All Questions
Tagged with fa.functional-analysis topological-vector-spaces
213 questions
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
2
votes
1
answer
188
views
Dual fixed point
Let $E$ be a Banach space, let $T:E\to E$ have norm $1$ and let $\nu\in E^*\setminus\{0\}$ be such that $T^*\nu=\nu$. Under which conditions there is $e\in E$ such that $Te=e$ and $\langle e,\nu\...
- 5,529
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
0
votes
0
answers
170
views
Limit of balls in $L^p$
Setup:
Let $\mu$ be a measure on a measurable space $(X,\Sigma)$, such that for every $p ,q\in [1,\infty)$, $L^p_{\mu}(\Sigma)\subseteq L^q_{\mu}(\Sigma)$ if $p\geq q$. Furthermore, the inclusions ...
- 5,405
11
votes
5
answers
801
views
Colimits in the category of (not necessarily locally convex) topological vector spaces
Do colimits in the category of (not necessarily locally convex) topological vector spaces (over R, C, respectively) exist in general?
If no, is there a well-known condition of when they exist?
If ...
- 419
3
votes
1
answer
456
views
Duality of Topological Vector Spaces
Let $K$ be a topological field. Let $\text{top-} K \text{-vect}$ be the category of topological $K$-vector spaces $V$, so that the maps $\cdot : K \times V \rightarrow V$ and $+ : V \times V \...
user30211
0
votes
2
answers
344
views
subspace topology and strong topology
Suppose $X$ is a locally convex space and $Y$ is a subspace of the strong dual of $X$, is the induced topology on Y equivalent to the strong topology $b(Y,Y')$ on $Y$? If this is not correct, then on ...
- 71
9
votes
6
answers
838
views
Open mapping theorem for complete non-metrizable spaces?
The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's ...
2
votes
3
answers
230
views
Every linear topological space embeds into the Tychonoff product of linear metric spaces
I need a reference to the following (known?)
Fact. Every topological vector space $X$ over the field of real numbers is topologically isomorphic to a linear subspace of the Tychonoff product of ...
- 41.8k
6
votes
2
answers
355
views
Complete dual of bornological space
A bornologigal topological vector space is such that any bounded linear function on it is continuous. It is a standard result [Jarchow, Locally convex spaces, 1981] that if the dual $E'$ of a Mackey ...
- 103
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
8
votes
1
answer
687
views
When does the dual to the space $K(X)$ of compact operators consist of nuclear functionals?
Let $X$ be a Banach space and $B(X)$ be its space of all (bounded) operators. A nuclear functional on $B(X)$ is a linear functional $u:B(X)\to{\mathbb C}$ that can be represented in the form
$$
u(A)=\...
- 7,426
4
votes
1
answer
294
views
When is a totally bounded set of an inductive limit contained in a component of this limit?
A. P. Robertson and W. Robertson in their "Topological Vector Spaces" VII, 1.4, (and H.Jarchow in "Locally convex spaces", 4.6, Theorem 2) prove the following proposition:
Let $E=\lim_{n\to\infty}...
- 7,426
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
3
votes
1
answer
132
views
Openness of invertibility in Fréchet spaces for families parameterized by compact spaces
Consider the following setup. Let $K$ be a compact topological space, $X$ a Fréchet space and $T:K \times X \to X$ a continuous family of linear maps (i.e. $T$ is a continuous map and $T_k \equiv T(k, ...
- 5,824
5
votes
1
answer
333
views
Is the compact-open topology on the dual of a separable Frechet space sequential?
Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the ...
- 41.8k
2
votes
1
answer
352
views
The completeness of spaces of continuous functions with the compact-open topology
For a Tychonoff space $X$ let $C_k(X)$ denote the space of continuous real-valued functions on $X$, endowed with the compact-open topology.
Problem. Is the space $C_k(X)$ Polish if it is Polishable ...
- 41.8k
4
votes
2
answers
443
views
On convergent sequences in locally convex topological vector spaces
Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and ...
- 41.8k
6
votes
1
answer
567
views
Is restriction a closed map?
Originally asked on MSE.
Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ ...
- 5,529
1
vote
1
answer
124
views
Compactness of operators and norming sets
Originally asked on MSE.
Let $T$ be a linear map from a normed space $E$ into a Banach space $F$.
Let $D\subset \overline{B}_{F^{\ast}}$ be norming, i.e., there is $r>0$ such that $\sup\limits_{v\...
- 5,529
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
3
votes
1
answer
199
views
Are second-countable subsets of topological vector spaces metrizable?
Let $X$ be a topological vector space of size $\mathfrak{c}$. Assume that there exists a countable union $X=\cup X_n$ such that all subsets $X_n$'s are relatively second countable.
Q. Does there ...
- 4,058
2
votes
1
answer
151
views
Boundedness of Dirac deltas
Suppose that $X$ is a metric space and let $C_k(X)$ denote the space of real functions on $X$ with the topology of uniform convergence on compact sets. Then $C_k(X)$ is a topological vector space. Let ...
- 25
4
votes
1
answer
394
views
Separable Lindelöf locally convex spaces that are not second-countable
A Lindelöf space is a topological space in which every open cover has a countable subcover.
Does there exists a Lindelöf locally convex space which is not second countable?
I am also looking for a ...
- 4,058
2
votes
1
answer
236
views
Closure in the strong dual topology
Originally asked on MSE.
Let $E$ be a metrizable locally convex topological vector space and let $E^{*}$ be its dual space endowed with the strong topology = topology of uniform convergence on (...
- 5,529
8
votes
5
answers
2k
views
Topological vector space textbook with enough applications
(Sorry for my bad English.)
For "applications", I mean applications in math, not real-life.
There are many textbooks about topological vector space, for example, GTM269 by Osborne, Modern Methods in ...
11
votes
1
answer
258
views
Bilinear product of two summable families
Consider the following statement, which I suspect is false as written:
Let $E,F,G$ be (Hausdorff) topological vector spaces (over $\mathbb{R}$), let $\varphi\colon E\times F\to G$ be continuous and ...
- 32.5k
3
votes
1
answer
214
views
Recognizing locally convex spaces on which all bounded linear functionals are continuous
Is it possible to characterize the Hausdorff locally convex spaces on which all bounded linear functionals are continuous?
It is known that a space is bornological if and only if the space is Mackey ...
- 5,407
1
vote
1
answer
144
views
When is the strict topology bornological?
Let $X$ be a completely regular Hausdorff space. Are there known conditions under which the algebra of bounded continuous functions on $X$, endowed with the strict topology, is bornological?
(Of ...
- 5,407
1
vote
1
answer
183
views
Criterion of reflexivity 2
Originally I meant to ask this question here, but got confused and ended up asking another question, which had some mathematical meaning, but was not what I vaguely had in mind.
Let me restate the ...
- 5,529
5
votes
2
answers
673
views
When are the closed convex subsets countable intersections of halfspaces
For what kind of topological vector spaces (separable maybe?) are the closed convex subsets countable intersections of halfspaces.
I've seen somewhere that it's true for separable Hilbert spaces, ...
- 506
1
vote
1
answer
220
views
Criterion of reflexivity
Let $E$ be a Banach space.
It is known that if for any equivalent norm on $E^*$ the closed unit ball of $E^*$ is weakly* closed, then $E$ is reflexive (a very short proof is in the book by Fabian, ...
- 5,529
7
votes
1
answer
402
views
Is any dual metrizable locally convex space a Frechet space?
[I have posted this question on MSE some time ago, but received no answer.]
The title basically says all of it.
If a normed space $F$ is a dual of a normed space $E$, then $F$ is a Banach space. I ...
- 5,529
2
votes
2
answers
125
views
Topologies of pairs and closed bounded convex sets
[I have posted this question on MSE some time ago, but received no answer.]
It is known, that if two locally convex topologies on a vector space determine the same collection of continuous linear ...
- 5,529
0
votes
1
answer
235
views
Criterion for weak compactness
Let $F$ be a metrizable locally convex space (you may assume it is a Banach space), and let $E$ be a complete locally convex space (you may assume it is a Frechet space). Let $T$ be a continuous ...
- 5,529
2
votes
2
answers
374
views
A criterion for norming sets
Let $F$ be a Banach space with the closed unit ball $B$. Let $E\subset F^*$ be a total subspace such that $B$ is complete with respect to the norm $|||f|||=\sup \limits_{e\in E,~e\ne 0} \frac{|\left&...
- 5,529
0
votes
1
answer
82
views
Semi-embeddings and weak compactness
Let $F$ and $H$ be normed spaces and let $E$ be a locally convex space.
Let $T:F\to H$ and $S:H\to E$ be linear operators, such that $\|T\|= 1$, $S$ is an injective semi-embedding (i.e. $S\overline{B}...
- 5,529
2
votes
1
answer
365
views
Why is an inductive limit of bornological spaces bornological?
Let $(E_\alpha,\tau_\alpha,g_\alpha)$ be a family of bornological (locally convex) topological vector spaces $(E_\alpha,\tau_\alpha)$, where a LCTVS $E$ is said to be bornological if every circled, ...
- 1,247
6
votes
3
answers
507
views
Proof of the Schauder Lemma
Schauder's Lemma in functional analysis states the following:
Let $E$ and $F$ be metrizable locally convex topological vector spaces, and let $E$ be Fréchet. Then if the linear continuous map $A:E\...
- 1,247
3
votes
1
answer
80
views
DF-algebras and DF-modules
Recall Lemma 0.5.1 from the Helemskii's monograph "The homology of Banach and Topological Algebras":
$\textbf{Lemma}$ Let $\phi\colon X\to Y$ be an injective map between Banach spaces with dense ...
- 375
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
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
3
votes
1
answer
228
views
The sheaf of generalized functions on compact subsets
For $K\subseteq \mathbb{R}^d$ compact, let $C_{\mathrm{c}}^{\infty}(K)$ denote the space of smooth functions on (an open neighborhood of) $K$ with compact support contained in $K$ with the usual ...
- 1,270
4
votes
1
answer
177
views
DF-spaces and F spaces
It is well known that when $E$ is a $DF$-space and $F$ is a Fréchet space, the space $\mathcal{L}_{b} (E,F)$ is Fréchet. The converse, that is the fact that $\mathcal{L}_{b} (F,E)$ would be $DF$, is ...
- 103
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
17
votes
1
answer
759
views
Classification of non-Hausdorff topological vector spaces
It is well-known that up to topological isomorphism there is exactly one Hausdorff topological vector space (say, over $\mathbb{C}$) of a given dimension $n$, namely $\mathbb{C}^n$ with the euclidean ...
- 5,482
1
vote
1
answer
998
views
Subspaces of Quotient Spaces
Let $X$ be a topological vector space (not necessarily Hausdorff), with topology $\tau$, and $M, N$ linear subspaces of $X$. Let $\pi:X \rightarrow X/N$ be the quotient map, which associates to each $...
- 640
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
4
votes
1
answer
180
views
Productivity of certain sequential subcategories of topological vector spaces
Consider the usual sequential modifications of topologies (spaces) in the categories of topological spaces $\text{Top}$, topological vector spaces $\text{TVS}$ and locally convex spaces $\text{LCS}$ :
...
- 1,773
5
votes
2
answers
1k
views
Are bounded sets always weakly metrizable in reflexive separable spaces?
It is known that if a Banach space is reflexive and separable, its unit ball is weakly metrizable.
My question is about the generalization of this property :
1) Is it true that for all reflexive ...
- 549