$C^\infty$-vectors in general representations of Lie groups on locally convex spaces

Question

This question is related to this one. Let $G$ be a real Lie group (I should emphasize I only care about ordinary Lie groups, not Lie groups modeled on locally convex spaces or anything like that). In the representation theory of $G$, one usually considers representations $G\to\mathrm{Aut}_\mathbf{C}(V)$ where $V$ is a Banach space and the map $G\times V\to V$ is continuous, but in practice one also considers (at the very least) continuous representations on Fréchet spaces like $C^\infty(G)$. The question referenced above is essentially about the definition of a $C^\infty$-vector in a continuous representation of $G$ on a Fréchet space. My interpretation of the answers given is that perhaps this is not completely standardized. However, on page 2 of the book Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups by Borel and Wallach, they introduce the notation $C^\infty(M;V)$ for "the space of $C^\infty$ functions of $M$, with values in $V$, endowed with the $C^\infty$-topology." Here $M$ is a smooth manifold which is "countable at infinity" (I'm not sure what that means but I'm sure I'm happy to assume it for my Lie group $G$, if it's not automatic) and $V$ is any Hausdorff quasi-complete locally convex space over $\mathbf{R}$ or $\mathbf{C}$. There is no explanation of this space's definition, or of its topology, which leads me to believe (perhaps mistakenly) that this is something standard. (One would I think care about this for defining the notion of a $C^\infty$-vector in a general continuous representation of $G$ on a locally convex space.)

Is the notion of a $C^\infty$-map from a smooth manifold to a Hausdorff quasi-complete locally convex space over $\mathbf{R}$ or $\mathbf{C}$, and of the (presumably locally convex) topology on the space of such functions, actually standard? If so, is there a convenient reference where this generality is explained?

(I don't think this appears in any Bourbaki volume because I don't think any of them really combine Lie groups with TVS, although if there were such a volume, I don't doubt that it would have the answer to my question.)

Whether or not one cares about such generality in practice is not really my motivation for asking the question (my efforts to thoroughly read Borel-Wallach have been mostly abortive and I honestly don't know whether or not they use this beyond the case of $V$ Fréchet). I'm really curious about this because, in the case where $k$ is a $p$-adic field, $M$ is a paracompact locally $k$-analytic manifold, $V$ is an arbitrary Hausdorff locally convex space over a $p$-adic field, and smooth maps are replaced by locally analytic ones, spaces of functions and their topologies in this generality are both defined and genuinely important in practice (namely in the representation theory of locally $k$-analytic groups on $p$-adic locally convex spaces that aren't just Banach spaces or even Fréchet spaces). But I'm not sure that mimicking the definition in the non-Archimedean case (which uses Banach subspaces of $V$ and the already-defined case of Banach spaces) would really work in the case of smooth manifolds, because somehow defining smooth maps $M\to V$ in extreme generality seems harder to me than defining (locally) analytic ones as there is no issue of second derivatives, etc. (this is perhaps supported by the Wikipedia page on the Gâteaux derivative, which contains the statement "...for Gâteaux differentials...there are several inequivalent ways to formulate their continuous differentiability.")

Answer 1

Yes, the notions are pretty standard. Grothendieck gives a brief summary of differentiable vector-valued functions in Chapter III.8 of his Topological vector spaces.

As starting point, I would suggest Garth Warner's Harmonic Analysis on Semi-simple Lie groups, I, especially Appendix 2. There you'll find a ton of references - there is a useful guide to the literature accompanying the list of references. Most of the theory is due to Laurent Schwartz's school.

The notion of differentiability poses no real problems: note that we map from a finite-dimensional manifold into $V$. It is a theorem of Grothendieck that (if $V$ is complete) then smoothness of a function $f: M \to V$ is equivalent to smoothness of all scalar maps $p \mapsto \langle f(p), \varphi \rangle$, where $\phi$ runs through all continuous linear functionals on $V$.

The topology on $C^{\infty}(M,V)$ is "uniform convergence on compact subsets" (of the functions and all their derivatives), so if $K \subset M$ is compact, $D$ is a differential operator of finite order and $\lvert \cdot \rvert$ is a continuous seminorm on $V$, then $$ \sup_{p \in K} \lvert Df(p)\rvert $$ is a continuous seminorm on $C^{\infty}(M,V)$ (and these norms define the topology).

I sympathize with your experience with Borel-Wallach, whose style I found pretty brutal (but precise). If you read French, you can try Guichardet's book (Cohomologie continue ...) for a somewhat gentler treatment which might prepare you for reading B.-W.

"Countable at infinity" is the Bourbaki way of saying that a locally compact space is $\sigma$-compact (there is a countable basis of the neighborhood filter of the point at infinity in the one-point compactification).

Answer 2

A bit late to the party, but:

In addition to @user56365's good answer...

Yes, the notion of "quasi-complete" (also known as "locally complete") is standard "in certain circles", and turns out to be the correct/viable notion of completeness in topological spaces beyond Frechet spaces. (Of course with local convexity... not to mention Hausdorff-ness.) The definition is that bounded (in the TVS sense) Cauchy nets converge. Yes, in general, this is strictly stronger than sequential completeness, but/and in general is strictly weaker than "full" completeness (convergence of all Cauchy nets). In fact, the weak dual of a (not-finite-dimensional) Hilbert space is quasi-complete, but not complete in the strongest sense. No, this is not trivial to see.

(My relatively recent books "Modern Analysis of Automorphic Forms, by Example", with Cambridge University Press, fairly carefully explain much of this, and relevance to automorphic forms... and to other things... not assuming any functional analysis background. Also, these are freely available at my website, for non-commercial use, per my contract with CUP.)

As it turns out (!), quasi-completeness is a sufficient assumption to prove things about Gelfand-Pettis integrals, about holomorphic vector-valued functions, and about differentiability of vector-valued functions. (My books talk about these things, assuming few prerequisites, exactly because they arise in representation theory of real Lie groups, and elsewhere...)

EDIT: letting the other shoe drop... spaces of distributions with the weak dual topology, spaces of operators on Hilbert spaces with the strong or weak operator topologies, etc., are (provably) quasi-complete, although they're not complete-metric. And the case of test functions, an LF-space but not complete-metric. These examples are quite sufficient to think seriously about completeness issues on "fancier" topological vector spaces than Frechet. :)