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I do not know if it is good form for MO to cite one's own papers when answering a question, but I will take the chance. This matter is addressed in quite a bit of detail in my joint paper with Romeo B …

I will leave to Yemon Choi discussing the answer from Gelfand-Raikov-Shilov's book (Commutative Normed Rings, I suppose?), and restrict myself to more recent discussions on the matter... There is an …

A proof of the Poincaré lemma with optimal regularity for (non-integer order) Hölder and (nonnegative-order $L^p$, $2\leq p<\infty$) Sobolev forms is provided by Theorem 8.3, pp. 148-149 of the book b …

Two (edit: now four) not-so-usual examples come to my mind: There is the proof of the central limit theorem using Fourier analysis, as done in Chapter 7 of Hörmander's book. It's a cornerstone of pr …

$\DeclareMathOperator\Ann{Ann}\DeclareMathOperator\Tr{Tr}$My answer is somewhat complementary to Nik Weaver's, and admitedly more focused on Question 2 since I have nothing more to add to the latter r …

For the $L^\infty$ norms, nothing at all apart from Alexander Eremenko's answer, because of Borel's theorem: any sequence of real (resp. complex) numbers can be realized as the sequence of Taylor coef …

Let $0\leq f\in\mathscr{D}(\mathbb{R}^n)$. As shown e.g. by J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232 (2006) 137-14 …

The answer is no. In what follows (see OP's comments below) we assume that the arrows of the category of lctvs are Michal-Bastiani smooth maps. Recall that a map $\Phi:E\rightarrow F$ from a lctvs $E$ …

Since $\mathscr{S}'$ is the dual of an infinite-dimensional Fréchet space, the weak-* topology in $\mathscr{S}'$ is not even first countable. What people usually do is to define probability measures i …

Besides Hodge and index theories, mentioned in Qiaochu Yuan's comment above as applications of functional analysis to (complex) algebraic geometry and algebraic topology respectively, I believe that a …

Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is convenient (also called locally complete, Mackey-complete or $c^\infty$-complete) if, given …

This is easy and comes from the fact that $V$, being a Fréchet space, is barrelled, that is, the locally convex topology of $V$ coincides with its strong topology $\beta(V,V')$ (where $V'$ = topologic …

Yes, there is. The (Constant) Rank Theorem for Banach spaces is Theorem 2.5.15 of the book of R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications (3rd. edition, Springer …

I'll try to explain what Igor meant in his comments in a different way, maybe it helps. Of course, any tempered distribution is a distribution in the broader sense - more precisely, any compactly sup …

This is not at all a complete answer, but rather an expanded update on my above comment. I shall start with a few general considerations: If $\{f_n\ |\ n\in\mathbb{N}\}$ is any Schauder basis for eit …