Some Minimax Results on Dense Sets and Dense Families of Functionals in C(K) and B(K)

TL;DR

This paper investigates minimax theorems in function spaces, demonstrating limitations of existing results on dense sets and introducing self-segment-dense sets to establish new minimax results and functional denseness.

Contribution

It introduces the concept of self-segment-dense sets and extends minimax results to these sets, advancing the understanding of functional denseness in C(K) and B(K).

Findings

Minimax results of Fan and Sion do not extend to usual dense sets.

New minimax results are established for self-segment-dense sets.

Application to denseness of certain families of functionals in function spaces.

Abstract

In this paper, by an example we show that the general minimax results of Fan and Sion cannot be extended on usual dense sets. Nevertheless, we obtain some new minimax results on a special type of dense set that we call self-segment-dense. We apply our results in order to obtain denseness of some family of functionals in function spaces.