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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" ?

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Yes. When you consider continuity in general, you are concerned with the topologies in both the domain and the codomain. But when the domain are theh real or complex numbers one tacitly assumes the natural topologies on them.

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