All Questions
44 questions
3
votes
1
answer
228
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:...
- 7,426
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
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(...
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
3
votes
1
answer
352
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 $x=\lim x_{n_k}$. ...
- 4,058
1
vote
1
answer
232
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 $\{...
- 4,058
0
votes
1
answer
231
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 ...
- 4,058
3
votes
1
answer
156
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 ...
- 7,426
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
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
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
6
votes
1
answer
316
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 ...
- 6,018
0
votes
1
answer
86
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
7
votes
1
answer
754
views
Closed convex hull in infinite dimensions vs. continuous convex combinations
tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$?
I am essentially asking for the most general, infinite-dimensional analogue of ...
- 71
5
votes
1
answer
219
views
Are linear continuous mappings open on totally bounded sets?
Let $X$ and $Y$ be locally convex spaces, and $\varphi: X\to Y$ a linear continuous mapping. Suppose first that $S$ is a compact set in $X$. Then $\varphi$, being considered as a mapping from $S$ to $\...
- 7,426
4
votes
4
answers
796
views
On Köthe sequence spaces
I asked this a week ago at math.stackexchange, but without success.
As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his "...
- 7,426
6
votes
1
answer
117
views
Infra-Pták space that is not Pták
From reading the literature of the 1970s heyday of locally convex spaces, it seems that it was an important open question whether there is an infra-Pták (i.e. $B_r$-complete) space that is not Pták (i....
- 3,816
8
votes
2
answers
385
views
Metrizability of a topological vector space where every sequence can be made to converge to zero
This is a follow-up to this answer.
If $E$ is a (real or complex) topological vector space, we say that a sequence $\{x_n\}_{n=1}^\infty$ in $E$ can be made to converge to zero if there exists a ...
1
vote
1
answer
203
views
Continuous function on colimit
Let $X$ be a Banach space and $f:X\rightarrow \mathbb{R}$ be continuous. Suppose that $\{X_n\}_{n \in \mathbb{N}}$ is a strictly nested sequence of sub-Banach spaces, for which $\cup_{n \in \mathbb{N}...
- 5,405
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
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
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
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
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
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
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
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
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
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
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
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)$, ...
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
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
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
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
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
5
votes
1
answer
510
views
The space $H(D)$ of holomorphic functions.
A very natural example of a nuclear Montel space is the space $H(D)$ of all holomorphic functions on the open disc topologized by the family of seminorms
$$p_n(f)=\sup\{|f(z)|\colon |z|\leq 1-\tfrac{...
- 51