Locally bounded adapted process

Question

The definition for Locally bounded is utilized in the following link. Which states that for the process $\Phi:[0,T]\times\Omega\to H$ ($H$ is a Hilbert space), $\Phi$ is locally bounded provided $$ \sup_{\Omega} \big\Vert \Phi_t(\omega)\big\Vert_{H}<\infty, \ \text{ for all }\ t$$

But I could'nt find any reference/books stating such definition. I'm new in this stuffs, any help is highly appreciated.

How can we prove that $\langle\int_0^t\Phi_s{\rm d}W_s,x\rangle_H=\sum_{n\in\mathbb N}\int_0^t\langle\sqrt{λ_n}\Phi_se_n,x\rangle_H{\rm d}B_s^{(n)}$?