Reference for : a Fréchet nuclear space is Montel

I'm looking for a reference to cite regarding the property presented in the title: "Closed and bounded sets of a nuclear Fréchet space are compact"

Thank you in advance for the help!

Have a look at Proposition 50.2 in

F. Treves: Topological Vectors Spaces, Distributions and Kernels, Academic Press 1995 or Dover 2006

Statement (50.12) in that proposition is precisely what you need.

does Wikipedia qualify as a reference? for a historical overview, see A pedagogical history of compactness

• Wikipedia is the place where I actually stumbled across the result... the problem is that I doubt wikipedia qualifies as an academic source in an article. Anyway, I prefer not. Thanks anyway! Commented Sep 20, 2012 at 13:03

Maybe not in a single theorem, but you can go for Cor1 in Section 33 and Cor3 in Section 50 in Treves book.

• Thanks for your answer, but I guess Liviu narrowed down a more precise reference, I guess! Commented Sep 20, 2012 at 13:04
• and he was a couple of second faster :) Commented Sep 20, 2012 at 13:05
• By accident, I had Treves' book right by my side. Commented Sep 20, 2012 at 13:12

[Sch99, §III.7.2, Corollary 2]: every bounded subset of a nuclear space is precompact.

[Pie, §4.4.7]: In each nuclear or dual nuclear locally convex space $$E$$ all bounded subsets are precompact.

References.

[Pie72]: Albrecht Pietsch, Nuclear Locally Convex Spaces (1972), Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin.

[Sch99]: H.H. Schaefer, M.P. Wolff (translator), Topological Vector Spaces, Second Edition (1999), Springer Graduate Texts in Mathematics, Springer, New York.

Since the garden variety Fréchet nuclear spaces (the Schwartz space, the space of smooth functions on the torus, the space of smooth fuctions with support in a given compact set, etc., Edit: see also Jochen's comment below) are isomorphic as topological vector spaces to $$\mathscr{s}$$, the space of rapidly decaying sequences, and since the proof of the Montel property for $$\mathscr{s}$$ is very easy, I might as well explain it here in a self-contained answer.

I will use the convention $$\mathbb{N}=\{0,1,2,\ldots\}$$ and denote by $$\mathscr{s}$$ the space of sequences $$x=(x_n)_{n\in\mathbb{N}}$$ such that $$||x||_{\infty,k}:=\sup\limits_{n\ge 0}\ \langle n\rangle^k|x_n|<\infty$$ for all $$k\in\mathbb{N}$$. Here $$\langle n\rangle:=\sqrt{1+n^2}$$. The topology is the one given by the collection of seminorms $$||\cdot||_{\infty,k}$$, $$k\ge 0$$. An equivalent system of seminorms defining the same locally convex topology is $$||x||_{1,k}:=\sum_{n\ge 0}\langle n\rangle^k|x_n|\ .$$ I will denote by $$\mathscr{s}_{+}$$ the subset of $$\mathscr{s}$$ made of sequences with only nonnegative entries. Given a subset $$A\subset \mathscr{s}$$, I will define the envelope of $$A$$, or $${\rm env}(A)$$, as the sequence $$z=(z_n)$$ given by $$z_n:=\sup_{x\in A}|x_n|\ .$$ This sequence a priori belongs to $$[0,\infty]^{\mathbb{N}}$$. Using the $$||\cdot||_{\infty,k}$$ seminorms, the following is trivial.

Proposition: $$A$$ is a bounded subset of $$\mathscr{s}$$ if and only if $${\rm env}(A)\in\mathscr{s}_{+}$$.

Conversely, given $$\omega\in\mathscr{s}_{+}$$, let me define the subset $${\rm box}(\omega)$$ made of all sequences $$x$$ such that, for all $$n$$, $$|x_n|\le \omega_n$$.

The wanted Montel property immediately follows from the following result.

Proposition: For any $$\omega\in\mathscr{s}_{+}$$, we have that $${\rm box}(\omega)$$ is a compact subset of $$\mathscr{s}$$.

Proof: (in the real case, the complex case only needs disks instead of intervals) Let $$K:=\prod_{n\ge 0}[-\omega_n,\omega_n]$$ with the (metrizable) product topology. Of course, by the countable Tychonov Theorem, $$K$$ is compact. Let $$\tau:K\rightarrow \mathscr{s}$$ be the obvious tautological inclusion map. Then the compactness of $${\rm box}(\omega)=\tau(K)$$ would follow from the continuity of the map $$\tau$$. Since the spaces are metrizable, it is enough to show sequential continuity. Now switch to the $$||\cdot||_{1,k}$$ seminorms for $$\mathscr{s}$$. The needed sequential continuity is immediate from the discrete Dominated Convergence Theorem (DCT). Indeed, let $$(x^{(m)})_{m\ge 0}$$ be a sequence in $$K$$ converging to some element $$x\in K$$. This means that for all $$n$$, one has the pointwise convergence $$\lim_{m\rightarrow\infty}x_{n}^{(m)}=x_n$$. Once mapped inside $$\mathscr{s}$$, we have, for any $$k\ge 0$$, $$||\tau(x^{(m)})-\tau(x)||_{1,k}=\sum_{n\ge 0}\langle n\rangle^k|x_{n}^{(m)}-x_n| \longrightarrow 0$$ by the DCT and the use of the dominating function $$n\mapsto 2\langle n\rangle^k \omega_n$$.

• The Komura-Komura theorem implies that every nuclear Fréchet space is isomorphic to a closed subspace of $s^{\mathbb N}$. Thus knowing that $s$ is Montel implies that every nuclear Fréchet space is Montel -- another example of mathematical overkill. Commented Jul 31, 2020 at 15:01
• I suspected there was a result like Komura-Komura, but I didn't want to venture beyond what I knew. I found I think a good reference to this result which solved a conjecture by Grothendieck, Thm 3.18 in www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf BTW, what is the mathematical overkill you are referring to? I didn't quite get that. Commented Jul 31, 2020 at 17:04
• Komura-Komura is much deeper than the fact that nuclear Frechet spaces are Montel. Commented Jul 31, 2020 at 19:10