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2
votes
2answers
103 views

Generic topology on a field

I'm wondering if there is some generic topology that can be put on any field of characteristic zero which is similar to those induced by a norm on the field. I know that for vector spaces you can take ...
0
votes
2answers
101 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 ?
0
votes
0answers
60 views

Open ideally convex sets

Background Recall that a subset $A \subseteq X$ of a Banach$^{1}$ space $X$ is said to be ideally convex if, for every bounded sequence $(x_n)_{n \in {\mathbb N}}$ in $A$ and every sequence ...
0
votes
1answer
108 views

Commutativity of convex hulls and closed balls

Let $X$ be a closed convex subset of a Banach space, and let $A\subseteq X$ be a Borel set. Denote $$ B_r(A) :=\{y\in X:\exists x\in A \text{ such that }\|x-y\|\leq r\} $$ and by $H(A)$ the convex ...
4
votes
3answers
213 views

Convex hulls of families of probability measures

Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$. In this paper for any family of probability ...
2
votes
1answer
103 views

Tensor product of topological abelian groups with the reals

Given an abelian group A, the tensor product $A \otimes R$, where R are the reals, is naturally an R-vector space. Now suppose that A is a topological abelian group (if necessary, we can assume it ...
4
votes
1answer
144 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 ...
5
votes
2answers
275 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 ...
10
votes
2answers
297 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 ...
7
votes
1answer
146 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 ...
2
votes
1answer
195 views

Browder's fixed point theorem in non-Hausdorff topological vector spaces

Browder proved the following fixed point theorem in his 1968 Mathematische Annelen paper (Theorem 1): Theorem. Let $K$ be a non-empty compact convex subset of a topological vector space $E$ (where we ...
3
votes
3answers
252 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 ...
5
votes
1answer
116 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 ...
2
votes
1answer
175 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 ...
4
votes
0answers
107 views

Hilbert metric of a sum of cones

Suppose that $K_{1}$ and $K_{2}$ are pointed closed cones in a finite-dimensional space $V$ whose Hilbert metrics $d_{1},d_{2}$ are known. Is there a way to express the Hilbert metric of $K_{1}+K_{2}$ ...
10
votes
0answers
155 views

Is it possible to take “limits up to homotopy”?

Before I begin, an apology: it's been a while since I've done much analysis, so I might be misusing (or just missing) terminology. I have a chain complex $(V_\bullet,\partial)$ of topological vector ...
4
votes
1answer
141 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 ...
2
votes
0answers
164 views

Is it true that a solid, minihedral cone in infinite dimensions cannot be regular?

Background Consider a real Banach$^1$ space $V$. We'll call a subset $V_+ \subseteq V$ a cone if $V$ is closed, $\alpha V_+ \subseteq V_+$ for every $\alpha \geqslant 0$ and $V_+ \cap (-V_+) = ...
4
votes
1answer
233 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 $X$, ...
4
votes
4answers
311 views

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", ...
2
votes
0answers
136 views

Density of adjoint operators

I am interested in operators on a non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{\ast\ast})$ by ...
2
votes
2answers
284 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), ...
4
votes
2answers
281 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 ...
6
votes
2answers
280 views

non-artificial examples of non-smooth and non-admissible representations of GL_2

Let $F$ be a finite degree extension over $\mathbf{Q}_p$ and consider the locally profinite group $G:=GL_2(\mathbf{Q}_p)$. P1: Give an interesting example (non-artificial one, i.e., one that arises ...
2
votes
0answers
203 views

questions about the Riemann-Lebesgue integral

(Assume Countable Choice.) Generalizing arxiv.org/pdf/math/9903103 and www.um.es/beca/papers/BirkhoffBourgain.pdf, the Riemann-Lebesgue integral can be defined as follows: For $V$ a real ...
2
votes
0answers
603 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 ...
5
votes
1answer
407 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 ...
5
votes
1answer
312 views

Extreme points of a compact convex set are a $G_\delta$?

Dear All, I'm reading a paper (Residuality of Dynamical Morphisms by Burton, Keane and Serafin) that makes a claim that I've been unable to verify or find a reference for. The claim is made that the ...
8
votes
4answers
2k views

Grothendieck on Topological Vector Spaces

In the short biography article on the Alexander Grothedieck http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf , it is mentioned that after Grothendieck submitting his first thesis on ...
4
votes
3answers
836 views

Topological vector spaces that are isomorphic to their duals

After reviewing the (locally convex) topological vector spaces that I know, the only examples I could find where there is an isomorphism from the space to its (anti)dual, are Hilbert spaces. So my ...
3
votes
1answer
103 views

Independence of the axiomatics of metric cones

A metric cone $C$ is a nonempty metric space (whose distance is denoted $d$) together with a map $\cdot\colon \mathbf{R}\times C \mapsto C$ satisfying these axioms: $a\cdot(b\cdot x) = (ab)\cdot x$ ...
3
votes
2answers
413 views

Sequences of linear combinations of measures

Let $X$ be a Polish space. Let $J\in\mathbb{N}$. Let $\lbrace a^n_1\rbrace_n,\dots,\lbrace a^n_J\rbrace_n$ be $J$ sequences of reals. Let $\lbrace \mu^n_1\rbrace_n,\dots,\lbrace \mu^n_J\rbrace_n$ be ...
11
votes
1answer
416 views

Status of the compact AR problem?

The so-called "compact AR Problem" reads: Is every compact convex set in a metrizable topological vector space an absolute retract? It is open according to the chapter by T. Banakh, R. Cauty ...
5
votes
1answer
401 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
vote
1answer
784 views

Strong topology

Let $E$ and $F$ be a locally convex topological vector spaces (LCS) and let $E^{\star}$ and $F^{\star}$ denote the strong duals of $E$ and $F$, respectively. A dual of $E^{\star}$ given by the ...
-2
votes
2answers
588 views

Question on Linear Operators

Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is: $$ \forall v \in V \quad ...
1
vote
1answer
290 views

Compact sets in TVS

Let $K$ be a compact subset of a Hausdorff topological vector space. Is it true that $\bigcap_{n\in \mathbb{N}}\frac{1}{n}K$ is either empty of or is a set consisting of the origin only?
9
votes
2answers
503 views

Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?

Motivation A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed ...
4
votes
1answer
219 views

Are all continuous linear operators on the space of entire functions “simple”?

Let $\langle \operatorname{Ent},+,\cdot \rangle$ be the (complex) vector space of entire functions. For all members $n$ of $\{1,2,3,...\}$, define $||\cdot ||_n : \operatorname{Ent} \to \mathbb{R}$ ...
2
votes
3answers
392 views

General theory for p-normed spaces

Hello, in functional analysis and operator theory, you encounter several (at first glance, at least) similar constructions of normed spaces that can be indexed with some $ p \in [1,\infty]$, and ...
2
votes
4answers
539 views

On locally convex (and compactly generated) topological vector spaces

Part 1: How big is the category $TVS_{loc.conv.}$ of locally convex topological vector spaces (and continuous maps)? In other words (and less cheekily), is there a free locally convex TVS having any ...
4
votes
3answers
582 views

Topology on the set of linear subspaces

Hello, let $X$ be a seperable Hilbert space. Let $(e_i)_i$ be a Hilbert basis, and for each index let $E_i = \langle e_1,\dots,e_i \rangle \subset X$ the span of the first $i$ basis vectors. For any ...
3
votes
4answers
561 views

An example of a non-paracompact tvs (over the reals, say)

What is an example of a non-paracompact topological vector space? I'm aware of this question, but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that ...
2
votes
3answers
756 views

Sequential topological vector spaces

Since I'm dealing with the distinction between sequential continuous and continuous maps at the moment I came to ask myself once again what can be said about spaces where these two notions agree ...