Three characterizations of the Mosco topology for convex functions (Q1823422)

Let \(\Gamma(X)\) be the proper lower semicontinuous convex functions on a reflexive Banach space X. We show that the Mosco topology is the weakest topology \(\tau\) on \(\Gamma(X)\) such that the epigraphical multifunctions \(f\to \text{epi }f\) and \(f\to \text{epi }f^*\) are both \(\tau\)-lower semicontinuous. This leads to a characterization of the topology in terms of continuous selections for these multifunctions. In addition, we show that the topology is the weakest topology \(\tau\) on \(\Gamma(X)\) with all of the following properties: (i) \(f\to \inf_ Vf\) is \(\tau\)-upper semicontinuous for each norm open subset \(V\) of \(X\); (ii) \(f\to \inf_ Kf\) is \(\tau\)-lower semicontinuous for each weakly compact subset \(K\) of \(X\); (iii) for each \(y\) in \(X^*\), \(f\to f+<y,\cdot >\) is a \(\tau\)-continuous operator.