MOSCO CONVERGENCE IN LOCALLY CONVEX-SPACES

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{lunate} 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{lunate} E. © 1992.