Hilbert Space Direct Sum is Hilbert Space

Theorem
Let $\sequence {H_i}_{i \mathop \in I}$ be a $I$-indexed family of Hilbert spaces over $\Bbb F \in \set {\R, \C}$.

Let $H = \ds \bigoplus_{i \mathop \in I} H_i$ 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 Two-Step Vector Subspace Test it follows that it is to be shown that:
 * $(1): \quad \forall h_1, h_2 \in H: \ds \sum \set {\norm {\map {\paren {h_1 + h_2} } i}^2_{H_i}: i \in I} < \infty$
 * $(2): \quad \forall \lambda \in \Bbb F, h \in H: \ds \sum \set {\norm {\map {\paren {\lambda h} } i}^2_{H_i}: i \in I } < \infty$

Considering $(1)$, have the following:

For $(2)$, observe that:

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

$\innerprod \cdot \cdot$ is an Inner Product
It suffices to check well-definedness of $\innerprod \cdot \cdot$, and subsequently the five properties of an inner product.

Well-definedness
It is necessary to verify that for $g, h \in H$, in fact $\innerprod g h \in \Bbb F$.

That is, it is required to show that $\innerprod g h = \ds \sum \set {\innerprod {\map g i} {\map h i}_{H_i}: i \in I}$ converges in $\Bbb F$.

Absolutely Convergent Generalized Sum Converges applies to the Banach space $\Bbb F$ and the $I$-indexed subset $\innerprod {\map g i} {\map h i}_{H_i}$ of $\Bbb F$.

Hence it will suffice to show that $\ds \sum \set {\size {\innerprod {\map g i} {\map h i}_{H_i} }: i \in I}$ converges in $\R$.

For brevity, denote already $\norm h^2$ for the expression $\ds \sum \set {\norm {\map h i}_{H_i}^2: i \in I}$.

Define $g' \in H$ by $\map {g'} i = \begin{cases} \map g i & \text{if } \norm {\map g i}_{H_i} \ge \norm {\map h i}_{H_i} \\ \mathbf 0_{H_i} & \text{otherwise} \end{cases}$.

Note that $\norm {g'}^2 \le \norm g^2$ by Generalized Sum Preserves Inequality.

Similarly, let $h' \in H$ be defined by $\map {h'} i = \begin{cases} \map h i & \text{if } \norm {\map h i}_{H_i} > \norm {\map g i}_{H_i} \\ \mathbf 0_{H_i} & \text{otherwise} \end{cases}$

By Generalized Sum Preserves Inequality again, we have:
 * $\norm {h'}^2 \le \norm h^2$

More significantly, by construction of $g', h'$:
 * $(3): \quad \norm {\map g i}_{H_i}, \norm {\map h i}_{H_i} \le \norm {\map {\paren {g' + h'} } i}_{H_i}$

As $H$ is a vector space, $g' + h' \in H$, and we can establish:

Hence, for all $g, h \in H$, $\innerprod g h \in \Bbb F$ by the comment on Generalized Sum Preserves Inequality.

Conclusion
$\innerprod \cdot \cdot$ is checked to be a mapping from $H \times H$ to $\Bbb F$, satisfying the five conditions for an inner product.

That is, $\innerprod \cdot \cdot$ is an inner product on $H$.

$H$ is complete
A Hilbert space is a complete inner product space.

Thus, it remains to verify that $H$ is complete.

Suppose $\sequence {h_n}_{n \mathop \in \N}$ is a Cauchy sequence in $H$.

Let $N \in \N$ such that $n, m \ge N \implies \size {h_n - h_m} < \epsilon$.

That is:
 * $\ds \sum \set {\norm {\map {\paren {h_n - h_m} } i}_{H_i}^2: i \in I} < \epsilon^2$.

From Generalized Sum is Monotone obtain that, for all $i \in I$:
 * $\norm {\map {\paren {h_n - h_m} } i}_{H_i}^2 < \epsilon^2$

It follows that $\sequence {\map {h_n} i}_{n \mathop \in \N}$ is a Cauchy sequence in $H_i$.

$H_i$ is a Hilbert space, hence complete.

Hence there is some $h_i \in H_i$ such that $\ds \lim_{n \mathop \to \infty} \map {h_n} i = h_i$.

Now let $h$ be defined by $\map h i = h_i$; it is the only candidate for $\ds \lim_{n \mathop \to \infty} h_n = h$.

It remains to be shown that indeed $\ds \lim_{n \mathop \to \infty} h_n = h$, and then that $h \in H$.

So, for any $\epsilon > 0$, an $N \in \N$ is to be found such that for all $n \ge N$:
 * $(4): \quad \ds \sum \set {\norm {\map {\paren {h_n - h} } i}_{H_i}^2: i \in I} < \epsilon^2$

To this end, let $N \in \N$ be such that:
 * $(5): \quad n, m \ge N \implies \size {h_n - h_m}^2 < \frac {\epsilon^2} 2$

Such an $N$ exists as $\sequence {h_n}_{n \mathop \in \N}$ is a Cauchy sequence.

Now observe that, for any finite $G \subseteq I$ and $n \ge N$:

The last inequality follows from $(5)$, as $m \ge N$ eventually when $m \to \infty$.

From Bounded Generalized Sum Converges, it now follows that:
 * $\ds \sum \set {\norm {\map {\paren {h_n - h} } i}_{H_i}^2 : i \in I} \le \frac {\epsilon^2} 2 < \epsilon^2$

This precisely establishes the inequality desired in $(4)$ for $n \ge N$.

It follows that $\ds \lim_{n \mathop \to \infty} h_n = h$.

To show that $h \in H$, it is to be shown that $\norm h^2 < \infty$.

This is done as follows:

The latter sum converges by Square-Summable Indexed Sets Closed Under Addition, yielding convergence of $\norm h^2$.

Therefore, $h \in H$.

That is, every Cauchy sequence in $H$ converges to a limit in $H$, hence $H$ is complete.

By definition, $H$ is a Hilbert space.