Hilbert Space Direct Sum is Hilbert Space

Theorem
Let $\left({H_i}\right)_{i \in I}$ be a $I$-indexed family of Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$.

Let $H = \displaystyle \bigoplus_{i \in I} H_i$ be their Hilbert space direct sum.

Then $H$ is a Hilbert space.

$H$ is a Vector Space
From the definition of Hilbert space direct sum, we see that $H$ is a nonempty subset of a vector space (namely, the direct sum of the $H_i$ as vector spaces).

From the Vector Subspace Test it follows that it is to be shown that:


 * $(1): \qquad \forall h_1, h_2 \in H: \displaystyle \sum \left\{{ \left\Vert{ \left({h_1 + h_2}\right) \left({i}\right) }\right\Vert^2_{H_i}: i \in I }\right\} < \infty$
 * $(2): \qquad \forall \lambda \in \Bbb F, h \in H: \displaystyle \sum \left\{{ \left\Vert{ \left({\lambda h}\right) \left({i}\right) }\right\Vert^2_{H_i}: i \in I }\right\} < \infty$

Considering $(1)$, have the following:

For $(2)$, observe that:

Thus, by the Vector Subspace Test, $H$ is a vector space.