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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questions with no upvoted or accepted answers
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Admissible relations in a Banach algebra
Suppose that $\mathbb{C}\left\langle x, y \right\rangle = R$ is a free (associative and unital) algebra and $f \in R$. I wonder whether there exists a (unital) Banach algebra $A$ and a non-zero pair $...
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290
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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 ...
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132
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Does every locally convex space with a Schauder basis have the approximation property?
For Banach spaces, the existence of a Schauder basis implies that this space has the approximation property.
Since both the notion of Schauder bases and of the approximation property are well ...
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5
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1
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401
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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 ...
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144
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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 $X^{*}$ ...
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5
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0
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188
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Useful notion for locally convex spaces - well known?
In my current work the following property of maps between locally convex spaces showed up at several places and proved to be useful. It seems quite elementary to me, so I would like to know whether it ...
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126
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Is the Baireness a 3-space property of topological groups
It is known that the product of two Baire spaces can be meager.
On the other hand, by a recent result of Li and Zsilinszky the product of two Baire spaces is Baire if one of the spaces is countably ...
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317
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The second dual of $C(X)$ with the compact-open topology
Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...
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5
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796
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Dual of representation is irreducible implies the representation is irreducible?
Suppose $V$ is an infinite dimensional $\mathbb{Q}_p$-representation of Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_p$. If its dual representation $V^{\prime}$ is irreducible then, is it always true ...
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205
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A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff
We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...
- 1,220
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208
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Tensors and Nuclear/Fredholm Operators
For a locally convex Hausdorff spaces $E$, consider the canonical map
$$\overline{\psi}:E^\prime \hat{\otimes}_\pi E \longrightarrow L(E_\sigma)$$
that maps the projective tensor product to the space ...
- 9,591
5
votes
0
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187
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Topology for bounded operators quotiented by Schatten ideal
I saw this particular question on stackexchange. Since there has been zero answers and since I've been interested in this question myself I want to ask it here.
Given the $C^{\ast}$-algebra of bounded ...
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4
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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....
- 1,262
4
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A characterization of nuclear functionals in terms of continuity with respect to some special topologies on $B(X)$?
I think, nuclear functionals on the space of operators $B(X)$ (on a Banach space $X$) must have a characterization in terms of some special continuity. I would be grateful if somebody could help me ...
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113
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point-wise approximation of the identity in hereditary Lindelof spaces
Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.
Q. Can we concluded that $X$ is hereditery ...
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4
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140
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A point concerning Fremlin's example on Borel sets in non-separable Banach spaces
Let $E$ be a Banach space. Let us consider the following three sigma algebras on $E$.
$~~~~\mathcal{B}$= The $\sigma$-algebra coming from the norm topology.
$~~\mathcal{M}$= The sigma algebra ...
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4
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0
answers
315
views
Compactly supported distributions as a projective G-module
For a Lie group $G$ and a locally convex space $V$ let $\mathcal{E}(G,V)$ be the locally convex space of smooth functions from $G$ to $V$, and accordingly $\mathcal{E}_c^\prime(G,V)$ the space of ...
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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 ...
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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-...
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369
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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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0
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112
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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 ...
- 7,274
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0
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82
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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$ ...
- 1,468
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150
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Elements of vector-valued $L^1$-spaces
Let $E$ be a complete locally convex space and let $(X, \Sigma, \mu)$ be a measure space where $\mu$ is a Radon measure. Then the space $L^{1}(X,E)$ is defined as a the completion of the space $S(X,E)$...
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317
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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(...
2
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0
answers
82
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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{\...
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2
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1
answer
287
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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}$. ...
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79
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Associated barrelled topology of norm topology on $C_c(X)$
Let $X$ be a locally compact Hausdorff space, $C(X; K)$ the Banach space of continuous functions on $X$ with support in $K$, for compact $K \subseteq X$, and $C_c(X) = \lim_K C(X; K)$ the locally ...
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70
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Gluing together mixed normed vector spaces with mixed topologies
This is a variant of this question.
Definitions/Facts
$Ball_1(X)$ denotes the unit ball (about $0$) in a normed vector space $X$.
MixTop of triples of pairs $(X,\tau)$ of normed vector spaces $X$ ...
- 4,969
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0
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117
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A new topology on the dual of a locally convex space?
Working with the separable quotient problem for locally convex spaces we (with Saak Gabriyelyan) arrived to an interesting topology on the dual of a locally convex space and we would like to know if ...
- 40.9k
2
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58
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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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2
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0
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83
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An equivalent condition for second countable locally convex spaces
Let $(X,\tau)$ be a locally convex topological vector space.
Assume for any arbitrary topological base $\mathcal{E}$ of $\tau$ we have that: the Borel sigma algebras coming from $\mathcal{E}$ and $\...
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answers
355
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Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?
Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with
weak-$*$ topology (weak topology induced by the continuous functions).
Consider a ...
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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 ...
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1
vote
0
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95
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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 \...
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1
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0
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55
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 $k\in \mathbb{F}$ ...
- 4,969
1
vote
0
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50
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 ...
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58
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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 ...
- 1,468
1
vote
1
answer
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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'...
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1
vote
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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 ...
1
vote
0
answers
153
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 $...
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 ...
1
vote
0
answers
107
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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 ...
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Refinement: Can $L^1_{loc}$ be represented as colimit?
Let $\mu$ be a $\sigma$-finite measure on a measure space $(\mathbb{R}^d,\Sigma)$. Can $L^1_{\mu,loc}$ be represented as an injective-limit in the category of LCS (locally convex spaces and ...
- 4,969
1
vote
0
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96
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Gluing together dense subset of Projective Limit in $Ban_1$
Let $(X_n,\pi_n^{m})$ be a countable projective system in the category Ban$_1$ of Banach spaces and short linear maps (is (continuous) linear constructions). Then (co)-completeness of this category ...
- 4,969
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vote
0
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51
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A different kind of weighted Hardy space
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$, let $\mathcal{A}\left(\mathbb{D}\right)$ denote the vector space of all complex-valued functions which are holomorphic on $\mathbb{D}$, and ...
- 1,266
1
vote
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67
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Working in coordinates with topologies on the algebra of continuous functions
Let $X$ be a Hausdorff completely regular topological space, and let $C_b (X)$ be its algebra of continuous bounded functions. Endow $C_b (X)$ with a topology given by some seminorms, that contains ...
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1
vote
0
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47
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Analogues of properties (DN) and (Ω) for more general locally convex spaces
In the structure theory of Fréchet spaces, especially results around splitting short exact sequences, the properties (DN) and (Ω) play a major rôle. There are many variants, but they are phrased in ...
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1
vote
0
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950
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The dual of the space of smooth functions that vanish at infinity
Let $U \subset \Bbb R ^n$ be an open subset and let $\mathcal C$ be the space of the smooth functions on $U$ that vanish at infinity, endowed with the seminorms $p_\alpha (f) = \sup \limits _{x \in U} ...
- 5,282
1
vote
0
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161
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Example of locally convex space such that its weak and initial topology coincide
If $X$ is a normed space than it is well known that its norm topology and its weak topology coincide if and only if $X$ is finite-dimensional.
Now I asked myself the same question about general ...
- 2,926
1
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0
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368
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Adjoint operators in LCS
Before my main question let me start with the following notions.
Let $X$ and $Y$ be locally convex spaces and let $T \colon X \rightarrow Y$ be a linear mapping. The adjoint of $T$ is an operator
$T^...
- 145