Questions tagged [topological-vector-spaces]

A topological vector space is a vector space $V$ over a topological field $\mathbb{K}$ (typically $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$), together with a topology on $V$ such that vector addition and scalar multiplication are both continuous. Hilbert spaces and Banach spaces are examples of topological vector spaces.

Filter by
Sorted by
Tagged with
5
votes
1answer
61 views

The tensor product of two topological complexes with closed range

A Künneth formula by Grothendieck/Schwartz states the following: Let $A, B$ be chain complexes of nuclear Fréchet spaces. If the differentials $d_A, d_B$ are topological homomorphisms (meaning in ...
1
vote
1answer
102 views

What's the size of non standard monad for weak topology?

There have been several works characterizing weak topology by nonstandard analysis, which give rise to the following monad ($X$ is a Hilbert space): $$\mu(0) = \{y\in{}^{*}X: \forall x\in X ~~ \...
6
votes
0answers
280 views

Infinite-dimensional affine space in algebraic geometry and algebraic topology

In topology, there are of course many different infinite-dimensional topological vector spaces over $\mathbb R$ or $\mathbb C$. Luckily, in algebraic topology, one rarely needs to worry too much about ...
1
vote
1answer
64 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 ...
0
votes
2answers
97 views

$(ST)_{[13]}= S_{[13]}T_{[13]}$ for $S,T \in B(\mathcal{H}\otimes \mathcal{H}).$

Let $T\in B(\mathcal{H} \otimes \mathcal{H})$ where $\mathcal{H}$ is a Hilbert space. We can define operators $$T_{[12]}= T \otimes 1;\quad T_{[23]}= 1 \otimes T$$ and if $\Sigma: \mathcal{H} \otimes \...
5
votes
2answers
224 views

Is there a topology that makes every basic sequence null?

Let $E$ be a Banach space. Let $F$ be the collection of all $f\in E^*$ such that $\left<f,e_n\right>\to 0$, for every normalized basic sequence $\{e_n\}$. It is easy to see that $F$ is a closed ...
1
vote
1answer
197 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 ...
3
votes
0answers
39 views

Continuous differentiations of functional algebras

Let $A$ be some algebra (infinite-dimensional) of analytic functions on $\mathbb{C}^n$, and $D$ be some derivation of $A$, i.e. $D(fg)=Df \cdot g + f \cdot Dg)$ (so A may be considered as a ...
8
votes
1answer
182 views

Is a Banach lattice isomorphic to a Hilbert space in fact a Hilbert lattice?

The title says it all: Let $E$ be a Banach lattice, which is isomorphic to a Hilbert space (as normed spaces). Is there an equivalent Hilbert norm on $E$, which still makes it a Banach lattice with ...
5
votes
1answer
170 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
votes
1answer
160 views

A “proof” that all separately continuous maps on LF-spaces are continuous

Problem Consider the locally convex spaces $C^\infty(\mathbb{R})$ and $C^\infty_c(\mathbb{R})$, the former equipped with its standard Fréchet topology, the latter equipped with the inductive limit ...
1
vote
0answers
171 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
1answer
339 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
1answer
203 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 ...
3
votes
0answers
43 views

The Loday-Quillen-Tsygan theorem for topological (Fréchet) algebras

In "Additive K-theory" by Tsygan and Feigin, Section 0.4, a statement is given which seems to generalize (cohomological version of) the well-known Loday-Quillen-Tsygan theorem $$H_{\text{CE}}...
4
votes
1answer
143 views

Strict topology and $*$-strong toppology on $B(H)$ coincide

In the paper Woronowicz - $C^*$-algebras generated by unbounded elements, I read that the $*$-strong operator topology on $B(H)$ and the strict topology on $B(H)$ coincide. I believe this means the ...
4
votes
1answer
129 views

Is the unit ball of $A \odot B$ strictly dense in that of $M(A \otimes B)$?

Let $A$ and $B$ be $C^*$-algebras and let $A \otimes B$ their minimal tensor product and $M(A \otimes B)$ the associated multiplier algebra. On $M(A \otimes B)$, we consider the strict topology which ...
3
votes
2answers
154 views

smooth functions on closed intervals with values in infinite-dimensional spaces

There are three ways to define when a ($\mathbb{R}$-valued) function on a closed interval is smooth: $f$ can be extended to a smooth function on $(a - \epsilon, b + \epsilon)$ for some $\epsilon > ...
3
votes
1answer
96 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 ...
0
votes
0answers
40 views

Composition mapping for smooth maps with restricted domain

I'm studying convenient calculus. One of the results is that the composition mapping is smooth. $$ comp:C^{\infty }(F,G)\times C^{\infty }(E,F)\to C^{\infty }(E,G) $$ I'm interested if similar ...
3
votes
2answers
91 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 ...
0
votes
0answers
49 views

Closed image in direct sum

Let $K$ be a complete non-archimedean field and \begin{equation*} V \stackrel{f}{\longrightarrow} \bigoplus_{i=1}^n W_i \end{equation*} be a morphism of normed $K$-vector spaces, not necessarily ...
4
votes
1answer
176 views

Convenient vector space and its locally convex structure

I'm trying to understand convenient vector spaces, but I'm unsure about the definition of the topology on smooth maps. A map $f : E \rightarrow F$ between locally convex vector spaces $E$ and $F$ is ...
0
votes
1answer
64 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 $\...
3
votes
0answers
68 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 ...
0
votes
0answers
50 views

Topological constraints on a compact convex set admitting a strictly convex and subdifferentiable real function

It is a theorem of Hervé that A compact convex set $K$ admits a strictly convex and continuous real function only if $K$ is metrizable. (The converse is also true.) I'm wondering if any results of ...
3
votes
0answers
90 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 ...
3
votes
0answers
62 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\...
0
votes
0answers
113 views

Weak topology on spaces of measures and Borel sets

Let $K$ be a compact Hausdorff space (not necessarily metric or even separable). Let $M(K)$ be the space of all Radon measures on $K$ (that is, finite signed regular Borel measures) endowed with the ...
14
votes
0answers
397 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 "...
6
votes
2answers
370 views

A question on Grothendieck space

A Banach space $X$ is said to be Grothendieck if the weak and the weak* convergence of sequences in $X^{*}$ coincide. I have the following two questions. Question 1. A Banach space $X$ is Grothendieck ...
5
votes
1answer
270 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 ...
2
votes
1answer
61 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 ...
0
votes
0answers
42 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 ...
1
vote
0answers
70 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 ...
0
votes
0answers
41 views

An example of a certain continuous and strictly convex function

Let $X$ be a locally convex topological vector space. I am looking for an example of a function $f: X \times X \to [0,\infty]$ with the following properties: (1) For all $x,y \in X$, $f(x,y) = 0$ if ...
1
vote
0answers
193 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)|...
5
votes
0answers
81 views

Continuity vectors of log-concave measures

Let me begin by recalling some definitions and setting some notation I'll be using. As a complete reference I point to the book Bogachev, Differentiable measures and Malliavin Calculus (essentially ...
5
votes
1answer
334 views

A set of questions on continuous Gaussian Free Fields (GFF)

As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
6
votes
1answer
490 views

What is the role of topology on infinite dimensional exterior algebras?

Wedge products and exterior powers are discussed in W. Greub's book Multilinear algebra as follows. Definition: Let $E$ be an arbitrary vector space and $p \ge 2$. Then a vector space $\bigwedge^{p}E$ ...
0
votes
0answers
67 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
votes
1answer
184 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 $\...
5
votes
1answer
284 views

Reformulation - Construction of thermodynamic limit for GFF

I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to ...
1
vote
0answers
44 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 $...
6
votes
1answer
184 views

If $X$ is separable space then $X^∗$ is separable in all topologies $\tau$ such that $(X^∗,\tau)^∗ =X$?

Let $(X,\|.\|_{X})$ be a separable Banach space and the associated dual space is denoted by $X^*$. By $w^*$ we shall indicate the weak$-*$ topology on $X^*$. Let $B_{X^∗}= \{x^∗ \in X^∗ : \|x^∗\|_{X^∗...
4
votes
4answers
255 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 "...
1
vote
0answers
52 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\...
2
votes
0answers
41 views

A complete metric space with some convex-type property

Let $(X,d)$ be a complete metric space with this property: for each $x∈X$, $r>0$ and $y∈X$ with $d(x,y)<r$, there exists $z∈X$ such that $d(x,y)+d(y,z)=d(x,z)=r$. I want to know if the family ...
6
votes
1answer
107 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....
2
votes
0answers
34 views

On the equivalence of the two definitions of Hadamard differentiability

Sorry if this question is not eligible. I have posted it on Math.SE, but hasn't received any responses. Let $X$ be a Hausdorff locally convex space. As it is known [e.g., see Yamamuro S. (1974). ...

1
2 3 4 5