On weighted inductive limits of spaces of ultradifferentiable functions and their duals

TL;DR

This paper investigates the completeness and duality of weighted inductive limits of ultradifferentiable function spaces, providing characterizations based on convolution growth and conditions for ultrabornological properties.

Contribution

It offers new criteria for completeness, dual space characterization, and ultrabornological conditions for these function spaces, including applications to Gelfand-Shilov convolutors.

Findings

Established completeness of certain weighted inductive limits.

Characterized dual spaces via convolution average growth.

Provided conditions for ultrabornological properties.

Abstract

In the first part of this paper we discuss the completeness of two general classes of weighted inductive limits of spaces of ultradifferentiable functions. In the second part we study their duals and characterize these spaces in terms of the growth of convolution averages of their elements. This characterization gives a canonical way to define a locally convex topology on these spaces and we give necessary and sufficient conditions for them to be ultrabornological. In particular, our results apply to spaces of convolutors for Gelfand-Shilov spaces.