The Perfekt theory of $M$-ideals

TL;DR

This paper revisits Perfekt's framework for identifying biduality in function and sequence spaces, offering new proofs under weaker assumptions and extending the application to Lipschitz spaces.

Contribution

It provides new, weaker-assumption proofs of Perfekt's theory and applies these results to Lipschitz spaces, expanding the scope of the original framework.

Findings

New proofs under weaker assumptions

Extension of the theory to Lipschitz spaces

Enhanced understanding of biduality in function spaces

Abstract

We revisit some ideas of K.-M.~Perfekt who has provided an elegant framework to detect the biduality between function or sequence spaces defined in terms of some $o$- resp.\ $O$-condition. We present new proofs under somewhat weaker assumptions than before and apply the result to Lipschitz spaces.