All Questions
Tagged with fa.functional-analysis topological-vector-spaces
55 questions with no upvoted or accepted answers
17
votes
0
answers
677
views
Are dualizable topological vector spaces finite-dimensional?
Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product.
Every finite-dimensional ...
- 37.8k
14
votes
0
answers
860
views
strong topologies on $C_c^\infty$
UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
- 486
6
votes
0
answers
293
views
Looking for Mackey's PhD thesis, "The subspaces of the conjugate of an abstract linear space"
I'm looking for a copy of George Mackey's PhD thesis, The subspaces of the conjugate of an abstract linear space (Harvard Univ., 1942), but am currently struggling to find one online, with the only ...
- 11.8k
6
votes
0
answers
219
views
Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
- 370
6
votes
0
answers
1k
views
Condensed/liquid vector spaces and path integrals
[Edited to take into account comments.]
Background
One approach to the problem of making rigorous various measures on spaces of paths (for example, the Wiener or Feynman measure) is the time-slicing ...
- 61
6
votes
0
answers
272
views
Extension operators for topological vector space-valued smooth functions on closed sets
There are many known results about extension theorems for real-valued functions on closed sets, with varying levels of differentiability and so on, all very roughly following the Whitney approach. For ...
- 35.5k
6
votes
0
answers
697
views
distributions on Lie groups and representations
Let $G$ be a Lie group and $\pi$ a continuous action on $V$, a Fréchet space.
This action induces a representation of the space of compactly supported functions, $C_c(G)$, with convolution as product ...
- 342
5
votes
0
answers
211
views
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 ...
- 1,270
5
votes
0
answers
104
views
On the embedding of manifolds into infinite-dimensional spaces
Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
- 5,529
5
votes
0
answers
322
views
Differential operators acting on the Schwartz space
I asked a similar question on math stack exchange but didn't get an answer so I will try to ask it here. Any help/suggestion is most than welcome!
Let $D$ be a linear differential operator with ...
- 81
5
votes
0
answers
295
views
Convergence of convex combinations in topological vector spaces
I am studying certain quadratic forms on $L^0(m)$ equipped with the topology of (local) convergence in measure which in general is not locally convex. I am also interested in the situation where $m$ ...
4
votes
0
answers
101
views
There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions
Consider the space $\mathcal D'(M)$ of distributions on a manifold $M$.
Is there a ready reference for the fact that the Borel $\sigma$-algebra (for the strong dual topology) coincides with the weak ...
- 3,669
4
votes
0
answers
87
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 ...
- 398
4
votes
0
answers
108
views
Larger possible chain of closed subspaces in the dual of a Banach space
In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces.
My question is the following. If $X$ is an ...
- 153
4
votes
0
answers
176
views
Is the test function topology a Mackey topology?
I am a physicist, and I have lately been thinking about distributions as they appear in quantum field theory. In the standard development of the theory of distributions, one considers the space $C^{\...
- 41
4
votes
0
answers
481
views
Generalized Jensen's inequality for positively homogeneous functions
The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
- 519
4
votes
0
answers
147
views
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 ...
- 7,426
3
votes
0
answers
122
views
Analytic functions and Hyperfunction as TVS
I have several related questions on Analytic functions and Hyperfunction as topological vector spaces (I am mainly interested in questions 4,6,10):
For an open set $U\subset \mathbb C^n$ we can ...
- 2,639
3
votes
0
answers
120
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 ...
- 7,426
3
votes
0
answers
84
views
"Weakly" nuclear operators (terminology)
Recently, I'd come across the following kind of operators and I wonder if they have been considered before and given a name.
Let's say that a linear map $T:V\to W$ between locally convex topological ...
- 81
3
votes
0
answers
67
views
Non-linear weak*-continuous left inverses
Let $T\colon X\to Y$ be a continuous linear surjection between Frechet spaces. Then it is possible to use Michael's selection theorem to show that there exists a continuous (non-linear) function $g\...
- 31
3
votes
0
answers
59
views
Convergence of sesqui-holomorphic kernels on the diagonal
Let $X\subset \mathbb{C}^d$ be a domain.
A function (kernel) $K:X\times X\to \mathbb{C}$ is called sesqui-holomorphic if it is holomorphic in the first variable, and anti-holomorphic in the second ...
- 5,529
3
votes
0
answers
125
views
Commutative discrete cyclic operator groups on topological vector spaces
Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
- 1,959
3
votes
0
answers
98
views
How Universal is the Topological $\mathbb K$-algebra $C(\Omega, \mathbb K)$?
For $\Omega$ an arbitrary set the family $C(\Omega, \mathbb K)$ of all functions $\Omega \to \mathbb K$ becomes a complete topological $\mathbb K$-algebra under the topology of uniform convergence. ...
- 1,955
3
votes
0
answers
373
views
Closed Graph Theorem and Spaces Of Continuous Functions
Let $X$ be a (Tychonoff) topological space. Consider $C\left(X\right)$ being a topological vector space of all continuous scalar-valued functions with the compact-open topology.
Assume that $Y$ is a ...
- 5,529
2
votes
0
answers
320
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
0
answers
103
views
Schwartz kernel theorem for restricted operators
Let $(M,g)$ be a smooth Riemannian manifold. The celabrated kernel theorem of Schwartz shows that for any linear and continuous operator $A:C_{c}^{\infty}(M)\to C^{\infty}(M)$, there exists a ...
- 1,171
2
votes
0
answers
119
views
Fréchet and DF spaces
Is there a canonical way to make a DF-space Fréchet while keeping the same vectorial structure? Or the converse? I've been looking in the classical books for locally convex spaces but haven't found ...
- 61
2
votes
0
answers
211
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 $...
- 109
2
votes
0
answers
1k
views
Bounded weak and weak-$\star$ topologies and metrics
Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $\overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$). Set
$$
d(x,y) := \sum_{n=1}^\infty 2^{-n} |(x-y, h_n)|...
- 349
2
votes
0
answers
98
views
Smooth functions with values in bornological vector space
Let $U$ be an open set in $\mathbb{R}^n$ (or more generally, a manifold) and let $V$ be a separated bornological vector space. Do we have
$$C^\infty(U, V) \cong C^\infty(U) \,\hat{\otimes}\, V,$$
as ...
- 9,853
2
votes
0
answers
459
views
Weak topology on subsets of a normed space
I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset.
When is the norm a continuous function on $E$?
When is the metric induced by the ...
- 5,529
2
votes
0
answers
787
views
Regarding a proof in Bourbaki's Topological Vector Spaces
On Bourbaki's TVS Chapter IV pages 33-34, the last part of Proposition 2 can be formulated as follows:
Notations:
$K$ - The underlying field which is the real or complex number field;
$X$ - A ...
- 21
1
vote
0
answers
157
views
Density of Schwartz distributions in the space of distribution
Let $\mathscr S(\mathbf R^3)$ and $\mathscr D(\mathbf R^3 )$ be the space of Schwartz function and test function respectively, $\mathscr S'(\mathbf R^3)$ and $\mathscr D'(\mathbf R^3)$ be their duals....
1
vote
0
answers
54
views
Minimal F-semi-norms
There are conflicting terminologies in the literature on this subject, so let me define an F-semi-norms on a real vector space $E$ to be a subadditive function $\rho:E\to[0,+\infty)$ such that $\rho\...
- 5,529
1
vote
0
answers
220
views
Reference for Schwartz kernel theorem on vector bundles
In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has
Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...
1
vote
0
answers
190
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
70
views
Consequences of having unbounded points in a bornology
For a set $X$ a bornology $\mathcal{B}$ is essentially an ideal in the power set $\mathcal{P}(X)$. Many sources including Wikipedia state additional property that $X = \bigcup \mathcal{B}$. Call it a ...
- 519
1
vote
0
answers
142
views
Splitting of a topological vector space (TVS) into an (a) countable sum and (b) direct integral of subspaces
I thought that this would be a simpe question, and placed it here at the Mathematics Stackexchange. Now have to elevate it to Mathoverflow.
LANGUAGE
TVS = topological vector space. Any subspace of a ...
- 603
1
vote
0
answers
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 ...
- 11
1
vote
0
answers
48
views
Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?
Let $X$ be a Banach space equipped with the Mackey topology $m(X,X^*)$. We suppose that $\big(X,m(X,X^*)\big)$ is separable space. Let $H$ be a countable, $m(X,X^*)$-dense subset with $(H=-H)$.
Let $...
- 117
1
vote
0
answers
89
views
Do we have $M\hat{\otimes}_A N\cong M\otimes_A N$ if $M$ is a finitely generated projective $A$-module over a nuclear Frechet algebra $A$?
Let $A$ be a nuclear Frechet algebra with unit. Let $M$ be a right Frechet $A$-module and $N$ be a left Frechet $A$-module. Both $M$ and $N$ are assumed to be non-degenerate. We can define the ...
- 9,019
1
vote
0
answers
122
views
Mackey topology characterising property
Let $V$ be a topological $k$-vector space.
Let $V^{\star}$ denote the space of all linear functionals $V \rightarrow k$ and $V' \subset V^{\star}$ the subspace of all continuous linear functionals.
...
- 11
1
vote
0
answers
66
views
Characterization of the weak completion of $L^2(\mathbb{R}^d)$
The completion $\overline{L^2_w(\mathbb{R}^d)}$ of $L^2_w(\mathbb{R}^d)$ (i.e. the completion of $L^2(\mathbb{R}^d)$ endowed with the $\sigma(L^2(\mathbb{R}^d),L^2(\mathbb{R}^d))$ topology) is ...
- 488
1
vote
0
answers
218
views
Is the closed ball of a normed space closed in any Hausdorff locally convex topology, weaker than the norm topology?
Assume that we have a normed space $X$ and a subspace $Y$ of $X^{*}$ such that $Y^{\perp}=\{0\}$. They form a non-degenerate dual pare.
Moreover, $\|y\|=\sup_{x\in B_{X}}|\langle x,y\rangle|$, where $...
- 11
0
votes
0
answers
119
views
Boundedness of 2 times the unit ball
Suppose that $X$ is a topological vector space where the topology is given by a metric $d$ on $X$. Assuming that the unit ball
$$
B(0, 1) := \{x \in X : d(0, x) < 1\} \neq X,
$$
is it necessarily ...
0
votes
0
answers
97
views
Heine-Borel property for (probability) measures on $\mathcal{S}'$?
For Euclidean space $\mathbb{R}^n$, it is a well-known fact that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is a Montel space. Moreover, the space of tempered distributions $\mathcal{S}'(\mathbb{R}...
- 3,477
0
votes
0
answers
235
views
Analogue of $\ell^2(X)$ over an arbitrary Banach ring
Let $X$ be a set. Over the Banach fields $F=\mathbb{R}$ or $F=\mathbb{C}$ we can define the Banach space$$\ell^2(X)=\{\xi\colon X\to F\mid \sum_{x\in X}|\xi(x)|^2<\infty\}$$which satisfies a list ...
- 1,485
0
votes
0
answers
192
views
Reference request: an introduction to nuclear spaces
I am looking for a short introduction to nuclear spaces and nuclear operators. I am interested in these spaces as they often arise in mathematically rigorous quantum field theories. I have read the ...
- 721
0
votes
0
answers
75
views
Goldstine theorem in quasi-Banach spaces
A classical theorem of Goldstine is the following: Let $X$ be a Banach space and $J \colon X \to X''$ the natural inclusion. Then $J(B_X)$ is $\sigma(X'', X')$-dense in $B_{X''}$, where $B_Y$ is the ...
