All Questions
Tagged with gn.general-topology topological-vector-spaces
73
questions
3
votes
1
answer
279
views
Pointwise convergence and disjoint sequences in $C(K)$
Let $K$ be a Hausdorff compact space and let $C(K)$ be the space of continuous real-valued functions on $K$. A sequence $(h_n)$ in $C(K)$ is called almost disjoint if there is a sequence $(g_n)$ with ...
- 5,275
2
votes
1
answer
197
views
Parametrization of topological algebraic objects
There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is ...
- 5,275
1
vote
0
answers
47
views
Neighborhoods of idempotents in topological inverse semigroups
In a topological group, for any neighborhood $U$ of the origin, there is another such neighborhood with the property that $V.V\subseteq U.$ I conjecture a similar property for topological inverse ...
- 1,007
3
votes
1
answer
314
views
Topological vector spaces in direct sum
A year ago, I asked this question here at Mathematics Stackexchange, but no one there managed to answer it. So I am elevating it to MathOverflow.
This question had emerged as an offshoot of a bigger ...
- 593
2
votes
1
answer
181
views
On the equality $\{f\in C(X), f|_A=0\}+\{f\in C(X), f|_B=0\}=\{f\in C(X), f|_{A\cap B}=0\}$
Let $X$ be a Tychonoff space, let $A,B\subset X$ be closed. Let $J_A$ be the set of all continuous on $X$ real-valued functions which vanish on $A$.
For which $X$'s is it true that $J_A+J_B=J_{A\cap ...
- 5,275
1
vote
0
answers
75
views
Morphism in commutative square strict?
Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism.
Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...
- 463
1
vote
1
answer
228
views
Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?
I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below).
It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be
written as an ...
- 113
2
votes
1
answer
271
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 $x=\lim x_{n_k}$. ...
- 3,898
1
vote
1
answer
213
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 $\{...
- 3,898
5
votes
2
answers
334
views
Product of inductive limit topologies on $C_c(X)\times C_c(X)$
I have a stupid question about a topology on $C_c(X)$. Here $X$ is locally compact Hausdorff. Can assume $\sigma$-compact if it helps.
Definition (topology on $C_c(X)$): For each compact $K \subset X$,...
- 655
6
votes
1
answer
231
views
Is the projectivization of a topological vector space Tychonoff?
Let $E$ be a locally convex topological vector space over $\mathbb{R}$. The projectivization $PE$ is the quotient of $E\backslash\{0_{E}\}$ with respect to the equivalence relation $e\sim f$ if $e=\...
- 5,275
0
votes
1
answer
337
views
About the normability of the space of continuous functions
Let $A$ be a subset of $\mathbb{R}^n$, and denote by $C(A)$ the space of complex-valued continuous functions defined on $A$. We know that if $A$ is compact then we can define a norm on $C(A)$ so that ...
- 1,017
8
votes
0
answers
183
views
On "linearly independent" metric spaces
Urysohn's universal metric space $\Bbb U$ satisfies the following surprising property:
Whenever $i\colon\Bbb U\to E$ is an isometric embedding into a normed vector space such that $0\not\in i(\Bbb U)$...
- 2,672
3
votes
2
answers
766
views
Topologies on space of compactly supported continuous functions
Let $X$ be a locally compact Hausdorff space. As far as I understand, the space $C_c(X) = C_c(X; \mathbb{C})$ of compactly supported continuous complex-valued functions on $X$ is (most?) often ...
- 89
5
votes
0
answers
174
views
Logarithm on formal power series continuous?
Denote $V:=\mathbb{R}^d$ and consider the Cartesian product $V^\infty:=\prod_{k=0}^\infty V^{\otimes k}$ together with its canonical projections $\pi_k : V^\infty\rightarrow V^{\otimes k}, (v_0, v_1, \...
- 411
2
votes
1
answer
144
views
Is $C_0(X) $ Frechet-Urysohn with respect to the compact-open topology?
Recall that a topological space is called Frechet-Urysohn if the operations of closure and sequential closure coincide.
Let $X$ be a locally compact Hausdorff space. It is known that $C(X)$ is not ...
- 5,275
1
vote
1
answer
217
views
Is a topology sandwiched between two norms compactly generated?
Recall that a Hausdorf topological space $X$ is called compactly generated if any set whose intersections with compacts are compact is closed. Locally compact and first countable spaces are compactly ...
- 5,275
5
votes
1
answer
196
views
If a subspace $F$ is contained in a subspace $G$, and $H$ is close to $G$, can we choose a subspace of $H$ close to $F$?
Let $E$ be a Banach space. Recall that the collection of all closed linear subspaces of $E$ can be turned into a metric space in a number of ways. In particular, consider the notion of a gap: if $G$ ...
- 5,275
1
vote
0
answers
242
views
Complete topological groups in which all subgroups are closed
My previous question has been answered by YCor; so I am asking a new one with a reasonable additional assumption. See the previous question for the background and motivation.
General question: does ...
7
votes
1
answer
628
views
Topological groups in which all subgroups are closed
General question: does there exist a nondiscrete topological group $G$ such that all subgroups of $G$ are closed? Or, does there exist a nondiscrete topological vector space $V$ such that all vector ...
5
votes
1
answer
361
views
Is the filtered colimit topology on the space of signed Radon measures linear and locally convex?
Let $X$ be a compact Hausdorff space. In chapter 3 of Peter Scholze's Lectures on Analytic Geometry he considers the space of signed Radon measures on $X$ equipped with the filtered colimit (aka ...
- 153
3
votes
2
answers
207
views
Sufficent condition for strict morphism of normed vector spaces
Let $K$ be a non-archimedean field of char 0 and a morphism $f:V \rightarrow W$ of normed $K$-vector spaces given. The map $f$ is said to be strict if $V/\ker(f)$ with the quotient topology is ...
- 463
3
votes
0
answers
144
views
Can every contractible space be embedded as a convex subset of a vector space?
Given a contractible topological space $X$, is there (or what are some conditions for the existence of) a continuous embedding $\iota:X\hookrightarrow V$ into some topological vector space $V$ such ...
- 22.4k
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
2
votes
1
answer
67
views
Equicontinuity-like property of a convex compact set
Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$.
Is there an ...
- 5,275
0
votes
0
answers
45
views
Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
- 5,001
1
vote
0
answers
100
views
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 ...
- 839
0
votes
0
answers
90
views
Can a quotient space of a locally convex space have finer topology that its domain?
The following is related to this post.
Let $X=X'$, as sets, and let $T:X \rightarrow X'$ be a surjective map from a countably infinite-dimensional LCS $X$ to itself and equip $X'$ with the final ...
- 5,001
1
vote
0
answers
53
views
Spaces that are comparable with their compacts
This is an outgrowth of this question.
For a (metrizable) space $X$ consider the following increasingly strong properties:
(i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\...
- 5,275
0
votes
0
answers
46
views
Show that $\big(s(. |C_n)\big)_n$ is equicontinuous on $X^*$
Let $(X,\|.\|)$ be a separable Banach space with dual $X^*$. $\mathcal{P}_{wkc}(X)$ be the class of nonempty, weakly-compact and convex subsets of $X$. For any $C\in\mathcal{P}_{wkc}(X)$ we define ...
- 1
3
votes
1
answer
397
views
Criterion for weak convergence of sequences
Let $E$ be a normed space and let $F\subset E^{*}$. It is known that $F$ is dense if and only if the restriction of $\sigma(E,F)$ on $B_E$ coincides with the weak topology.
Hence, if $F$ is dense and ...
- 5,275
4
votes
0
answers
159
views
Pointwise vs. local homotopy equivalences of continuous and smooth complexes of real vector bundles
Let $(E^\bullet,d_E)$ and $(F^\bullet,d_F)$ be two complexes of real vector bundles on a topological manifold $X$, and let $f^\bullet\colon E^\bullet\to F^\bullet$ be a morphism of complexes, i.e. a ...
- 6,679
3
votes
4
answers
480
views
Inductive limit of $\mathbb R^n$s is Hausdorff and second countable?
When dealing with infinite jet bundles, one can consider the topological vector space $\mathbb R^\infty$ obtained by taking the projective limit of the inverse system $(\mathbb R^n,\pi^n_m)$, where $\...
- 2,317
0
votes
2
answers
318
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
-4
votes
1
answer
73
views
Two notions of boundedness in metrizable topological vector space [closed]
In a metrizable topological vector space X with the metric d, a subset A is said to be bounded if it can be absorbed by any neighbourhood of 0 and a subset A is said to be d-bounded if its diameter ...
- 93
3
votes
1
answer
106
views
Is each cometrizable space a subspace of a cometrizable topological group?
Following Gruenhage we call a topological space $X$ cometrizable if $X$ admits a weaker metrizable topology such that every point $x\in X$ has a (not necessarily open) neighborhood base consisting of ...
- 40.2k
2
votes
0
answers
69
views
Invariant compact in division ring
Let $K$ be a discrete valued (with discrete valuation $v$) complete local division ring with ring of valuation $V$. Let $F$ be a compact subset of $V$. Suppose that for all $x\in F$ and for all $y\in ...
- 3,623
1
vote
0
answers
120
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
4
votes
1
answer
299
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 ...
- 40.2k
2
votes
1
answer
323
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 ...
- 40.2k
6
votes
1
answer
521
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 $X$ ...
- 5,275
3
votes
1
answer
194
views
Is each closed subgroups of $\mathbb R^\omega$ isomorphic to a Tychonoff product of locally compact Abelian groups?
It is known that any closed linear subspace of $\mathbb R^\omega$ is topologically isomorphic to $\mathbb R^n$ for some $n\in\omega$.
Problem 1. Is each closed subgroup of $\mathbb Z^\omega$ (or ...
- 40.2k
4
votes
1
answer
235
views
Approximation of the identity by finite range functions in topological vector spaces
Let $X$ be a topological vector space. Assume that there exists a sequence $\phi_n:X\to X$ of finite range measurable functions with $\lim\phi_n(x)=x$ for every $x\in X$. Can we concluded there exists ...
- 3,898
3
votes
1
answer
187
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 ...
- 3,898
2
votes
1
answer
149
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
365
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 ...
- 3,898
2
votes
1
answer
100
views
Measurability of the product on particular topological vector spaces
Let $X$ be a topological vector space. Let us say that $X$ has property P if there exists a sequence of closed subsets $\{X_n\}$ such that
1- $X=\bigcup X_n$
2- The relative topology is both ...
- 3,898
10
votes
1
answer
304
views
Closed vector subspaces of large powers of R
By a large power of $\mathbb R$ is meant a topological vector space which is the product of infinitely many copies of the real line.
Is every closed subspace of such a TVS linearly homeomorphic to ...
- 101
2
votes
1
answer
339
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, ...
- 1,197
3
votes
0
answers
153
views
Topology of the Hamel basis in a TVS
Let $V$ be a complex topological vector space, and let $I$ be a Hamel basis of it. Then as a subset $I\subset V$ acquires an induced topology, becoming a topological space. For a topological space $X$ ...
- 1,807