I was wondering about how to prove that
$$ \mathcal{H}^{(n)}:= \mathcal{H} \oplus \mathcal{H} \oplus \dots \oplus \mathcal{H} $$
is still a Hilbert space with norm and inner product $$\|(x_1,x_2, \dots, x_n)\|^2 = \|x_1\|^2 + \dots+\|x_n\|^2 \hspace{6mm} \langle(x_1,x_2, \dots, x_n),(y_1,y_2, \dots, y_n)\rangle = \sum_{i=1}^n \langle x_i,y_i\rangle_{\mathcal{H}} $$
How to prove that such a space is complete having that $\mathcal{H}$ is Hilbert?