$. We equip the real vector space $A(G)$ of continuous group homomorphisms $\mathbb{R}\to G$ with the topology of uniform convergence on compact subsets ...
Hans's user avatar
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Colimits of locally convex spaces in the categories of topological vector spaces vs locally convex spaces

Let $S$ be a set and let $V_s$ be a family of locally convex topological vector spaces (LCSs) indexed by $s \in S$. Let $V$ be a vector space (without topology) and let $T_s:V_s \to V$ be a family of ...
James Tener's user avatar
1 vote
1 answer
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Bounded $C_0$-semigroups on barrelled spaces are equicontinuous

I have the following question: Let $X$ be a barrelled locally convex space (every absolutely convex, absorbing and closed set is a neighborhood of zero) and let $(T(t))_{t\geq0}$ be a $C_0$-semigroup, ...
Sonam Idowu's user avatar
1 vote
1 answer
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Local completion of bornological space

I recently stumbled across this old publication which defines the local completion of a Hausdorff locally convex space. The construction is as follows: A Hausdorff locally convex space $E$ is locally ...
iolo's user avatar
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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(...
CoffeeArabica's user avatar
2 votes
1 answer
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Is the projective limit $\mathcal{D}(\mathbb{R})$ separable?

Let $\mathscr{D}(\mathbb{R})$ be the set $C_0^\infty(\mathbb{R})$ of smooth functions with compact support endowed with the following topology: The initial topology with respect to the family maps $(\...
CoffeeArabica's user avatar
3 votes
0 answers
251 views

Equality of topologies in the spaces of section of a vector bundle

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 ...
amilton moreira's user avatar
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Quasi-completion of a locally convex space as a space of linear functionals on its dual

A locally convex topological vector space is said to be quasi-complete if its closed bounded subsets are complete. Given a locally convex space $E$, one can show the existence of a Hausdorff quasi-...
P. P. Tuong's user avatar
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1 answer
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Global control of locally approximating polynomial in Stone-Weierstrass?

Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials. Given $\varphi\in C_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that $$\...
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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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1 vote
1 answer
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Takesaki lemma: existence Gelfand-Pettis integral

Consider the following fragment from Takesaki's second volume of "Theory of operator algebras" (lemma 2.4, chapter VI "Left Hilbert algebras"). In another post, it was explained ...
Andromeda's user avatar
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1 answer
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Large ideally convex sets

Let $E$ be a Banach space. A set $C \subseteq E$ is called ideally convex if for every bounded sequence $(x_n)$ in $C$ and for every sequence $(\lambda_n)$ in $[0,1]$ that sums up to $1$ the vector $\...
Jochen Glueck's user avatar
5 votes
2 answers
523 views

Separate continuity implies (joint) continuity

I believe that the following fact is true and I am looking for a reference. Let $X$ be a locally compact Hausdorff topological space (may be assumed to be metrizable). Let $V$, $W$ be Fréchet spaces. ...
asv's user avatar
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4 votes
1 answer
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Why Gateaux derivative is a distribution?

Thanks to Jan Bohr answer and comment I edited this question. 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 $E$....
amilton moreira's user avatar
1 vote
0 answers
185 views

Is the strong topology the strongest?

Let $X$ be a topological vector space. We know that the weak topology $\sigma(X,X^*)$ is the weakest locally convex topology in $X$ that make every $f \in X^*$ continuous. Consequently, if $\tau$ is ...
Nicolay Avendaño's user avatar
1 vote
0 answers
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Seminorms ported by a compact

Let $X$ be a Banach space, and $U\subset X$ an open balanced set. A seminorm $\rho$ in $\mathcal{H}(U)$ is said to be ported by a compact $K\subset U$ if for all open set $V$ such as $K\subset V \...
Nicolay Avendaño's user avatar
4 votes
1 answer
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Weak* bounded and strong bounded are the same?

I have this problem at the moment which the strong topology $\beta (E;E^* )$ is defined, when $E$ is a locally convex space. This topology is generated by the basic open sets: $$U=\{x \in E : \sup_{f \...
Nicolay Avendaño's user avatar
2 votes
0 answers
82 views

Pullbacks of LCS-valued distributions

Suppose $X$ is a locally convex space. Since the distributions $\mathcal{D}'\!(M)$ ($M$ a manifold) are a nuclear space, there is a canonical meaning to the topological tensor product $X\,\widehat{\...
J_P's user avatar
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1 vote
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When is the metric on a Fréchet space homogeneous

Let $(F,d)$ be a Fréchet space over $\mathbb{F}\in \{\mathbb{C},\mathbb{R}\}$. Are there conditions under which, there exists some $C,d>0$ such that: for every $f\in F$ and every $k\in \mathbb{F}$ ...
ABIM's user avatar
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3 votes
0 answers
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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)\...
Lorenzo Cecchi's user avatar
5 votes
1 answer
431 views

Properties of $C_B(X)$ equipped with the strict topology

Let $X$ be a Polish space. $C_B(X)$ is the space of bounded continuous functions $X\to\mathbb{R}$ equipped with the strict topology, which is the finest locally convex topology that agrees with the ...
Alex Appel's user avatar
2 votes
1 answer
290 views

Sequential separability on $C_p(X)$

Definition. Let $E$ be a topological space. Suppose that $E$ contains a sequence $\{x_n\}$ such that for every $x\in E$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x=\lim x_{n_k}$. ...
ABB's user avatar
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6 votes
1 answer
171 views

Some special sequence in $C(\mathbb{R})$

Let us consider $C(\mathbb{R})$, the space of continuous functions on the reals. Q. Does there exist a sequence $\{f_n\}$ in $C(\mathbb{R})$ such that for every $f\in C(\mathbb{R})$ one may find a ...
ABB's user avatar
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1 vote
1 answer
223 views

An approximation property in a separable topological vector space

Let $X$ be a topological vector space. Let us say that $X$ enjoys sequential separablity if there exists a sequence $\{x_n\}$ in $X$ such that for every $x\in X$ there exists a subsequence of $\{...
ABB's user avatar
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0 votes
1 answer
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Borel sigma algebra coming from the weak topology on TVS

Let $(X,\tau)$ be a topological vector space. Suppose that, there is a sequence of subsets $X_n\subseteq X$ with, For every $n\in \mathbb{N}$, the topology $\tau$ on $X_n$ is second countable and ...
ABB's user avatar
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3 votes
1 answer
153 views

$\varepsilon$-product in Bierstedt's paper

I am reading K.D.Bierstedt's paper Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. I. Journal für die reine und angewandte Mathematik 259 (1973): 186-210. It is ...
Sergei Akbarov's user avatar
5 votes
0 answers
147 views

Bochner–Minlos Theorem for locally convex spaces and their duals

Let $(X,\tau)$ be a locally convex space and $(X^{*},\tau_{s})$ be its topological dual space equipped with the strong topology. Denote by $S(X,X^{*})$ the collection of operators from $X$ to $X^{*}$ ...
JustWannaKnow's user avatar
3 votes
0 answers
113 views

Approximation of a linear functional by linear continuous functionals

Let $X$ be a locally convex space, $T$ a balanced convex compact set in $X$, and $f:X\to\mathbb{C}$ a linear functional which is (not necessarily continuous on $X$, but) continuous on $T$. It is not ...
Sergei Akbarov's user avatar
4 votes
1 answer
218 views

Reference for Choquet-like theorem

While reading a paper, I encountered the following statement: Let $K$ be a convex compact subset of a locally convex topological vector space. If $\mu \in P(K)$ is a Radon probability measure on $K$, ...
Andromeda's user avatar
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1 vote
0 answers
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Nested nets of closed bounded star-shaped sets in a semi-reflexive space

Among Hausdorff locally convex spaces, semi-reflexivity is characterized by the weak topology having the Heine-Borel property. It follows that, in a semi-reflexive space, every nested net of closed ...
Alcen's user avatar
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1 vote
0 answers
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Smooth representations of a direct product of groups

Let $G_1, G_2$ be Lie groups, and let $E_i$, $i=1,2$ be a smooth representation of $G_i$ in a locally convex complete Hausdorff TVS. Then $E_1\hat\otimes E_2$ is a smooth representation of $G_1\times ...
Peter Kravchuk's user avatar
3 votes
0 answers
82 views

Bilinear maps on smooth vectors of unitary representations

Let $G$ be a connected semi-simple real Lie group with finite center. Let $R_i$ ($i=1,2,3$) be unitary irreducible representations of $G$. Let $R_i^\infty$ be the corresponding representations of $G$ ...
Peter Kravchuk's user avatar
1 vote
1 answer
187 views

Hyperplane separation of a concave functional and a point, in domain theory

Problem: Let $D$ be an $\omega$-BC domain, and $[D\to[0,\infty]]$ be the space of Scott-continuous nonnegative functions on $D$, equipped with the obvious ordering and the Scott-topology. EDIT: I don'...
Alex Appel's user avatar
1 vote
0 answers
90 views

Dual of essentially compactly supported functions on a hemi-compact Radon space

Let $X$ be a hemicompact Radon space and fix a $\sigma$-finite Radon measure $\mu$ on $X$. Let $L(X_n)$ denote the subspace of $L_{\mu}^1(X)$ of "functions" which are $\mu$-essentially ...
Catologist_who_flies_on_Monday's user avatar
1 vote
0 answers
166 views

Effect of dualization of density

Let $D\subset X$ be a dense subset of a complete separable locally convex space $X$ over $\mathbb{R}$. Though the question seems simple enough, I can't seem to find the answer in the literature: If $...
John_Algorithm's user avatar
3 votes
2 answers
985 views

Topologies on space of compactly supported continuous functions

Let $X$ be a locally compact Hausdorff space. As far as I understand, the space $C_c(X) = C_c(X; \mathbb{C})$ of compactly supported continuous complex-valued functions on $X$ is (most?) often ...
epitaph's user avatar
  • 89
2 votes
1 answer
235 views

Semi-norms on LCS inductive limit of Banach Spaces

Let $(E_n,i_n)_{n\in\mathbb{N}}$ be an direct system of Banach spaces in the category of locally convex spaces (LCSs) with continuous linear maps and let $E_{\infty}$ by their inductive limit. What ...
ABIM's user avatar
  • 4,809
7 votes
0 answers
292 views

Weaker version of the Borel lemma for vector-valued functions

Borel's lemma for Frechét-spaces $V$ says: (i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists a smooth $f: \mathbb{R} \to V$ such that $$f^{(j)}(0) = v_j.$$ For general locally ...
Jannik Pitt's user avatar
  • 1,280
4 votes
1 answer
276 views

Compatibility of inductive and projective limits with dualization in functional analysis

Assume $(A_i)_{i \in I}$ is a family of locally convex topological vector spaces which are all moreover assumed to be Banach spaces. We assume moreover that $(A_i)_{i \in I}$ has additional structure ...
user267839's user avatar
  • 5,938
2 votes
1 answer
95 views

Subspaces of quasi-complete locally convex spaces

Let $V$ be a quasi-complete Hausdorff locally convex space. (By quasi-complete, one means that every bounded closed subset of $V$ is complete.) For a bounded closed absolutely convex subset $B$, ...
Chris 's user avatar
  • 303
0 votes
1 answer
77 views

Are bounded sets in second duals of locally convex spaces weak* pre-compact?

Let $X$ be a locally convex Hausdorff space. Then $X$ injects into $X^{**}$ via the canonical map $\kappa: X\to X^{**}$. Now, $X^{**}$ carries the weak* topology. Let $B$ be a bounded set in $X$. Is $\...
Jean R.'s user avatar
1 vote
0 answers
72 views

Good source for Jordan Fréchet algebras

Is there any good source for Jordan Fréchet (or more generally, Jordan locally convex) algebras? I'm looking for something on the level similar to the level of the book "Banach and Locally Convex ...
Jaŭhien Piatlicki's user avatar
3 votes
1 answer
92 views

Characterizing 'very homogeneous' finitely valued stochastic processes

Fix a positive integer $n$. Let $X = \{X_i\}_{i \in \mathbb{N}}$ be a discrete time stochastic process such that each $X_i$ is a $\{0,\dots,n-1\}$-valued random variable. Suppose that the joint ...
James E Hanson's user avatar
7 votes
1 answer
657 views

Closed convex hull in infinite dimensions vs. continuous convex combinations

tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$? I am essentially asking for the most general, infinite-dimensional analogue of ...
user163625's user avatar
1 vote
0 answers
109 views

Is the Vietoris topology on compact subsets of $\mathbb R^n$ locally convex?

The title question says it all really. If the question is negative for compact subsets of $\mathbb R^n$, is it affirmative for compact and convex subsets of $\mathbb R^n$? How about for all nonempty ...
aduh's user avatar
  • 839
0 votes
1 answer
76 views

Ultrabornological representation for the space of uniformly continuous functions?

Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space $$ C_{\omega_i}(\mathbb{R}^n,\...
ABIM's user avatar
  • 4,809
1 vote
1 answer
815 views

Known dense subset of Schwartz-like space and $C_c^{\infty}$?

After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...
ABIM's user avatar
  • 4,809
5 votes
1 answer
207 views

Are linear continuous mappings open on totally bounded sets?

Let $X$ and $Y$ be locally convex spaces, and $\varphi: X\to Y$ a linear continuous mapping. Suppose first that $S$ is a compact set in $X$. Then $\varphi$, being considered as a mapping from $S$ to $\...
Sergei Akbarov's user avatar