Weakly web-compact Banach spaces C(X),and Lip0(M),F(M)over metric spaces M

The class of web-compact spaces (in sense of Orihuela), which encompasses a number of spaces, like Lindel & ouml;f Sigma-spaces (called also countably determined), Quasi-Suslin spaces, separable spaces, etc., applies to distinguish a class of weakly web-compact Banach spaces E whose dual unit ball is weak & lowast;-sequentially compact, consequently Banach spaces without quotients isomorphic to l infinity.We prove however that for a Banach space E the space Ew(i.e.E with the weak topology) is web-compact if and only if Ew is a Lindel & ouml;f Sigma-space if and only if Ew contains a web-compact total subset. Consequently, for compact X the space C(X) w is web-compact if and only if X is Gul'ko compact if and only if C(X)w is a Lindel & ouml;f-space if and only if Cp(X)contains a web-compact total subset. If X is compact and C(X)w is web-compact, then Cp(X)contains a complemented copy of the space(c0)p={(xn)is an element of R-omega:xn -> 0}with the topology ofR omega but does not admit quotients isomorphic to(l infinity)p={(xn)is an element of R-omega:supn|xn|<infinity}. We characterize weakly web-compact Banach spacesLip0(M)of Lipschitz functions on metric spaces M and their predual F(M).In fact,Lip0(M)w is web-compact if and only if M is separable. Illustrating examples are provided.