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Topological vector space with a locally convex topology, i.e. induced by a system of seminorms.

1 vote

Weak convergence of probability measures on weak versus strong dual

Just after posting this MO answer Generalization of Lévy's continuity theorem for nuclear spaces I went and looked again at Fernique's remarkable article and he answered my question above in the affir …
Abdelmalek Abdesselam's user avatar
1 vote

On Köthe sequence spaces

Good question. I think these sequence spaces deserve to be better known because they provide a rich bank of concrete examples for things related to the theory of topological vector spaces which can be …
Abdelmalek Abdesselam's user avatar
16 votes
Accepted

Why is multiplication on the space of smooth functions with compact support continuous?

You can spare yourself the functional analytic abstract nonsense by using an explicit set of seminorms on $\mathcal{D}(\mathbb{R}^d)=C_{c}^{\infty}(\mathbb{R}^d)$ which, unfortunately, are not well-kn …
Abdelmalek Abdesselam's user avatar
1 vote

Topologies on space of compactly supported continuous functions

A bit long for a comment so I will post it as an answer. I did not think of $C_c(\mathbb{N})$ as $\mathscr{D}(M)$ for some zero-dimensional manifold $M$ (with countably many connected components), but …
Abdelmalek Abdesselam's user avatar
4 votes
Accepted

Known dense subset of Schwartz-like space and $C_c^{\infty}$?

I don't know about "well known" or canonical answers to this question, but it is easy to construct an $X$ that works as follows. Using the definition of Hermite polynomials given by $$ H_n(x)=(-1)^n e …
Abdelmalek Abdesselam's user avatar