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7
votes
1answer
261 views

Error in Maurins proof for the nuclear spectral theorem?

I am currently studying the nuclear spectral theorem as presented in K. Maurins Monograph "General Eigenfunction Expansions and Unitary Representations of Topological Groups", second chapter or ...
3
votes
2answers
161 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 ...
1
vote
0answers
122 views

Reference for a General Theory of Sequences?

Since decades, mathematicians are studying function spaces, discovering new structures more and more adapted for a general theory of functional analysis. In that works, sequence spaces are generally ...
4
votes
2answers
275 views

Wiener measure and Bochner Minlos

I am reading probability theory and I have a question. The Bochner-Minlos theorem roughly says that if we have $E \subset H \subset E^*$ where $H$ is a Hilbert space, then there is a unique measure ...
7
votes
2answers
325 views

Measure theory in nuclear spaces

Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
4
votes
1answer
492 views

What functions can be obtained as a convolution of a Schwartz function and a tempered distribution?

Let $\mathcal S (\mathbb R)$ denote the space of Schwartz functions on $\mathbb R$ and $\mathcal S^* (\mathbb R)$ denote the dual space of Schwartz (a.k.a tempered) distributions. We consider ...
13
votes
6answers
2k views

Grothendieck on Topological Vector Spaces

In a short biography article on Alexander Grothendieck, it is mentioned that after Grothendieck submitted his first thesis on Topological Vector Spaces (TVS), apparently, he told Bernard Malgrange ...
5
votes
1answer
430 views

Schwartz Kernel theorem for tempred functions

Let $T(R)$ denote the space of tempered functions on the line, i.e. the smooth functions that give Schwartz function after a multiplication by any Schwartz function, equipped with the natural nuclear ...
4
votes
0answers
278 views

Exactness of completed tensor product of nuclear spaces

Let $0 \to V \to W \to L \to 0$ be a strict short exact sequence of (complete) nuclear spaces, i.e. it is a short exact sequence of (complete) nuclear spaces, all the maps are continuous, the map ...
1
vote
0answers
207 views

Hom of Nuclear spaces

Let V be a Nuclear space (not necessary Frechet, but complete). Let V^* be the dual space considered with the weak topology (i.e. pointwise convergence). Is it true that $V^*$ is also nuclear? Is it ...
14
votes
0answers
617 views

Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?

This is a followup to my previous question, which asked whether the category of commutative or noncommutative C*-algebras or von Neumann algebras is equivalent to the category of commutative or ...
12
votes
2answers
526 views

Are extensions of nuclear Fréchet spaces nuclear?

Consider the category of Fréchet spaces, the morphisms being continuous linear maps with closed image. Suppose that we have a short exact sequence in that category: $0 \rightarrow V_1 \rightarrow ...