If $R : H \to \mathbb{R}$ be a functional with $H$ be a Hilbert space. I want to how convexity of $R$ (If it is convex) is related with weak lower semi continuity??
By definition weak lower semi continuity : Let $(x_n)_n$ a sequence such that $x_n \to x$ in $H$, then $x_n \to x$ weakly in $H$, then $$R(x) \leq \liminf_{n \to \infty} R(x_n).$$ where topology on $H$ is weak topology.
Best Answer
Convexity plus (strong) lower semicontinuity implies weakly lower semicontinuity.
Just argue by using epigraphs: If $R$ is convex and strong lower semicontinuous, then its epigraph is convex and closed, hence weakly closed, hence $R$ is weakly lower semicontinuous.