# Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III. Hilbert spaces and Universality

## TL;DR

This paper investigates the structure of smooth function spaces generated by Hilbert space elements, establishing their perfect sequence space nature, tensor structures, and demonstrating their universality on compact manifolds.

## Contribution

It extends previous work by characterizing these function spaces as perfect sequence spaces and proving their universality on compact manifolds.

## Key findings

- Spaces are perfect sequence spaces
- Tensor structure of sequential mappings characterized
- Universality of these spaces on compact manifolds proven

## Abstract

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.