2

I want to work in the most general settings. Let us say we have a normed space $\mathcal{B}$. To what extent can we capture properties on the space by studying the functionals on this space?

  • 1
    It's only vaguely related to what you have in mind, but the question reminds me of the standard result that for any compact manifold $X$, the maximal ideals of $C(X)$ are of the form $\mathfrak{m}_x = {f\in C(X):, f(x) = 0}$ for $x\in X$. – anomaly Mar 19 '20 at 22:15
  • Interesting and ambitious question... Why especialy simple normed spaces ? Banach spaces are much richer. On the other side, Schwartz space (of distributions) is based on semi-norms... – Jean Marie Mar 19 '20 at 22:21
  • For sure I would love to understand that as well ! –  Mar 19 '20 at 22:22
  • 1
    There's plenty of nonisomorphic spaces with isomorphic duals so probably not very much. You can say something, for example if the dual is separable so is the starting space, but you can't even tell if $\mathcal B$ is complete by looking at the dual, so there's probably very little to be said – Alessandro Codenotti Mar 19 '20 at 22:23
  • @JeanMarie I would like to know more on Schwartz space (of distributions). –  Mar 19 '20 at 22:48
  • I advise you to begin gently by the historical context, such as in this article (http://sites.mathdoc.fr/OCLS/pdf/OCLS_1982__63__11_0.pdf) – Jean Marie Mar 19 '20 at 23:01
  • @JeanMarie Thank you! –  Mar 20 '20 at 00:00
  • Does anyone know if functionals are dense in certain space then we can capture a lot of information about the original space given this fact? I have my reason to believe some theorem like that might exist. –  Mar 20 '20 at 03:17

0 Answers0