Let \({\mathcal C}(X)\) denote the space of closed convex sets in a Banach space. A sequence \((C_ n)\) in \({\mathcal C}(X)\) is said to be Mosco convergent to a closed set \(C\) if (1) every \(c\in C\) is a strong limit of a sequence \((c_ n)\), \(c_ n\in C_ n\); (2) if a vector \(x\) is the weak limit of some sequence \(c_ k\in C_{n(k)}\), where \(n(k)\) is an increasing sequence of indices, then \(x\in C\).
This convergence can be topologized, and that is called the Mosco topology \(\tau_ M\). In a reflexive Banach space, \(\tau_ M\) contains the Wijsman topology \(\tau_ W\) (\(\tau_ W\) is the weakest topology on \({\mathcal C}(X)\) making the distance functionals \(C \mapsto d(x,C)\) continuous, for every \(x\in X\)). Besides displaying various properties of the Mosco topology, the article compares \(\tau_ M\) and \(\tau_ W\). In particular, Theorem 2.5 says that, in a reflexive Banach space, \(\tau_ W=\tau_ M\) if and only if the dual norm on \(X^*\) is a Kadec norm (any weak\(^*\) convergent net in the dual sphere converges strongly). Additional characterizing conditions in terms of continuity of certain transformations of convex sets (e.g., polar set, translation, etc.) are also given.
Let \(\Gamma(X)\) denote the space of proper lower semicontinuous convex functions (identifiable with epigraphs \(\text{epi\,}f =\{(x,a):x\in X, a\in {\mathbf R}, a \geq f(x)\}\)), equipped with the Mosco topology. In the reflexive setting this ensures the lower semicontinuity of certain multifunctions \(\Delta\), related to the Fenchel transform, on \(\Gamma(X)\). The important fact is that \(\tau_ M\) is the weakest topology with this property (Lemma 3.4 and Theorem 3.5).