Weak continuity from evolution-space bounds
= Weak continuity from evolution-space bounds
If $u\in L^\infty(0,T;H)$ and $u_t\in L^q(0,T;V')$ for a dense continuous embedding $V\hookrightarrow H$ and some $q>1$, then $u$ has a representative in $C([0,T];H_{\rm weak})$. Indeed, its pairing with each element of a dense subset of $V$ is absolutely continuous, and the uniform $H$ bound extends continuity to every element of $H$.