Weakly continuous path

Let $I$ be a real interval and $X$ a topological vector space. A map $u:I\to X$ is weakly continuous, written $u\in C_w(I;X)$ , if it is continuous for the weak topology on $X$ . Equivalently, $t\mapsto\ell(u(t))$ is continuous for every $\ell$ in the continuous dual $X'$ .

Hilbert spaces

For a Hilbert space this means continuity of $t\mapsto\langle u(t),v\rangle$ for every fixed $v$ . It specifies a value at every time, unlike an equivalence class in a time Lebesgue space. Weak continuity need not imply norm continuity.