Let $V,H,V^*$ be a Gelfand-Triple, $\phi\colon V \to \mathbb{R}$ convex, lower semicontinuous and proper. There exists a so called Moreau-Enveloppe $\phi_j$, which is Gateâux-differentialable. It's derivative $\phi_j'\colon V \to V^*$ is demicontinuous and maps bounded sets to bounded sets in $V^*$. Now my question: Is the mapping $tilde{\phi_j} \colon L^2(0,T,V) \to L^2(0,T,V^*)$ defined by $\tilde{\phi_j}(u)(t) \colon = \phi_j(u(t))$ well-defined? And are in $L^2(0,T,V)$-bounded sets mapped to bounded sets in $L^2(0,T,V^*)$?