A quantitative version of Krein’s theorems for Fréchet spaces

Abstract

For a Banach space E and its bidual space E ′′, the following function \({k(H) : = {\rm sup}_{y\in\overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}} {\rm inf}_{x\in E} \|y - x\|}\) defined on bounded subsets H of E measures how far H is from being σ(E, E′)-relatively compact in E. This concept, introduced independently by Granero [10] and Cascales et al. [7], has been used to study a quantitative version of Krein’s theorem for Banach spaces E and spaces C p (K) over compact K. In the present paper, a quantitative version of Krein’s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E the above function k(H) reads as follows \({k(H) := {\rm sup}\{d(h, E) : h \in \overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}\},}\) where d(h, E) is the natural distance of h to E in the bidual E ′′. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds \({k(coH) < (2^{n+1} - 2) k(H) + \frac{1}{2^{n}}}\) for all \({n \in \mathbb{N}}\). Consequently this yields also the following formula \({k(coH) \leq \sqrt{k(H)}(3 - 2\sqrt{k(H)})}\). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein’s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet space. We also define and discuss two other measures of weak non-compactness lk(H) and k′(H) for a Fréchet space and provide two quantitative versions of Krein’s theorem for both functions.