Let V be a seperable, reflexive Banach space with V* as its dual.

Let A : V -> V* be a bounded mapping. Take sequence (u_k) such that u_k -> u in V (strongly).

Let A(u_k) -> f (weakly) for some f in V*.

If we can show <A(u), u - v> <= lim inf<A(u_k), u_k - v> = <f, u - v> for any v in V,

how does it follow from this that A(u) = f?

Thanks