A NEW LOOK AT COMPACTNESS VIA DISTANCES TO FUNCTION SPACES

Many classical results about compactness in functional analysis can be derived from suitable inequalities involving distances to spaces of continuous or Baire one functions: this approach gives an extra insight to the classical results as well as triggers a number of open questions in different exciting research branches. We exhibit here, for instance, quantitative versions of Grothendieck's characterization of weak compactness in spaces C(K) and also of the Eberlein–Šmulyan and Krein–Šmulyan theorems. The above results specialized in Banach spaces lead to several equivalent measures of non-weak compactness. In a different direction we envisage a method to measure the distance from a function f ∈ ℝ^X to B₁(X) — space of Baire one functions on X — which allows us to obtain, when X is Polish, a quantitative version of the well-known Rosenthal's result stating that in B₁(X) the pointwise relatively countably compact sets are pointwise compact. Other results and applications are commented too.