Compactness and Distances to Spaces of Continuous Functions and Fréchet Spaces

Abstract

In recent years, several quantitative counterparts for several classical such as Krein-S̆mulyan, Eberlein-S̆mulyan, Grothendieck, etc. have been proved by different authors. These new versions strengthen the original theorems and lead to new problems and applications in topology and analysis. In this survey, we present several of these quantitative versions of theorems about compactness in Banach spaces with the weak topology, Fréchet spaces with the weak topology and spaces of continuous functions with the pointwise convergence topology. For example if $H$ is a subset of a Banach space $E$, and $w^*$ is the weak* topology in $E^{\prime \prime }$, the index $k(H):=\sup \{d(x^{**},E),x^{**}\in \overline{H}^{w^{*}}\}$ is zero if and only if $H$ is relatively compact in $(E,w)$. Then $k(H)$ measures how far is $H$ from being relatively compact in $(E,w)$. The following inequalty $k(co(H))\le 2 k(H)$ is a quantitative version of the Krein-S̆mulian theorem about the $w$-relative compactness of the convex hull of a weakly compact set.