Let $J$ be a functional over a reflexive Banach space $X$.
Is it true that the sequentially weakly lower semicontinuity is equivalent to convexity and continuity for the functional $J$?
I think the direction convexity+continuity $\Rightarrow$ s.w.l.s.c. is true while the other way is false. This would mean that given a sequentially wekly compact set the functional attains its minimum but only if the functional is convex I have that the minimum is a global one.
Am I correct?