Let $C_{w} ( [0,M], H)$ be the space of weakly continuous functions from $[0,M]$ into Hilbert space $H$. Then does " $u \in C_w( [0,M], H)$ " mean $$\text{for any}\; \{ x_n \} \subset [0,M] \;\text{satisfying} \; \lim_{n \to \infty} x_n = x, \\ u(x_n) \longrightarrow u(x ) \;weakly \;\;\; \text{in} \;\;\;H \; ?$$ Is this the definition of "weakly continuous" ?