$D(X) < 1$ or $\hat{D}(X) < 1$ imply Property($K$)

TL;DR

The paper introduces two new Banach space moduli involving weakly convergent sequences, demonstrating that if either is less than 1, the space possesses Property(K).

Contribution

It presents novel Banach space moduli related to weak convergence and links their values to Property(K), advancing understanding of Banach space geometry.

Findings

If $D(X) < 1$, then the Banach space has Property(K).

If $ ilde{D}(X) < 1$, then the Banach space has Property(K).

Introduction of new moduli involving weak convergent sequences.

Abstract

Two new Banach space moduli, that involve weak convergent sequences, are introduced. It is shown that if either one of these moduli are strictly less than 1 then the Banach space has Property($K$)