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 {f_n} on E is said to be Mosco-convergent to another function f₀, denoted f_n M → f₀, if for every v ε{lunate} E, lim sup_{n → t8} f_n(v_n) ≤ f₀(v) for some sequence v_n t → v, and lim inf_{n → ∞} f_n(v_n) ≥ f₀(v) for every sequence v_n δ → v. In this paper it is shown that if F is a separable Fréchet space, {f_n: n ≥ 1} a sequence of proper, lower semicontinuous convex functions, and f_n^* the convex conjugate of f_n, then f_n M → f₀ ⇒ f^*_n M → f^*_n if f₀(v) < ∞ for some v ε{lunate} E.