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# 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.

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Properties on morphism of locally convex vector spaces

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $U,V,W,W'$ $K$-vector spaces, such that $U$ is a Banach-space and $W,W'$ are finite dimensional. Further we have an (algebraic) short exact ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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:...
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### $\sigma$-compactness of some locally compact Hausdorff topological groups

Is the topological group $(\mathbf{Q}_p/\mathbf{Z}_p)^{\oplus k}$, $k\ge 1$, a $\sigma$-compact topological group when endowed with its natural $p$-adic topology? More generally, I'm looking for a ... 1 vote
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### 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 ...
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### Reference on inductive (direct) limit of ordered vector spaces and vector lattices

I looked in all textbooks on vector lattices (Riesz spaces) as well as ordered vector spaces, but couldn't find any mentions of neither inductive nor projective limit for these structures. Googling ...
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### 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 $x=\lim x_{n_k}$. ...
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### Why is the space of smooth sections complete?

This page is about the space of sections: Let $E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$ be a smooth vector bundle. On its real vector space $\Gamma_{\Sigma}(E)$ of smooth sections consider the ...
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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 $\{... 2 votes 1 answer 137 views ### Extending a$\sigma$-weakly continuous map: Takesaki IV.5.13 Consider the following fragment from chapter IV in Takesaki's book "Theory of operator algebra I": Why is the boxed line true? Takesaki argues that $$\theta_0: \mathscr{M}_1\otimes_{\... 2 votes 1 answer 275 views ### Predual theorem proof in Takesaki's volume I Consider the following fragment from Takesaki's book "Theory of operator algebra I" (Section III.3 ,p133-134). Why is the boxed line true? I can see that \epsilon: \widetilde{A}\to A is ... 0 votes 1 answer 169 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 ... 3 votes 1 answer 148 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 ... 5 votes 2 answers 324 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,... 3 votes 0 answers 105 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 ... 6 votes 1 answer 217 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 votes 1 answer 998 views ### Definition of infinite-dimensional Gaussian random variable For infinite-dimensional Gaussian measures, we often see the definition of Gaussian random variables like this: Let H(\Omega;\mathbb{R}) be a separable Hilbert space. A random variable u \in H is ... 4 votes 1 answer 249 views ### A Fréchet space characterization of smooth structures on topological spaces? For a compact manifold M the space of smooth functions C^{\infty}(M) is a Fréchet space where the seminorms are the suprema of the norms of all partial derivatives. Is there some way to ... 6 votes 0 answers 130 views ### Cech cohomology on product covers & Fréchet sheaves My question is about the paper [Ka67]. Let S, T be sheaves of nuclear Fréchet spaces over paracompact topological spaces X, Y, respectively; in particular, if V \subset U are open subsets in X,... 5 votes 0 answers 941 views ### Condensed/liquid vector spaces and path integrals [Edited to take into account comments.] Background One approach to the problem of making rigorous various measures on spaces of paths (for example, the Wiener or Feynman measure) is the time-slicing ... 1 vote 0 answers 45 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 ... 1 vote 1 answer 159 views ### Uniqueness of the predual of a W*-algebra Consider the following fragments in Takesaki's "Theory of operator algebras" (volume I): Question: So, we have an abstract Banach space F with A \cong F^*. In Lemma 3.6, one considers ... 2 votes 1 answer 97 views ### About \sigma strong^*-functionals and seminorms I'm reading the book "Theory of operator algebras" by Takesaki. In this book, the \sigma-strong^* topology on the space B(H) (bounded operators on the Hilbert space H) is defined (... 0 votes 1 answer 243 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 ... 2 votes 1 answer 249 views ### Strict topology on the multiplier algebra Let A be a C^*-algebra. Let M(A) be its multiplier C^*-algebras. The strict topology on M(A) is given by$$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to ... 1 vote 0 answers 98 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$...
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 $0\not\in i(\Bbb U)$...