Mosco convergence in locally convex spaces
Given a dual pair \(E\), \(F\) of locally convex spaces, each with its corresponding weak topology \(\sigma\) and Mackey topology \(\tau\), one says that a sequence \(\{f_ n\}\) of functions \(E\to [-\infty,\infty]\) (or \(F\to [-\infty,\infty]\)) is Mosco-convergent to a function \(f_ 0\) if the following conditions are satisfied for each \(v\) in \(E\) (or in \(F\)): (i) \(\liminf f_ n(v_ n)\geq f_ 0(v)\) for every sequence \(\{v_ n\}\) \(\sigma\)-convergent to \(v\), (ii) \(\limsup f_ n(v_ n)\leq f_ 0(v)\) for some sequence \(\{v_ n\}\) \(\tau\)-convergent to \(v\). Mosco showed that if \(E\) is a reflexive Banach space, and \(F\) is its dual, and if each \(f_ n\) is a proper lower-semicontinuous convex function, then \(\{f_ n\}\) is Mosco-convergent to \(f_ 0\) if and only if \(\{f_ n^*\}\) is Mosco-convergent to \(f^*_ 0\), the convex conjugate of \(f_ 0\). After some relevant exposition, the second of these implications is extended to the case in which \(F\) is a separable Fréchet space dual to a locally convex space \(E\). Some related results are discussed.
proper lower-semicontinuous convex function
scientific article; zbMATH DE number 220682
scientific article