The topological-vector-spaces tag has no usage guidance.

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### from finite to $\sigma$-finite measure space [migrated]

This might be rather elementary. I have put it at MSE for a while without getting any answers.
Here is the question:
In the proof of the following theorem, would anyone explain how the general case ...

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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 ...

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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. ...

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281 views

### Two vector spaces with homeomorphic open subsets are isomorphic?

Is it true that if $ E,F$ are two topological vector spaces (or say Banach spaces) over $\mathbb{R}$ such that they have nonempty open subsets $U\subset E, V\subset F$ which are homeomorphic, then the ...

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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 $m$ ...

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### Real analysis on vector-valued spaces, $L^{p}(\mathbb{R}^N,E)$ ,$H^{s}(\mathbb{R}^N,E)$

I am dealing with some vector-valued Sobolev spaces $H^{s}(\mathbb{R}^N,E)$ where $E$ is a Banach space.
I am looking for references about results for the scalar case ...

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265 views

### Is the unitary group of a pre Hilbert space contractible?

I already posted my question on mathstackexchange
For a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for ...

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### Do locally convex topological vector spaces embed into diffeological spaces?

The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results ...

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175 views

### Examples of TVS with no non-trivial open convex subsets

I give here the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.
...

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### Schwartz space of functions with values in a Frechet space

While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in ...

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### Can you pair $H^s(\Omega)$ and $H^{-s}(\Omega)$ on a domain $\Omega$?

Consider the fractional Sobolev spaces on $\mathbb R^n$
$H^s(\mathbb R^n) := \left\{ u \in \mathcal S'(\mathbb R^n) \; : \; ( 1 + |\xi|^2 )^{s/2} \hat u \in L^2(\mathbb R^n) \right\}$.
Let $\Omega$ ...

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### Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)'$ and $\substack{\text{colim} \\ i \rightarrow } H_i$

Let $\substack{\text{lim} \\ \leftarrow i} H_i$ be a nuclear space, considered as the limit of the codirected diagram $$... \to H_2 \to H_1 \to H_0,$$
with $f_{ji}:H_i \to H_j$ being the trace class ...

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### Inner product spaces without symmetry/hermitian axiom

Consider a vector space $X$ over $\mathbb R$ and a bilinear form
$ \langle \cdot, \cdot \rangle : X \times X \rightarrow \mathbb R$.
We assume furthermore that for any $x \in X$ there exists $y \in ...

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### Weak convergence of probability measures on weak versus strong dual

The space of temperate distributions $S'(\mathbb{R}^d)$ is often equipped with the weak-$\ast$ or with the strong topology. When defining the notion of a probability measure on $S'(\mathbb{R}^d)$, ...

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### Weak topology on subsets of a normed space

I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset.
When is the norm a continuous function on $E$?
When is the metric induced by the ...

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129 views

### Is a Fréchet Montel space distinguished?

Based on a couple of references, it seems that the answer is yes, see for example
Boneta-Dierolf, 1992 and Bierstedt-Bonet, 1989.
However, from a comment to the answer of this MO question, I infer ...

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### Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...

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242 views

### Base of a cone in a vector space: can one always choose a convex base?

Let $C$ be a pointed convex cone in a vector space $V$. This means that $C$ satisfies the three following axioms:
$C + C \subset C$,
$\mathbb{R}_+ \cdot C \subset C$, and
$C \cap (-C) = \{ 0 \}$. ...

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### Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?

Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully).
Suppose $g$ is holomorphic on ...

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### Is the space $S'(\mathbb{N})$ of slowly increasing sequences the projective limit of Hilbert sequence spaces?

Let $S(\mathbb{N})$ be the space of rapidly decreasing sequences and $S'(\mathbb{N})$ its topological dual, the space of sequences bounded by a polynomial.
For $m\in \mathbb{Z}$, we also define ...

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### Invariant generalized sections of dual vector bundles

Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My ...

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### Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...

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### Examples of topologies compatible with a given dual pair

Let $\langle X, Y \rangle$ be a pair of vector spaces put in duality by a non-degenerate bilinear form $\langle \cdot, \cdot \rangle: X \times Y \to \mathbb{R}$. A topology $\tau$ on $X$ is called ...

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### $C^\infty$-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 ...

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### Stronger version of linearity for functions of measures

Let $X$ be a standard Borel space, and $P(X)$ be space of Borel probability measures on $X$. It is also a standard Borel space if endowed with the topology of weak convergence, so we can integrate ...

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### Measures, orthogonal to holomorphic functions

Let $G$ be a domain in $\mathbb{C}^{d}$ and let $H\left(G\right)$ be the space of all holomorphic functions on $G$.
My question is how to characterize all such (Radon) measures $\mu$ on $G$, that ...

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### Bases for spaces of smooth functions

Let $S$ denote the space of rapidly decreasing sequences, which means sequences $a=(a_k)_{k=1}^\infty$ such that the numbers $p_d(a)=\sup\{k^d|a_k| : 1\leq k<\infty\}$ are finite for all ...

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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 ...

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### Frechet Derivative in General Topological Vector Space

If I have a two Hausdorff topological vector spaces, $E$ and $F$ and a mapping $f:E\to F$, is it possible to have a meaningful notion of the derivative of $f$ if the space cannot be endowed with a ...

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### Is the closed ball of a normed space closed in any Hausdorff locally convex topology, weaker than the norm topology?

Assume that we have a normed space $X$ and a subspace $Y$ of $X^{*}$ such that $Y^{\perp}=\{0\}$. They form a non-degenerate dual pare.
Moreover, $\|y\|=\sup_{x\in B_{X}}|\langle x,y\rangle|$, where ...

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### Is the space of smooth functions with compact support a DF space?

Is there a good criterion, when a (nuclear) LF space is DF (DFN)? What are references to find out about that? If not, is there a known way, to construct a projective resolution of LF spaces? Prosmans ...

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863 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 Scottish ...

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### When sequentially continuous linear functional is continuous?

Let $C^\infty(X)$ denote the space of infinitely smooth functions on a compact manifold $X$ (at the beginning one may assume that $X$ is a circle, though I need a more general case). Let ...

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### Extending affine maps defined on weakly closed sets to the whole topological space

Given $C$ a weakly closed convex subset of a (real) Banach space $B$, with $0\in C$ and $\varphi:C\longrightarrow \mathbb{R}$ weakly continuous, with $\varphi(0)=0$, can we extend $\varphi$ to a ...

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### Linear operators on distributions with different topologies

Denote by $\mathscr{D}^\prime$ and $\mathscr{D}^\prime_b$ the space of distributions on $\mathbb{R}^n$ equipped with the weak and the strong topology, respectively. Because the topology of ...

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### 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 ...

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### 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 ?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### $c^\infty$-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 ...

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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 $K_{1}+K_{2}$ ...

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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 ...