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3
votes
2answers
161 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 $E$. ...
5
votes
1answer
95 views

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 ...
3
votes
1answer
96 views

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 $\Omega$ ...
4
votes
1answer
96 views

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 ...
1
vote
1answer
94 views

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 ...
2
votes
1answer
138 views

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 $S'(\mathbb{R}^d)$, ...
2
votes
0answers
133 views

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 ...
1
vote
1answer
116 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 ...
1
vote
0answers
122 views

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 ...
2
votes
1answer
144 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 $C \cap (-C) = \{ 0 \}$. ...
5
votes
1answer
170 views

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 ...
7
votes
1answer
149 views

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 ...
2
votes
0answers
119 views

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 ...
5
votes
1answer
249 views

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 ...
5
votes
0answers
134 views

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 ...
5
votes
1answer
167 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 ...
2
votes
0answers
50 views

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 ...
1
vote
2answers
163 views

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 ...
14
votes
1answer
414 views

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 ...
3
votes
0answers
248 views

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 ...
2
votes
3answers
276 views

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 ...
1
vote
0answers
111 views

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 ...
3
votes
1answer
168 views

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 ...
33
votes
2answers
756 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 ...
3
votes
3answers
279 views

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 ...
0
votes
1answer
77 views

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 ...
3
votes
2answers
170 views

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 ...
2
votes
2answers
136 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
120 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
1answer
120 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
316 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
146 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
172 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
329 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 ...
9
votes
2answers
419 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 ...
6
votes
1answer
183 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
260 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 ...
5
votes
3answers
278 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
154 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
208 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
116 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 $K_{1}+K_{2}$ ...
9
votes
0answers
163 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
190 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 ...
3
votes
0answers
217 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
268 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
376 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
146 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
331 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
308 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 ...
7
votes
2answers
306 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 ...