Questions tagged [locally-convex-spaces]
Topological vector space with a locally convex topology, i.e. induced by a system of seminorms.
171 questions
1 vote
0 answers
69 views
Weak convergence of nets of measures in a locally convex space
Let $X$ be a locally convex space, let $(p_t)_{t\in T}$ be a net of Borel probability measures on $(X,\sigma(X,X^*))$, and let $p$ be a $\tau$-additive (in particular, Radon) with respect to the weak ...
- 291
1 vote
0 answers
117 views
Do bounded weakly continuous functions extend from $X$ to $\mathbb{R}^{T}$?
Let $X$ be a locally convex topological vector space and let $X^*$ denote its dual. Let $T \subset X^*$ be a Hamel basis of $X^*$. Consider the canonical map $$ \Phi\colon X \to \mathbb{R}^T, \qquad \...
- 291
7 votes
2 answers
405 views
When does the smooth D-topology on a sequential Hausdorff lctvs coincide with the original one?
Following on from my previous question, assume that we have a sequential Hausdorff lctvs $V$, and use the Michal–Bastiani calculus for notions of differentiability, if needed (though probably not for ...
- 36.8k
4 votes
0 answers
102 views
When is the quotient of an LF space by a closed subspace LF (Reference request)
I know that the quotient of a Fréchet space by a closed subspace is Fréchet (see for example https://math.stackexchange.com/questions/4735928/is-the-fr%C3%A9chet-quotient-space-given-by-the-induced-...
- 113
10 votes
1 answer
329 views
Density of test functions in space of distributions
Let $\Omega\subset\mathbb{R}^{d}$ be some open set. The following density result is very well-known in distribution theory and can probably be find in many resources: Proposition 1: For every $u\in\...
user199422
8 votes
2 answers
416 views
Strict inductive limits of sequentially complete spaces
PS: I already asked this question in MSE, and someone said I have better chances to get an answer here than there! (Keyword: Spaces of distribution people!) If $\{A_n\}$ is a strict inductive sequence ...
- 283
4 votes
0 answers
152 views
Closedness of the polynomials in the space of mean periodic functions given an inductive limit topology
This question has been cross-posted to MSE. This question stems from the article "Fourier transform for mean periodic functions" by Lázsló Székelyhidi, on which I had already made two MSE ...
- 91
4 votes
1 answer
277 views
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 ...
- 43
3 votes
1 answer
270 views
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:...
- 7,456
3 votes
1 answer
249 views
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 ...
- 3,081
4 votes
0 answers
127 views
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 ...
- 498
1 vote
1 answer
134 views
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, ...
- 129
1 vote
1 answer
120 views
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 ...
- 701
2 votes
0 answers
354 views
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(...
2 votes
1 answer
285 views
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 $(\...
3 votes
0 answers
293 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 ...
3 votes
0 answers
90 views
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-...
- 529
2 votes
1 answer
340 views
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 $$\...
- 453
4 votes
0 answers
153 views
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....
- 5,210
2 votes
1 answer
504 views
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 ...
- 257
5 votes
1 answer
222 views
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 $\...
- 13k
5 votes
2 answers
952 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. ...
- 23k
4 votes
1 answer
361 views
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 $E$....
1 vote
0 answers
223 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 ...
- 203
1 vote
0 answers
118 views
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 \...
- 203
4 votes
1 answer
690 views
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: $$U=\{x \in E : \sup_{f \...
- 203
2 votes
0 answers
112 views
Pullbacks of LCS-valued distributions
Suppose $X$ is a locally convex space. Since the distributions $\mathcal{D}'\!(M)$ ($M$ a manifold) are a nuclear space, there is a canonical meaning to the topological tensor product $X\,\widehat{\...
- 439
1 vote
0 answers
91 views
When is the metric on a Fréchet space homogeneous
Let $(F,d)$ be a Fréchet space over $\mathbb{F}\in \{\mathbb{C},\mathbb{R}\}$. Are there conditions under which, there exists some $C,d>0$ such that: for every $f\in F$ and every $k\in \mathbb{F}$ ...
- 4,942
3 votes
0 answers
533 views
What's the problem with the evaluation map not being continuous?
When introducing differentiable functions between locally convex spaces, many authors (e.g. Bastiani, Keller, Kriegl-Michor) notice that the evaluation map $$ E\times E^*\to\mathbb R,\qquad (x,L)\...
5 votes
1 answer
644 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 ...
- 353
4 votes
1 answer
410 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 $x=\lim x_{n_k}$. ...
- 4,140
6 votes
1 answer
209 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 ...
- 4,140
1 vote
1 answer
280 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 $\{...
- 4,140
0 votes
1 answer
265 views
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 ...
- 4,140
3 votes
1 answer
195 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 ...
- 7,456
5 votes
0 answers
197 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 $X^{*}$ ...
- 3,645
3 votes
0 answers
150 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 ...
- 7,456
4 votes
1 answer
277 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 $K$, ...
- 257
1 vote
0 answers
71 views
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 ...
- 11
2 votes
0 answers
85 views
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 ...
- 1,498
3 votes
0 answers
97 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 $G$ ...
- 1,498
1 vote
1 answer
265 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'...
- 353
1 vote
0 answers
116 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 ...
2 votes
0 answers
284 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 $...
- 109
4 votes
2 answers
2k 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 ...
- 99
3 votes
1 answer
346 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 ...
- 4,942
7 votes
0 answers
327 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 ...
- 1,574
6 votes
1 answer
419 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 ...
- 3,884
2 votes
1 answer
133 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 $B$, ...
- 303
0 votes
1 answer
120 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 $\...
- 1