Rosenthal-Banach compact

Definition

• A compact Hausdorff set K is called Rosenthal-Banach compact, if there is a polish space X and a Banach space E so that K is homeomorphic with a compact subset of B_1(X,E) in the topology of pointwise-weak convergence?, where B_1(X,E) is the set of functions which there is a sequence continuous for every n ∈ N, such that if t ∈ X then

Property

• Every classical Rosenthal compact is Rosenthal-Banach compact (put E=R).
• Every Eberlein compact is Rosenthal-Banach compact (take for M a one point set).
• Every Gul'ko compact is a Rosenthal-Banach compact.

Reference

• Mercourakis, S. and Stamati, E., Compactness in the first Baire class and Baire-1 operators. Serdica Math. J. 28 (2002), no. 1, 1--36.