6
$\begingroup$

Let $\pi : E \to M$ a smooth vector bundle of finite rank, where both $E$ and $M$ are finite dimensional smooth manifolds, and $M$ is compact. I already know that the space $\Gamma^\infty(M,E)$ of smooth sections of $E$ is naturally a Fréchet space (with the seminorms given in terms of multiple derivatives of sections).

Now suppose that all the objects above are real analytic and let $\Gamma^\omega(M,E)$ be the space of real analytic sections of $E$.

My question is: is $\Gamma^\omega(M,E)$ a Fréchet space in some “natural” way? If not, what is the “richest” structure one can put in this vector space? For instance, I would like to know if it is a Baire and metrizable topological vector space.

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ The space of real analytic functions, say on the line, doesn't have a natural Fréchet structure--the appropriate lc topology thereon is notoriously more complicated. It might pay to check out the work of, for example, Martineau, Vogt, Dománski. $\endgroup$
    – terceira
    Commented Jun 3, 2022 at 3:00
  • $\begingroup$ @terceira is it Baire, for instance? Their paper seem very complicated.. $\endgroup$
    – Eduardo Longa
    Commented Jun 3, 2022 at 23:17
  • 1
    $\begingroup$ Well, yes--the subject IS complicated. I am confident that it is NOT Baire (call this a conjecture since I have never sat down to write out a proof). The case of the real analytic functions on a compact interval (which is simpler) leads to a so-called Silva space (for which see Köthe's monograph) and these are never Baire, except in trivial situations. $\endgroup$
    – terceira
    Commented Jun 4, 2022 at 6:42
  • $\begingroup$ The upshot is that you have a projective limit of a countable spectrum of inductive limits of countable spectra of Banach spaces. These underlying spaces and the linking mappings are nice enough but mixing projective and inductive limits can lead to headaches. $\endgroup$
    – terceira
    Commented Jun 4, 2022 at 6:48

0

Reset to default

You must log in to answer this question.