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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Reference request for Grothendieck's work on "Integration with values in a topological group"
Disclaimer. This question was already asked in Mathematics Stack Exchange (see the link here). I wanted the question to be migrated here but I was told by a moderator that a question that old is ...
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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 "...
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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 ...
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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 ...
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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 $0\not\in i(\Bbb U)$...
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When is the sigma-algebra generated by closed convex sets the same as the Borel sigma-algebra
For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ?
More precisely, do we have ...
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Simultaneous Strong Law of Large Number classes?
Say that $C$ is a SSLLN class of subsets of some topological space $V$ provided that for every sequence of i.i.d.r.v.s $X_1,X_2,...$ with values in $V$, we almost surely have: For every $A$ in $C$, $\...
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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 $\...
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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 $X$,...
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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 ...
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Extension operators for topological vector space-valued smooth functions on closed sets
There are many known results about extension theorems for real-valued functions on closed sets, with varying levels of differentiability and so on, all very roughly following the Whitney approach. For ...
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The abstract kernel theorem implies Schwartz kernel theorem. How exactly?
Let me first give a little rapid background prior to formulating the question.
Let $\mathcal{D}$ be a Schwartz space of infinitely differentiable functions and $\mathcal{D}'$ is the space of ...
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distributions on Lie groups and representations
Let $G$ be a Lie group and $\pi$ a continuous action on $V$, a Fréchet space.
This action induces a representation of the space of compactly supported functions, $C_c(G)$, with convolution as product ...
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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 ...
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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 ...
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Topological localisations of algebras and solidification
Let $A$ be a (topologically discrete) commutative ring and consider the topological ring $A((z))$. Let $\underline{A((z))}$ denote the corresponding condensed ring, and let $a(z)$ be a non-zero ...
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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, \...
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Continuous and bornological Hochschild homology
As far as I understand, given a complex algebra $A$ with a locally convex topology $\mathcal{T}_A$ (e.g. $A=C^{\infty}(M, \mathbb C)$ for some manifold M), the topology induces a complete convex ...
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Extension of Vector Field in the $\mathcal{C}^r$ topology
This question was previously posted on MSE.
Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is ...
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409
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Spectral theory without topology
How much of spectral theory can be developed just working with vector spaces (finite or infinite dimensional) without referring to a choice of topology ?
Something along these lines, for example: ...
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References on categorical TVS theory
A survey by Castillo lists results on the category of Banach spaces and on Banach space constructions, such as:
Existence of limits in Banach spaces or suitable subcategories
Demonstrations of ...
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A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff
We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...
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On the embedding of manifolds into infinite-dimensional spaces
Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
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Continuous cohomology via model category
Is it possible to formulate notion of continuous cohomology in terms of model categories?
If yes, then is there a reference for this?
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296
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Differential operators acting on the Schwartz space
I asked a similar question on math stack exchange but didn't get an answer so I will try to ask it here. Any help/suggestion is most than welcome!
Let $D$ be a linear differential operator with ...
5
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285
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Convergence of convex combinations in topological vector spaces
I am studying certain quadratic forms on $L^0(m)$ equipped with the topology of (local) convergence in measure which in general is not locally convex. I am also interested in the situation where $m$ ...
4
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164
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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^{\...
4
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355
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Generalized Jensen's inequality for positively homogeneous functions
The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
4
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159
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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 ...
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A characterization of nuclear functionals in terms of continuity with respect to some special topologies on $B(X)$?
I think, nuclear functionals on the space of operators $B(X)$ (on a Banach space $X$) must have a characterization in terms of some special continuity. I would be grateful if somebody could help me ...
4
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Locally compact vector space over a finite field
In the wikipedia article titled "topological vector space", there is a line saying the following.
"Let $K$ be a locally compact topological field, for example to real or complex numbers. A ...
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Basic calculus on topological fields
Let $K$ be a a topological field (I am mainly interested in the cases when K is either an ordered field or a valued field, e.g. $K = \mathbb Q$ or $ \mathbb Q_p$).
1) Let $f: K^n \to K$ be a ...
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Question on this extension of abelian topological groups
A question on abelian topological groups:
Let $G$ be an abelian topological (Hausdorff) group and let $N$ be a closed subgroup of $G$. Assume that $N$ is (as a topological group) isomorphic to $(\...
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158
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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}$ ...
3
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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 ...
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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-...
3
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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 ...
3
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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 ...
3
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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}}...
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"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 ...
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146
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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
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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\...
3
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46
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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 ...
3
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57
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Convergence of sesqui-holomorphic kernels on the diagonal
Let $X\subset \mathbb{C}^d$ be a domain.
A function (kernel) $K:X\times X\to \mathbb{C}$ is called sesqui-holomorphic if it is holomorphic in the first variable, and anti-holomorphic in the second ...
3
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156
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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 $X$ ...
3
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Commutative discrete cyclic operator groups on topological vector spaces
Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
3
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97
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How Universal is the Topological $\mathbb K$-algebra $C(\Omega, \mathbb K)$?
For $\Omega$ an arbitrary set the family $C(\Omega, \mathbb K)$ of all functions $\Omega \to \mathbb K$ becomes a complete topological $\mathbb K$-algebra under the topology of uniform convergence. ...
3
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365
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Closed Graph Theorem and Spaces Of Continuous Functions
Let $X$ be a (Tychonoff) topological space. Consider $C\left(X\right)$ being a topological vector space of all continuous scalar-valued functions with the compact-open topology.
Assume that $Y$ is a ...
3
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393
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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
$V_+ \cap (-V_+) = \{0\}$...
2
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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(...