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There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions

Consider the space $\mathcal D'(M)$ of distributions on a manifold $M$. Is there a ready reference for the fact that the Borel $\sigma$-algebra (for the strong dual topology) coincides with the weak ...
Pierre PC's user avatar
  • 3,669
3 votes
0 answers
122 views

Analytic functions and Hyperfunction as TVS

I have several related questions on Analytic functions and Hyperfunction as topological vector spaces (I am mainly interested in questions 4,6,10): For an open set $U\subset \mathbb C^n$ we can ...
Rami's user avatar
  • 2,639
3 votes
1 answer
182 views

Tensor product of a slowly increasing smooth function and a tempered distribution converging to a co-located product

Let $T$ be a tempered distribution on $\mathbb{R}$ and $g$ be a smooth function on $\mathbb{R}$ whose derivatives of all orders are all polynomially bounded (a.k.a. slowly increasing). For any pair of ...
Isaac's user avatar
  • 3,477
2 votes
1 answer
49 views

Is any submetrizable linear topology linearly submetrizable?

Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$. Is ...
erz's user avatar
  • 5,529
0 votes
0 answers
119 views

Boundedness of 2 times the unit ball

Suppose that $X$ is a topological vector space where the topology is given by a metric $d$ on $X$. Assuming that the unit ball $$ B(0, 1) := \{x \in X : d(0, x) < 1\} \neq X, $$ is it necessarily ...
Chandan Biswas's user avatar
2 votes
1 answer
236 views

A sensible topology on the space of continuous linear maps between Fréchet spaces

Let $V_1$ and $V_2$ be Fréchet spaces. Let $\{ \lVert \cdot \rVert_{1,n} \}_{n \in \mathbb{N}}$ be a family of seminorms for $V_1$ and similarly $\{ \lVert \cdot \rVert_{2,n} \}_{n \in \mathbb{N}}$ ...
Isaac's user avatar
  • 3,477
0 votes
0 answers
97 views

Heine-Borel property for (probability) measures on $\mathcal{S}'$?

For Euclidean space $\mathbb{R}^n$, it is a well-known fact that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is a Montel space. Moreover, the space of tempered distributions $\mathcal{S}'(\mathbb{R}...
Isaac's user avatar
  • 3,477
1 vote
0 answers
156 views

Density of Schwartz distributions in the space of distribution

Let $\mathscr S(\mathbf R^3)$ and $\mathscr D(\mathbf R^3 )$ be the space of Schwartz function and test function respectively, $\mathscr S'(\mathbf R^3)$ and $\mathscr D'(\mathbf R^3)$ be their duals....
Alucard-o Ming's user avatar
3 votes
1 answer
228 views

Is compact-open topology stable with respect to injective limits?

Let $X$ be a locally convex space, and $\{Y_i;\ i\in I\}$ a covariant system of locally convex spaces over a partially ordered set $I$, i.e. a system of linear continuous mappings is given $\sigma^j_i:...
Sergei Akbarov's user avatar
4 votes
0 answers
87 views

Colimits of locally convex spaces in the categories of topological vector spaces vs locally convex spaces

Let $S$ be a set and let $V_s$ be a family of locally convex topological vector spaces (LCSs) indexed by $s \in S$. Let $V$ be a vector space (without topology) and let $T_s:V_s \to V$ be a family of ...
James Tener's user avatar
17 votes
0 answers
677 views

Are dualizable topological vector spaces finite-dimensional?

Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product. Every finite-dimensional ...
Dmitri Pavlov's user avatar
4 votes
0 answers
108 views

Larger possible chain of closed subspaces in the dual of a Banach space

In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces. My question is the following. If $X$ is an ...
Emerick's user avatar
  • 153
6 votes
0 answers
293 views

Looking for Mackey's PhD thesis, "The subspaces of the conjugate of an abstract linear space"

I'm looking for a copy of George Mackey's PhD thesis, The subspaces of the conjugate of an abstract linear space (Harvard Univ., 1942), but am currently struggling to find one online, with the only ...
Emily's user avatar
  • 11.8k
0 votes
1 answer
143 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 $\mathbb{R}^{d}$ ...
Math's user avatar
  • 509
0 votes
0 answers
235 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 ...
Luiz Felipe Garcia's user avatar
4 votes
3 answers
482 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 ...
Isaac's user avatar
  • 3,477
6 votes
2 answers
290 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 ...
Isaac's user avatar
  • 3,477
2 votes
1 answer
196 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 ...
olih's user avatar
  • 91
0 votes
0 answers
192 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 ...
CBBAM's user avatar
  • 721
6 votes
0 answers
219 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 ...
Brian Hepler's user avatar
4 votes
0 answers
176 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^{\...
Jon's user avatar
  • 41
0 votes
1 answer
242 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 ...
i like math's user avatar
1 vote
1 answer
113 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} ...
Isaac's user avatar
  • 3,477
2 votes
0 answers
320 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(...
CoffeeArabica's user avatar
3 votes
1 answer
198 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, ...
erz's user avatar
  • 5,529
1 vote
0 answers
54 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\...
erz's user avatar
  • 5,529
3 votes
1 answer
298 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 ...
erz's user avatar
  • 5,529
0 votes
0 answers
75 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 ...
Sebastian Bechtel's user avatar
2 votes
1 answer
143 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 ...
Nick's user avatar
  • 61
2 votes
0 answers
103 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 ...
B.Hueber's user avatar
  • 1,171
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)...
elmas's user avatar
  • 55
1 vote
1 answer
163 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_\...
A beginner mathmatician's user avatar
3 votes
1 answer
451 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 ...
Michael_1812's user avatar
1 vote
0 answers
220 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 ...
amilton moreira's user avatar
2 votes
0 answers
119 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 ...
Mar's user avatar
  • 61
0 votes
0 answers
105 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 ...
tom's user avatar
  • 143
1 vote
0 answers
190 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 ...
Nicolay Avendaño's user avatar
1 vote
1 answer
189 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^*...
Pietro Majer's user avatar
  • 60.5k
6 votes
1 answer
1k 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 ...
Rami's user avatar
  • 2,639
2 votes
1 answer
183 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 ...
erz's user avatar
  • 5,529
6 votes
1 answer
369 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 ...
Paolo Leonetti's user avatar
1 vote
0 answers
70 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 ...
Nik Bren's user avatar
  • 519
1 vote
0 answers
142 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 ...
Michael_1812's user avatar
2 votes
1 answer
332 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:...
Nik Bren's user avatar
  • 519
4 votes
0 answers
481 views

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}...
Nik Bren's user avatar
  • 519
1 vote
1 answer
214 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 $0$. ...
Zerox's user avatar
  • 1,543
2 votes
2 answers
250 views

Pontryagin-reflexivity of spaces of continuous functions

It is well-known that a Banach space $V$ is always Pontryagin-reflexive, i.e. the natural map $V\to \text{Hom}_\mathbb{R}(\text{Hom}_\mathbb{R}(V, \mathbb{R}), \mathbb{R})$, where $\text{Hom}_\mathbb{...
user avatar
1 vote
2 answers
204 views

When is a natural map between completions injective?

Let $X$ be a vector space equipped with a norm $p$ and a seminorm $q$. Denote the completion of $X$ with respect to $p$ with $X_p$ and with respect to $p+q$ by $X_{p+q}$. Then the induced map $\iota : ...
iolo's user avatar
  • 651
3 votes
1 answer
210 views

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 ...
erz's user avatar
  • 5,529
3 votes
1 answer
352 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 $x=\lim x_{n_k}$. ...
ABB's user avatar
  • 4,058

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