On the Krein-Šmulian theorem for weaker topologies (Q1419630)

The classical Krein-Šmulian theorem says that the \(w\)-closed convex hull of a \(w\)-compact subset \(H\) of a Banach space \(X\) is \(w\)-compact, where \(w\) is the weak topology. It is shown that for a WCG Banach space \(X\) the weak topology in this theorem can be replaced by any locally convex Hausdorff topology \(\tau\) weaker than the strong topology of \(X\). The WCG condition for the whole space \(X\) may be replaced by norm-boundedness together with norm-fragmentability of \((H, \tau)\). The proof uses an elegant application of recently developed Riemann-Lebesgue integration theory for vector-valued functions. A new proof of the following result is given: if \(X\) does not contain a copy of \(l_1[0,1]\), then the Krein-Šmulian theorem holds for every topology \(\tau\) induced by a norming set of functionals. This result implies the (known) positive answer to the Godefroy's boundary problem for this class of spaces. A positive solution of the boundary problem for \(C(K)\) and \(l_1(\Gamma)\) spaces is given as well.