Let $\operatorname{Lip}_0(M)$ be the space of Lipschitz functions on a complete metric $M$ that vanish at base point. We show every normal functional in $\operatorname{Lip}_0(M)^\ast$ is weak$^*$ continuous, answering question by N. Weaver.

A weak form of weak quasi-continuity, which we call subweak quasi-continuity, is introduced. It is shown that subweak quasi-continuity is strictly weaker than weak quasicontinuity. Subweak quasi-continuity is used to strengthen several results in the literature concerning weak quasi-continuity. Specifically, results concerning the graph, graph function, and restriction of a weakly quasi-continu...

In this note we prove that the weak topology of a Banach space has the Continuous Representability Property which means that every (weakly) continuous total preorder defined on a Banach space can be represented by a weakly continuous utility function. This shows that weak topologies are perfect to look for continuous utility functions that represent preferences on infinite-dimensional commodity...