Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality

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 papers (Dasgupta and Ruzhansky in Trans Am Math Soc 368(12):8481–8498, 2016) and (Dasgupta and Ruzhansky in Trans Am Math Soc Ser B 5:81–101, 2018). 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.