3
$\begingroup$

There are four most important functional spaces in analysis:

  • the space $\mathcal{C}(M)$ of continuous functions on a topological space,

  • the space $\mathcal{E}(M)$ of smooth functions on a smooth manifold,

  • the space $\mathcal{O}(M)$ of holomorphic functions on a complex manifold, and

  • the space $\mathcal{P}(M)$ of polynomials on an algebraic manifold.

In the first two cases the functionals on the spaces have well-known names:

  • continuous functionals $f:\mathcal{C}(M)\to\mathbb{C}$ are called measures, and

  • continuous functionals $f:\mathcal{E}(M)\to\mathbb{C}$ are called distributions.

What do people call continuous functionals on $\mathcal{O}(M)$ and on $\mathcal{P}(M)$?

(The space $\mathcal{O}(M)$ is endowed with the usual compact-open topology, and the space $\mathcal{P}(M)$ with the strongest locally convex topology, so each linear functional on $\mathcal{P}(M)$ is continuous.)

I saw the names analytic functionals for $f:\mathcal{O}(M)\to\mathbb{C}$, and currents for $f:\mathcal{P}(M)\to\mathbb{C}$, but both terms sound strange to me.

Is it possible that there are more or less convenient names for these objects?

Improve this question
$\endgroup$
6
  • 4
    $\begingroup$ In p-adic functional analysis it's conventional that the dual of the space of "[whatever] functions" gets called the space of "[whatever] distributions" (with measures being the exception to this convention). Not sure if this is a general convention though. I'd certainly use "algebraic distributions" for the dual of your $\mathcal{P}(M)$. $\endgroup$
    – David Loeffler
    Commented Nov 17, 2021 at 19:43
  • 1
    $\begingroup$ @DavidLoeffler yes, this sounds logical, but, to tell the truth, the terms "algebraic distribution", "holomorphic distribution", "smooth distribution", "continuous distribution" are strange as well... "Holomorphic distribution" does not seem to be holomorphic in any reasonable sense. Similarly, "algebraic distribution", etc. $\endgroup$
    – Sergei Akbarov
    Commented Nov 17, 2021 at 19:54
  • 2
    $\begingroup$ "Analytic functionals" is a standard name. See, Hormander, The analysis of linear partial differential operators I, Chap. 9. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 17, 2021 at 23:22
  • $\begingroup$ @AlexandreEremenko thank you. And what is the term for functionals on $\mathcal{P}(M)$? $\endgroup$
    – Sergei Akbarov
    Commented Nov 17, 2021 at 23:42
  • $\begingroup$ This I do not know. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 17, 2021 at 23:48

0

Reset to default

You must log in to answer this question.