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Part of me is afraid that this isn't a well-formed question, but try as I might, I can't seem to figure out anything reasonable on this topic. I'm hoping someone here can help.

In functional analysis, one defines the Schwartz space (on $\mathbb{R}^n$, say) as the collection of all smooth functions whose derivatives decay "rapidly" (in the very precise sense given, e.g., on Wikipedia). Over at the nLab (http://ncatlab.org/nlab/show/Schwartz+space), they have what appears to be a more general definition:

In functional analysis, a Schwartz space is a locally convex topological vector space $E$ with the property that whenever $U$ is an absolutely convex neighbourhood of $0$ then it contains another, say $V$, such that $U$ maps to a precompact set in the normed vector space $E_V$.

For the record, I'm an expert on neither of these spaces; however, my naive intuition seems to suggest that the nLab space is somehow "less concrete" than the typical definition involving rapidly-decreasing derivatives. I am unable to seek out more understanding on this, however, because no reference is given at nLab.

So are these notions of Schwartz spaces related? Are they the same? Is there any literature that mentions the nLab definition?

Any insight would be much appreciated.

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Disclaimer: I am not, and never have claimed to be, a functional analyst. However, I used functional analysis a lot in my work in differential topology and geometry. In particular, I read and re-read the mighty tome The Convenient Setting of Global Analysis extensively.

That book contains an appendix with a summary of all the functional analysis used. In particular, on page 585 (online edition), after Result 52.24, appears the definition:

A locally convex space $E$ is called Schwartz if each absolutely convex neighborhood $U$ of $0$ in $E$ contains another one $V$ such that the induced mapping $E_{ ( U )} \to E_{ ( V )}$ maps $U$ into a precompact set.

and this is the definition that appears on the nLab (unsurprisingly, given who is the original author of that page).

(Note: I think that the map $E_{(U)} \to E_{(V)}$ is actually the wrong way around. Reading elsewhere in that appendix, $E_{(U)}$ is the normed vector space formed by taking the Minkowski functional associated to $U$ as a semi-norm and making it a norm by quotienting out by the null space. Therefore an inclusion $V \to U$ should induce a mapping $E_{(V)} \to E_{(U)}$ and one would examine the image of $V$ in $E_{(U)}$ and want that to be precompact. I therefore think that it should read:

A locally convex space $E$ is called Schwartz if each absolutely convex neighborhood $U$ of $0$ in $E$ contains another one $V$ such that the induced mapping $E_{ ( V )} \to E_{ ( U )}$ maps $V$ into a precompact set.

)

The definition in The Convenient Setting ... is not directly cited, but the preceding result is about Schwarz spaces and is cited as Jarchow, 1981, 10.4.3, p202, Horvath, 1966, p277. So I would look there for more information.

The purpose of this definition is to try to capture the functional analytical essence of the Schwartz space (of rapidly decreasing functions). Thus anything one can say about the Schwartz space (in Functional Analysis) can be said about a Schwartz space (and, probably, vice versa). This fits in with a general theme of Functional Analysis wherein one says "This space has all these nice properties, what is it about this space which makes it have them? What other spaces have these properties? This looks like a useful class of spaces, let's give them a funky name.".

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  • $\begingroup$ How does it relate to the fact the usual Schwartz space is the intersection of the Banach spaces $B_{k,m}$ with norm $\|\phi\|_{B_{k,m}} = \sum_{n \le k} \sup_x (1+|x|^m) |\phi^{(n)}(x)|$ ? $\endgroup$ – reuns Aug 14 '17 at 20:52
  • $\begingroup$ @reuns In short, for each $(k,m)$ there will be an $(l,n)$ such that the natural map $B_{(l,n)} \to B_{(k,m)}$ maps the unit ball of $B_{(l,n)}$ into a precompact subset. $\endgroup$ – Andrew Stacey Aug 15 '17 at 10:21

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