On a certain class of $\K$ Banach spaces

TL;DR

This paper introduces the concept of strongly $ extbf{K}$ Banach spaces, a subclass of $ extbf{K}$ Banach spaces, exploring their properties and relationships with other classes like SWCG and SWKA spaces.

Contribution

It defines strongly $ extbf{K}$ Banach spaces using a new strengthening of the $ extbf{K}$ set concept and studies their connections to existing Banach space classes.

Findings

Strongly $ extbf{K}$ Banach spaces include subspaces of SWCG and Polish Banach spaces.

The class is related to strongly weakly $ extbf{K}$--analytic (SWKA) Banach spaces.

Connections between $ extbf{K}$, WKA, and strongly $ extbf{K}$ Banach spaces are established.

Abstract

Using a strengthening of the concept of $\K$ set, introduced in this paper, we study a certain subclass of the class of $\K$ Banach spaces; the so called strongly $\K$ Banach spaces. This class of spaces includes subspaces of strongly weakly compactly generated (SWCG) as well as Polish Banach spaces and it is related to strongly weakly $\mathcal{K}$--analytic (SWKA) Banach spaces as the known classes of $\K$ and weakly $\mathcal{K}$--analytic (WKA) Banach spaces are related.