Tameness in the Fr\'echet spaces of analytic functions

TL;DR

This paper explores the concept of tameness in Fréchet spaces of analytic functions on Stein manifolds, establishing a link between tameness and hyperconvexity.

Contribution

It characterizes tameness in nuclear Fréchet spaces with specific properties and relates it to the geometric property of hyperconvexity in Stein manifolds.

Findings

Tameness of O(M) is equivalent to hyperconvexity of M.

The study extends to nuclear Fréchet spaces with weak DN and Omega properties.

Provides a characterization of tameness in analytic function spaces.

Abstract

We investigate tameness in the Fr\'echet spaces O(M) of analytic functions on Stein manifolds M equipped with the compact open topology. Actually we will look into tameness in the more general class of nuclear Fr\'echet spaces with the properties weak DN and Omega, and then specialize to analytic function spaces. We show that for a Stein manifold M, tameness of O(M) is equivalent to the hyperconvexity of M.