Topologies Weaker Than the Weak Topology of a Banach Space

In this paper we investigate some properties of the compact subsets of Banach spaces X endowed with topologies of the kind σ(X, B) where B is a norming subset of the dual unit ball BX*. Assuming that BX* is sequentially compact we prove that the Krein-Smulian theorem holds for norm bounded σ(X, B)-compact subsets of X and we state that the convex σ(X, B)-compact subsets of X have the weak Radon-Nikodým property. When BX* is sequentially compact and X has either the separable complementation property or X is weakly Lindelöf (for instance, when BX* is Corson compact) we prove that the σ(X, B)-compact subsets (resp. σ(X, B)-compact convex subsets) of X are fragmented by the norm of X (resp. have the Radon-Nikodým property). So, if BX* is a Corson compact then the compact subsets of the space X[σ(X; B)] are Radon-Nikodým compact and thus sequentially compact. We apply the previous results to prove that if BX* is sequentially compact and B is assumed to be a boundary of BX*, then the norm bounded σ(X, B)-compact subsets of X are weakly compact, which partially answers a problem posed by G. Godefroy. We give, among others, applications to spaces of vector-valued Bochner integrable functions as well as to spaces of countably additive measures.