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?


Any continuous function on a finite-dimensional Banach space is sequentially weakly lower semicontinuous, but not convex.

The indicator function of a convex and closed set is sequentially weakly lower semicontinuous, but not continuous.


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