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
0 votes
1 answer
108 views

Is the space $C_0^{k}(\Omega)$ a Montel space?

I asked this question in the MathStackExchange, but I think I'm not get any answer. I'm trying to find a reference for the following result: Theorem: Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ ...
0 votes
0 answers
82 views

Analogue of $\ell^2(X)$ over an arbitrary Banach ring

Let $X$ be a set. Over the Banach fields $F=\mathbb{R}$ or $F=\mathbb{C}$ we can define the Banach space$$\ell^2(X)=\{\xi\colon X\to F\mid \sum_{x\in X}|\xi(x)|^2<\infty\}$$which satisfies a list ...
2 votes
1 answer
157 views

Factorization systems for vector bundles

Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
4 votes
3 answers
414 views

Does the uniform boundedness principle holds for multilinear maps as well?

This question has been motivated by weak* completeness of distributions. According to the answer in the above post, any barrelled locally convex topological vector space $E$ satisfies the uniform ...
6 votes
2 answers
249 views

If a Banach / Fréchet manifold $M$ happens to be a topological vector space, is $M$ just a Banach / Fréchet space?

In finite dimensions, if $M$ is a smooth manifold that happens to be a vector space, then it is indeed just the Euclidean space. I wonder if the same result holds valid in infinite dimensions. More ...
2 votes
1 answer
141 views

Topology of ${\mathcal D}(\Omega)$ (space of test functions)

I have seen two approaches to the topology of ${\mathcal D}(\Omega)$: (i) Let $K$ be a compact subset of $\Omega$ and consider the subset ${\mathcal D}_K(\Omega)$ of test functions with support ...
20 votes
3 answers
2k views

Duality between topology and bornology

I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way: Let $X$ be a set and let $\mathcal{P}(...
0 votes
0 answers
135 views

Reference request: an introduction to nuclear spaces

I am looking for a short introduction to nuclear spaces and nuclear operators. I am interested in these spaces as they often arise in mathematically rigorous quantum field theories. I have read the ...
5 votes
0 answers
206 views

Is the Taylor map continuous?

(Skip to the bolded theorem below for my question, if you'd like) Some context on asymptotic expansions and the Taylor map In the setting of irregular singularities of meromorphic connections on the ...
2 votes
1 answer
275 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 $x=\lim x_{n_k}$. ...
2 votes
1 answer
201 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 ...
4 votes
0 answers
163 views

Is the test function topology a Mackey topology?

I am a physicist, and I have lately been thinking about distributions as they appear in quantum field theory. In the standard development of the theory of distributions, one considers the space $C^{\...
0 votes
1 answer
113 views

When do the weak-star and compact convergence (compact-open) topology coincide on the dual of a Banach space?

In Measures Which Agree on Balls by Hoffmann-Jørgenson, it is claimed on page 323 that for an arbitrary Banach space $E$, if $\pi$ is the topology on $E^*$ of uniform convergence on compact subsets of ...
48 votes
4 answers
3k views

Is Schauder's conjecture resolved?

Schauder's conjecture: "Every continuous function, from a nonempty compact and convex set in a (Hausdorff) topological vector space into itself, has a fixed point." [Problem 54 in The ...
4 votes
4 answers
1k 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 ...
1 vote
1 answer
106 views

The Fourier projection mappings $\{ P_N \}$ form an equicontinuous family of linear maps on $E'(S^1)$ as well?

Let $S^1=\mathbb{R}/\mathbb{Z}$ and define the Fourier projection operator $P_N$ for each $N \in \mathbb{N}$ as \begin{equation} P_N(f)=\sum_{n=-N}^N \langle f, e_n \rangle_{L^2} e_n \end{equation} ...
2 votes
0 answers
317 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(...
5 votes
0 answers
202 views

Duality and compactness for pro vector spaces

I have a somewhat basic question which I haven't been able to piece together from the literature. Background. We work over a field $\bf{k}$. Consider the category, $\bf{Pro}_{k}$, of pro- vector ...
3 votes
1 answer
194 views

Do radially bounded sets form a bornology?

We call a subset $A$ in a real vector space $E$ radially bounded if it intersects every ray emanating from $0$ via a bounded set. It is easy to see that radially bounded sets in $E$ form a bornology, ...
3 votes
1 answer
285 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 ...
1 vote
0 answers
50 views

Minimal F-semi-norms

There are conflicting terminologies in the literature on this subject, so let me define an F-semi-norms on a real vector space $E$ to be a subadditive function $\rho:E\to[0,+\infty)$ such that $\rho\...
0 votes
0 answers
70 views

Goldstine theorem in quasi-Banach spaces

A classical theorem of Goldstine is the following: Let $X$ be a Banach space and $J \colon X \to X''$ the natural inclusion. Then $J(B_X)$ is $\sigma(X'', X')$-dense in $B_{X''}$, where $B_Y$ is the ...
9 votes
2 answers
818 views

$C^\infty$-vectors in general representations of Lie groups on locally convex spaces

This question is related to this one. Let $G$ be a real Lie group (I should emphasize I only care about ordinary Lie groups, not Lie groups modeled on locally convex spaces or anything like that). In ...
3 votes
0 answers
248 views

Equality of topologies in the spaces of section of a vector bundle

In this notes Geometric Wave Equations by Stefan Waldmann at page 7 he has Let $E \longrightarrow M$ be a vector bundle of rank $N$. For a chart $(U, \psi)$ we consider a compact subset $K \subseteq ...
3 votes
0 answers
41 views

Quasi-completion of a locally convex space as a space of linear functionals on its dual

A locally convex topological vector space is said to be quasi-complete if its closed bounded subsets are complete. Given a locally convex space $E$, one can show the existence of a Hausdorff quasi-...
2 votes
1 answer
122 views

Decomposition of weak* convergent nets into positive weak* convergent nets

Let $F$ be an order unit Banach space with order unit $e$ and topological dual space $F^*$ ordered by the dual cone. Let $E\subset F^*$ be a closed subspace that separates points of $F$ and such that ...
0 votes
0 answers
81 views

Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points

Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$ \begin{align} \max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...
2 votes
0 answers
100 views

Schwartz kernel theorem for restricted operators

Let $(M,g)$ be a smooth Riemannian manifold. The celabrated kernel theorem of Schwartz shows that for any linear and continuous operator $A:C_{c}^{\infty}(M)\to C^{\infty}(M)$, there exists a ...
2 votes
1 answer
103 views

Is $B \cap L^2\big( (0,T) \times (0,1)\big)$ closed in $L^2\big( (0,T) \times (0,1)\big)$?

I need help proving that the set $B \cap L^2\big( (0,T) \times (0,1)\big)$ is a closed subset of $L^2\big( (0,T) \times (0,1)\big)$, where $B$ is defined as: $$B=\Big\{x \in L^{\infty}\big(0,T;L^1(0,1)...
7 votes
3 answers
2k 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 ...
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 ...
6 votes
0 answers
177 views

Infinite-dimensional BRST reduction

Fix a base field $k$. First let me loosely describe the BRST reduction in the finite-dimensional setting. For a finite-dimensional Lie algebra $\mathfrak{n}$, we can form the Clifford algebra $\...
1 vote
1 answer
134 views

Complex interpolation of subspaces

Let $(X_0,X_1)$ be an interpolation couple of Banach spaces. Using complex interpolation we can form Banach spaces $X_\theta:=(X_0,X_1)_\theta$ where $0<\theta<1.$ Let $E_\theta\subseteq X_\...
3 votes
1 answer
361 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 ...
1 vote
0 answers
155 views

Reference for Schwartz kernel theorem on vector bundles

In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...
3 votes
4 answers
530 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 $\...
1 vote
0 answers
66 views

Projective limit of Fréchet reflexive spaces

I am reading this paper, constructing spaces of functions and distributions with exponential growth on Fréchet nuclear spaces and their dual. Un théorème de dualité entre espaces de fonctions ...
6 votes
1 answer
887 views

Under what conditions does a continuous linear map map a closed subspace to a closed subspace?

Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$? It is obviously satisfied if $W$ is ...
11 votes
5 answers
708 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 ...
2 votes
0 answers
99 views

Fréchet and DF spaces

Is there a canonical way to make a DF-space Fréchet while keeping the same vectorial structure? Or the converse? I've been looking in the classical books for locally convex spaces but haven't found ...
0 votes
0 answers
87 views

Is identity map on the space of smooth maps smooth?

I'm curious about the identity map on the space of all smooth maps (between two locally convex topological vector spaces in the sense of convenient calculi) when we equip the space with different ...
1 vote
0 answers
185 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 ...
1 vote
1 answer
164 views

Complemented subspaces in a dual Banach space

Let $Y$ be a complemented subspace in a dual Banach space $X$. Is it true that $Y$ is itself isomorphic to a dual? This is the case of a $w^*$-closed subspace $Y$, but a complemented subspace of $X^*...
6 votes
1 answer
326 views

Weakly sequentially closed convex cone which is not weakly closed

Let $V$ be the real vector space of finitely supported functions $f: \Omega\to \mathbf{R}$ such that $\sum_\omega f(\omega)=0$, where $\Omega$ is a given uncountable set. Endow $V$ with the weak ...
2 votes
1 answer
182 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 ...
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 ...
1 vote
0 answers
61 views

Consequences of having unbounded points in a bornology

For a set $X$ a bornology $\mathcal{B}$ is essentially an ideal in the power set $\mathcal{P}(X)$. Many sources including Wikipedia state additional property that $X = \bigcup \mathcal{B}$. Call it a ...
2 votes
1 answer
328 views

Are there "pathological convex sets" over ultravalued fields of char 2?

In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:...
1 vote
0 answers
131 views

Splitting of a topological vector space (TVS) into an (a) countable sum and (b) direct integral of subspaces

I thought that this would be a simpe question, and placed it here at the Mathematics Stackexchange. Now have to elevate it to Mathoverflow. LANGUAGE TVS = topological vector space. Any subspace of a ...
1 vote
1 answer
213 views

An explicit description for a certain type of infinite-dimensional homogeneous polynomials

This is a side question from Infinite-dimensional "algebraic varieties". Denote by $X_p$ ($1 \le p \le \infty$) the Banach spaces of complex sequences with finite $p$-norm and limit $0$. ...

1
2 3 4 5
7