Mosco convergence in locally convex spaces
Sandy L. Zabell · Journal of Functional Analysis · 1992
Let E and F be a pair of locally convex spaces in duality, with σ and τ the weak and Mackey topologies on E. A sequence of functions {fn} on E is said to be Mosco-convergent to another function f0, denoted fn M→ f0, if for every v ϵ E, lim supn → t8 fn(vn) ⩽ f0(v) for some sequence vn t→ v, and lim infn → ∞ fn(vn) ⩾ f0(v) for every sequence vn δ→ v. In this paper it is shown that if F is a separable Fréchet space, {fn: n ⩾ 1} a sequence of proper, lower semicontinuous convex functions, and fn∗ the convex conjugate of fn, then fn M→ f0 ⇒ f∗n M→ f∗n if f0(v) < ∞ for some v ϵ E.