All Questions
Tagged with fa.functional-analysis topological-vector-spaces
213 questions
6
votes
2
answers
3k
views
Closed convex bounded sets are weakly compact for which spaces?
It is known that for all reflexive Banach spaces, closed convex bounded sets are weakly compact (compact for the weak topology).
What is the general class of topological vector spaces for which this ...
- 549
4
votes
1
answer
615
views
Construction of a codimension 1 dense subspace without Zorn
Suppose $X$ is an infinite dimensional topological vector space
and $v\in X$ is non-zero. It is then not difficult to construct
a vector space $U\subset X$ so that
1) $U$ is dense in $X$.
2) $U+{...
- 462
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
8
votes
2
answers
590
views
Attempted Banachification of a space
In a recent blog post, Terry Tao mentioned the question of how to tell if a Hausdorff topological vector space admits a finer topological structure which happens to be the topology of a Banach space (...
- 148k
12
votes
2
answers
811
views
Nuclear operators/spaces and transfer operators
While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...
- 293
17
votes
3
answers
3k
views
Why is multiplication on the space of smooth functions with compact support continuous?
I asked the question
Why is multiplication on the space of smooth functions with compact support continuous? on M.SE
sometime ago but I didn't receive a satisfactory answer.
I was reading this ...
- 394
4
votes
1
answer
376
views
Is the topological dual of a Banach space weakly* closed in its algebraic dual?
The question is completely contained in the title :)
I can only add, that it is not difficult to give a counterexample for normed spaces, and also Banach-Steinhaus theorem implies the sequential ...
- 5,529
4
votes
1
answer
395
views
Linear extension operators for smooth functions: from compact sets to compact sets
I'm considering a situation where I have the linear restriction map of Fréchet spaces
$$
C^\infty(C_1) \to C^\infty(C_2)
$$
where $C_2 \hookrightarrow C_1$ are a pair of compact, connected subsets ...
- 35.5k
7
votes
1
answer
2k
views
Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$
We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset $\...
- 589
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
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
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 $m$ ...
5
votes
1
answer
491
views
Is the unitary group of a pre Hilbert space contractible?
I already posted my question on mathstackexchange
For a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for ...
- 51
12
votes
2
answers
647
views
Do locally convex topological vector spaces embed into diffeological spaces?
The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results ...
- 35.5k
3
votes
2
answers
326
views
Examples of TVS with no non-trivial open convex subsets
I give here the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.
...
- 1,355
8
votes
4
answers
811
views
Schwartz space of functions with values in a Frechet space
While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in ...
- 601
4
votes
1
answer
204
views
Can you pair $H^s(\Omega)$ and $H^{-s}(\Omega)$ on a domain $\Omega$?
Consider the fractional Sobolev spaces on $\mathbb R^n$
$H^s(\mathbb R^n) := \left\{ u \in \mathcal S'(\mathbb R^n) \; : \; ( 1 + |\xi|^2 )^{s/2} \hat u \in L^2(\mathbb R^n) \right\}$.
Let $\Omega$ ...
- 5,327
4
votes
1
answer
380
views
Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)'$ and $\substack{\text{colim} \\ i \rightarrow } H_i$
Let $\substack{\text{lim} \\ \leftarrow i} H_i$ be a nuclear space, considered as the limit of the codirected diagram $$... \to H_2 \to H_1 \to H_0,$$
with $f_{ji}:H_i \to H_j$ being the trace class ...
- 1,813
2
votes
1
answer
453
views
Weak convergence of probability measures on weak versus strong dual
The space of temperate distributions $S'(\mathbb{R}^d)$ is often equipped with the weak-$\ast$ or with the strong topology. When defining the notion of a probability measure on $S'(\mathbb{R}^d)$, ...
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
1
answer
269
views
Is a Fréchet Montel space distinguished?
Based on a couple of references, it seems that the answer is yes, see for example
Boneta-Dierolf, 1992 and Bierstedt-Bonet, 1989.
However, from a comment to the answer of this MO question, I infer ...
- 323
7
votes
1
answer
1k
views
Inductive/Projective Limits of Topological Algebras
It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...
- 71
5
votes
1
answer
608
views
Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?
Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully).
Suppose $g$ is holomorphic on $\mathbb{C}^n$...
- 29.7k
10
votes
2
answers
925
views
Isomorphisms between spaces of test functions and sequence spaces
I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
8
votes
1
answer
505
views
Examples of topologies compatible with a given dual pair
Let $\langle X, Y \rangle$ be a pair of vector spaces put in duality by a non-degenerate bilinear form $\langle \cdot, \cdot \rangle: X \times Y \to \mathbb{R}$. A topology $\tau$ on $X$ is called ...
- 5,824
2
votes
2
answers
252
views
Measures, orthogonal to holomorphic functions
Let $G$ be a domain in $\mathbb{C}^{d}$ and let $H\left(G\right)$ be the space of all holomorphic functions on $G$.
My question is how to characterize all such (Radon) measures $\mu$ on $G$, that $\...
- 5,529
15
votes
1
answer
2k
views
Bases for spaces of smooth functions
Let $S$ denote the space of rapidly decreasing sequences, which means sequences $a=(a_k)_{k=1}^\infty$ such that the numbers $p_d(a)=\sup\{k^d|a_k| : 1\leq k<\infty\}$ are finite for all $d\in\...
- 56.9k
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
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
3
votes
1
answer
860
views
Is the space of smooth functions with compact support a DF space?
Is there a good criterion, when a (nuclear) LF space is DF (DFN)? What are references to find out about that? If not, is there a known way, to construct a projective resolution of LF spaces? Prosmans ...
- 31
3
votes
3
answers
2k
views
When sequentially continuous linear functional is continuous?
Let $C^\infty(X)$ denote the space of infinitely smooth functions on a compact manifold $X$ (at the beginning one may assume that $X$ is a circle, though I need a more general case). Let $\mathcal{D}(...
- 21.8k
0
votes
1
answer
130
views
Extending affine maps defined on weakly closed sets to the whole topological space
Given $C$ a weakly closed convex subset of a (real) Banach space $B$, with $0\in C$ and $\varphi:C\longrightarrow \mathbb{R}$ weakly continuous, with $\varphi(0)=0$, can we extend $\varphi$ to a $\...
- 647
3
votes
2
answers
349
views
Linear operators on distributions with different topologies
Denote by $\mathscr{D}^\prime$ and $\mathscr{D}^\prime_b$ the space of distributions on $\mathbb{R}^n$ equipped with the weak and the strong topology, respectively. Because the topology of $\mathscr{D}...
- 9,853
1
vote
2
answers
326
views
Semi-reflexive dual
I am looking for an example of a semi-reflexive locally convex topological vector space, whose strong dual is not semi-reflexive. Is there some well-known example ?
- 61
4
votes
1
answer
286
views
Extending a Certain Result from Locally Convex Topological Vector Spaces to General Topological Vector Spaces
In this Math Stack Exchange post, I proved the following result.
Theorem: Let $ X $ be a locally convex topological vector space. Let $ x \in X $ and suppose that $ (x_{n})_{n \in \mathbb{N}} $ is ...
user36116
6
votes
2
answers
678
views
How general is the convergence of the exponential function's power series?
I asked essentially this over two weeks ago on MSE, and nothing was else was posted to that question.
Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$.
Let $\;\; \beta \: : \: V\times ...
user5810
10
votes
2
answers
2k
views
Pull-back of generalized functions
Let $f\colon X\to Y$ be a smooth map between smooth manifolds. Then the pull-back operation
$f^*\colon C^\infty(Y)\to C^\infty(X)$ is a linear continuous operator when $C^\infty$ is equipped with the ...
- 21.8k
6
votes
1
answer
353
views
Sequential continuity of linear operators
Let $u\colon L\to M$ be a linear map of locally convex linear topological vector spaces.
Assume that $u$ is sequentually continuous, i.e. maps convergent sequences to convergent ones.
(This notion is ...
- 21.8k
5
votes
3
answers
510
views
What is the definition of being smooth for a function from a Lie group to a Fréchet space?
In representation theory of real groups, one is confronted with the notion of smoothness for functions defined on a Lie group with values in a Fréchet space (e.g. see Wallach's Real Reductive Groups I,...
- 486
6
votes
1
answer
509
views
closed subspaces of locally convex inductive limits
It's a duplicate of this question, since I really want to get an explanation.
Let $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ be an inductive sequence of LCTV spaces. A locally convex ...
- 173
2
votes
1
answer
324
views
$c^\infty$-topologies on spaces of compactly supported sections and their products
Let $E$ be locally convex topological vector space. Let $c^\infty E$ denote the same
vector space equipped with the $c^\infty$-topology (i.e. the finest topology on it, s.t.
all smooth curves $\mathbb{...
- 55
4
votes
1
answer
908
views
Schwartz space defined on locally convex spaces
This is my first post here, so bear with me ;)
In wikipedia and other references, Schwartz space is defined as the set of infinitely differentiable functions on $\mathbb{R}^n$. On the other hand, A ...
- 43
4
votes
1
answer
520
views
Compactly generated Banach spaces
Suppose that $X$ is a Banach space (or more generally, Frechet space) such that $X$ is the closure of the span of a compact (in the original topology) subset $K$. Do we know anything "nice" about $X$, ...
- 3,115
6
votes
4
answers
1k
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Reference for integral of functions taking values in a topological vector space.
(Note that I am interested in the Gelfand-Pettis integral specifically, as opposed to, for example, the Bochner integral.) I have tried Googling things like "integral topological vector space", "...
- 1,270
3
votes
2
answers
599
views
Weak topology of WOT
Let $E$ be a reflexive Banach space and let $B(E)$ be the space of bounded operators on $E$ endowed with the weak operator topology. In particular, the unit ball of $B(E)$ is then WOT-compact. $(B(E), ...
- 115
4
votes
2
answers
670
views
Limits of von Neumann algebras
Consider (abstract) von Neumann algebras topologised by their weak*-topology arising from the unique predual.
In the theory of topological vector spaces, there is a natural notion of an inductive ...
- 115
8
votes
5
answers
545
views
Reference for : a Fréchet nuclear space is Montel
I'm looking for a reference to cite regarding the property presented in the title: "Closed and bounded sets of a nuclear Fréchet space are compact"
Thank you in advance for the help!
- 5,428
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
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
1
answer
525
views
A fact about finite-dimensional manifolds I fear does not hold for Frechet manifolds (what's new?)
Let $M$ be a manifold equipped with a pair of surjective submersions $N_1 \stackrel{p_1}{\leftarrow} M \stackrel{p_2}{\rightarrow} N_2$ where $dim N_1 = dim N_2 = n$. Then we can find, for any point $...
- 35.5k