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.
Proof
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$H$ 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:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left\Vert{ \left({h_1 + h_2}\right) \left({i}\right) }\right\Vert^2_{H_i}\) | \(=\) | \(\displaystyle \left\Vert{ h_1 \left({i}\right) + h_2 \left({i}\right) }\right\Vert^2_{H_i}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\le\) | \(\displaystyle \left({ \left\Vert{h_1 \left({i}\right)}\right\Vert_{H_i} + \left\Vert{h_2 \left({i}\right)}\right\Vert_{H_i} }\right)^2\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Triangle inequality for $\left\Vert{\cdot}\right\Vert_{H_i}$ | ||
| \(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \sum \left\{ { \left\Vert{ \left({h_1 + h_2}\right) \left({i}\right) }\right\Vert^2_{H_i}: i \in I }\right\}\) | \(\le\) | \(\displaystyle \sum \left\{ { \left({ \left\Vert{h_1 \left({i}\right)}\right\Vert_{H_i} + \left\Vert{h_2 \left({i}\right)}\right\Vert_{H_i} }\right)^2: i \in I }\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Generalized Sum Preserves Inequality | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(<\) | \(\displaystyle \infty\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | $h_1, h_2 \in H$, Square-Summable Indexed Sets Closed Under Addition |
For $(2)$, observe that:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left\Vert{ \left({\lambda h}\right) \left({i}\right) }\right\Vert^2_{H_i}\) | \(=\) | \(\displaystyle \left\vert{\lambda}\right\vert^2 \left\Vert{h \left({i}\right)}\right\Vert^2_{H_i}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | $\left\Vert{\cdot}\right\Vert_{H_i}$ is a norm | ||
| \(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \sum \left\{ { \left\Vert{ \left({\lambda h}\right) \left({i}\right) }\right\Vert^2_{H_i}: i \in I }\right\}\) | \(\le\) | \(\displaystyle \left\vert{\lambda}\right\vert^2 \sum \left\{ { \left\Vert{h \left({i}\right)}\right\Vert^2_{H_i}: i \in I }\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Generalized Sum is Linear | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(<\) | \(\displaystyle \infty\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | As $h \in H$ |
Thus, by the Vector Subspace Test, $H$ is a vector space.
$\Box$
$\left\langle{\cdot, \cdot}\right\rangle$ is an Inner Product
It suffices to check well-definedness of $\left\langle{\cdot, \cdot}\right\rangle$, and subsequently the five properties of an inner product.
Well-definedness
It is necessary to verify that for $g, h \in H$, in fact $\left\langle{g, h}\right\rangle \in \Bbb F$.
That is, it is required to show that $\left\langle{g, h}\right\rangle = \displaystyle \sum \left\{{ \left\langle{ g \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}: i \in I}\right\}$ converges in $\Bbb F$.
Absolutely Convergent Generalized Sum Converges applies to the Banach space $\Bbb F$ and the $I$-indexed subset $\left\langle{ g \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}$ of $\Bbb F$.
Hence it will suffice to show that $\displaystyle \sum \left\{{ \left\vert{ \left\langle{ g \left({i}\right), h \left({i}\right) }\right\rangle_{H_i} }\right\vert: i \in I}\right\}$ converges in $\R$.
For brevity, denote already $\left\Vert{h}\right\Vert^2$ for the expression $\displaystyle \sum \left\{{\left\Vert{h \left({i}\right)}\right\Vert_{H_i}^2: i \in I}\right\}$.
Define $g' \in H$ by $g' \left({i}\right) = \begin{cases}
g \left({i}\right) & \text{if } \left\Vert{g \left({i}\right)}\right\Vert_{H_i} \ge \left\Vert{h \left({i}\right)}\right\Vert_{H_i} \\
\mathbf{0}_{H_i} & \text{otherwise}
\end{cases}$.
Note that $\left\Vert{g'}\right\Vert^2 \le \left\Vert{g}\right\Vert^2$ by Generalized Sum Preserves Inequality.
Similarly, let $h' \in H$ be defined by $h' \left({i}\right) = \begin{cases} h \left({i}\right) & \text{if } \left\Vert{h \left({i}\right)}\right\Vert_{H_i} \ge \left\Vert{g \left({i}\right)}\right\Vert_{H_i} \\ \mathbf{0}_{H_i} & \text{otherwise} \end{cases}$.
By Generalized Sum Preserves Inequality again, have $\left\Vert{h'}\right\Vert^2 \le \left\Vert{h}\right\Vert^2$.
More significantly, by construction of $g', h'$:
- $(3): \qquad \left\Vert{g \left({i}\right)}\right\Vert_{H_i}, \left\Vert{h \left({i}\right)}\right\Vert_{H_i} \le \left\Vert{\left({g' + h'}\right) \left({i}\right)}\right\Vert_{H_i}$
As $H$ is a vector space, $g' + h' \in H$, and we can establish:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left\vert{ \left\langle{ g \left({i}\right), h \left({i}\right) }\right\rangle_{H_i} }\right\vert\) | \(\le\) | \(\displaystyle \left\Vert{g \left({i}\right)}\right\Vert_{H_i} \left\Vert{h \left({i}\right)}\right\Vert_{H_i}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Cauchy-Schwarz Inequality | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\le\) | \(\displaystyle \left\Vert{\left({g' + h'}\right) \left({i}\right)}\right\Vert_{H_i}^2\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Equation $(1)$ | ||
| \(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \sum \left\{ { \left\vert{ \left\langle{ g \left({i}\right), h \left({i}\right) }\right\rangle_{H_i} }\right\vert: i \in I}\right\}\) | \(\le\) | \(\displaystyle \sum \left\{ { \left\Vert{\left({g' + h'}\right) \left({i}\right)}\right\Vert_{H_i}^2: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Generalized Sum Preserves Inequality |
Hence, for all $g, h \in H$, $\left\langle{g, h}\right\rangle \in \Bbb F$ by the comment on Generalized Sum Preserves Inequality.
$\Box$
Property 1: $\left\langle{g, h}\right\rangle = \overline{\left\langle{h, g}\right\rangle}$
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left\langle{g, h}\right\rangle\) | \(=\) | \(\displaystyle \sum \left\{ { \left\langle{ g \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of $\left\langle{\cdot, \cdot}\right\rangle$ | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \sum \left\{ { \overline{\left\langle{ h \left({i}\right), g \left({i}\right) }\right\rangle_{H_i} }: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | $\left\langle{\cdot, \cdot}\right\rangle_{H_i}$ is an inner product | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \overline{ \sum \left\{ { \left\langle{ h \left({i}\right), g \left({i}\right) }\right\rangle_{H_i}: i \in I}\right\} }\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Corollary to Generalized Sum of Complexes | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \overline{ \left\langle{h, g}\right\rangle }\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of $\left\langle{\cdot, \cdot}\right\rangle$ |
$\Box$
Property 2: $\left\langle{\lambda g, h}\right\rangle = \lambda \left\langle{g, h}\right\rangle$
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left\langle{\lambda g, h}\right\rangle\) | \(=\) | \(\displaystyle \sum \left\{ { \left\langle{\left({\lambda g}\right) \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of $\left\langle{\cdot, \cdot}\right\rangle$ | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \sum \left\{ { \lambda \left\langle{ g \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | $\left\langle{\cdot, \cdot}\right\rangle_{H_i}$ is an inner product | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \lambda \sum \left\{ { \left\langle{g \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Generalized Sum is Linear | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \lambda \left\langle{h, g}\right\rangle\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of $\left\langle{\cdot, \cdot}\right\rangle$ |
$\Box$
Property 3: $\left\langle{g_1 + g_2, h}\right\rangle = \left\langle{g_1, h}\right\rangle + \left\langle{g_2, h}\right\rangle$
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left\langle{g_1 + g_2, h}\right\rangle\) | \(=\) | \(\displaystyle \sum \left\{ { \left\langle{\left({g_1 + g_2}\right) \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of $\left\langle{\cdot, \cdot}\right\rangle$ | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \sum \left\{ { \left\langle{g_1 \left({i}\right), h \left({i}\right)}\right\rangle_{H_i} + \left\langle{g_2 \left({i}\right), h \left({i}\right)}\right\rangle_{H_i}: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | $\left\langle{\cdot, \cdot}\right\rangle_{H_i}$ is an inner product | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \sum \left\{ { \left\langle{ g_1 \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}: i \in I}\right\} + \sum \left\{ { \left\langle{ g_2 \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Generalized Sum is Linear | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \left\langle{g_1, h}\right\rangle + \left\langle{g_2, h}\right\rangle\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of $\left\langle{\cdot, \cdot}\right\rangle$ |
$\Box$
Property 4: $\left\langle{h, h}\right\rangle \ge 0$
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left\langle{h, h}\right\rangle\) | \(=\) | \(\displaystyle \sum \left\{ { \left\langle{ h \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of $\left\langle{\cdot, \cdot}\right\rangle$ | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\ge\) | \(\displaystyle \sum \left\{ { 0: i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | $\left\langle{\cdot, \cdot}\right\rangle_{H_i}$ is an inner product, Generalized Sum Preserves Inequality | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle 0\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
$\Box$
Property 5: $\left\langle{h, h}\right\rangle = 0$ iff $h = \mathbf{0}_H$
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left\langle{h, h}\right\rangle\) | \(=\) | \(\displaystyle 0\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \iff \forall i \in I:\) | \(\displaystyle \) | \(\displaystyle \left\langle{ h \left({i}\right), h \left({i}\right) }\right\rangle_{H_i}\) | \(=\) | \(\displaystyle 0\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | $\left\langle{\cdot, \cdot}\right\rangle_{H_i}$ is an inner product, Generalized Sum is Monotone | ||
| \(\displaystyle \) | \(\displaystyle \iff \forall i \in I:\) | \(\displaystyle \) | \(\displaystyle h \left({i}\right)\) | \(=\) | \(\displaystyle \mathbf{0}_{H_i}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | $\left\langle{\cdot, \cdot}\right\rangle_{H_i}$ is an inner product | ||
| \(\displaystyle \) | \(\displaystyle \iff\) | \(\displaystyle \) | \(\displaystyle h\) | \(=\) | \(\displaystyle \mathbf{0}_H\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
$\Box$
Conclusion
$\left\langle{\cdot, \cdot}\right\rangle$ is checked to be a mapping from $H \times H$ to $\Bbb F$, satisfying the five conditions for an inner product.
That is, $\left\langle{\cdot, \cdot}\right\rangle$ is an inner product on $H$.
$\Box$
$H$ is complete
A Hilbert space is a complete inner product space.
Thus, it remains to verify that $H$ is complete.
Suppose $\left({h_n}\right)_{n\in\N}$ is a Cauchy sequence in $H$.
Let $N \in \N$ such that $n, m \ge N \implies \left\Vert{h_n - h_m}\right\Vert < \epsilon$.
That is, $\displaystyle \sum \left\{{ \left\Vert{ \left({h_n - h_m}\right) \left({i}\right) }\right\Vert_{H_i}^2: i \in I}\right\} < \epsilon^2$.
From Generalized Sum is Monotone obtain that, for all $i \in I$:
- $\left\Vert{ \left({h_n - h_m}\right) \left({i}\right) }\right\Vert_{H_i}^2 < \epsilon^2$
It follows that $\left({h_n \left({i}\right) }\right)_{n\in\N}$ 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 $\displaystyle \lim_{n\to\infty} h_n \left({i}\right) = h_i$.
Now let $h$ be defined by $h \left({i}\right) = h_i$; it is the only candidate for $\displaystyle \lim_{n\to\infty} h_n = h$.
It remains to be shown that indeed $\displaystyle \lim_{n\to\infty} h_n = h$, and then that $h \in H$.
So, for any $\epsilon > 0$, an $N \in \N$ is to be found such that for all $n \ge N$:
- $(4): \qquad \displaystyle \sum \left\{{ \left\Vert{ \left({h_n - h}\right) \left({i}\right) }\right\Vert_{H_i}^2: i \in I}\right\} < \epsilon^2$
To this end, let $N \in \N$ be such that:
- $(5): \qquad n, m \ge N \implies \left\Vert{h_n - h_m}\right\Vert^2 < \frac {\epsilon^2} 2$
Such an $N$ exists as $\left({h_n}\right)_{n\in\N}$ is a Cauchy sequence.
Now observe that, for any finite $G \subseteq I$ and $n \ge N$:
| \(\displaystyle \) | \(\displaystyle \forall i \in I:\) | \(\displaystyle \) | \(\displaystyle \left\Vert{ \left({h_n - h}\right) \left({i}\right) }\right\Vert_{H_i}^2\) | \(=\) | \(\displaystyle \lim_{m \to \infty} \left\Vert{ \left({h_n - h_m}\right) \left({i}\right) }\right\Vert_{H_i}^2\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of $h \left({i}\right)$ | ||
| \(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \sum_{i \in G} \left\Vert{ \left({h_n - h}\right) \left({i}\right) }\right\Vert_{H_i}^2\) | \(=\) | \(\displaystyle \sum_{i \in G} \ \lim_{m \to \infty} \left\Vert{ \left({h_n - h_m}\right) \left({i}\right) }\right\Vert_{H_i}^2\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \lim_{m \to \infty} \ \sum_{i \in G} \left\Vert{ \left({h_n - h_m}\right) \left({i}\right) }\right\Vert_{H_i}^2\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Sum Rule for Limits | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\le\) | \(\displaystyle \lim_{m \to \infty} \ \sum \left\{ {\left\Vert{ \left({h_n - h_m}\right) \left({i}\right) }\right\Vert_{H_i}^2 : i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Generalized Sum is Monotone | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \lim_{m \to \infty} \left\Vert{h_n - h_m}\right\Vert^2\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of $\left\Vert{\cdot}\right\Vert$ | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\le\) | \(\displaystyle \frac {\epsilon^2} 2\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Upper and Lower Bounds of Sequences |
The last inequality follows from $(5)$, as $m \ge N$ eventually when $m \to \infty$.
From Bounded Generalized Sum Converges, it now follows that:
- $\displaystyle \sum \left\{ {\left\Vert{ \left({h_n - h}\right) \left({i}\right) }\right\Vert_{H_i}^2 : i \in I}\right\} \le \frac {\epsilon^2} 2 < \epsilon^2$
This precisely establishes the inequality desired in $(4)$ for $n \ge N$.
It follows that $\displaystyle \lim_{n \to \infty} h_n = h$.
To show that $h \in H$, it is to be shown that $\left\Vert{h}\right\Vert^2 < \infty$.
This is done as follows:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left\Vert{h}\right\Vert^2\) | \(=\) | \(\displaystyle \sum \left\{ {\left\Vert{ h \left({i}\right) }\right\Vert_{H_i}^2 : i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Definition of $\left\Vert{\cdot}\right\Vert$ | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\le\) | \(\displaystyle \sum \left\{ {\left({ \left\Vert{ \left({h - h_n}\right) \left({i}\right) }\right\Vert_{H_i} + \left\Vert{ h_n \left({i}\right) }\right\Vert_{H_i} }\right)^2 : i \in I}\right\}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Triangle Inequality for all $\left\Vert{\cdot}\right\Vert_{H_i}$, Generalized Sum Preserves Inequality |
The latter sum converges by Square-Summable Indexed Sets Closed Under Addition, yielding convergence of $\left\Vert{h}\right\Vert^2$.
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.
$\blacksquare$
Sources
- John B. Conway: A Course in Functional Analysis (1990)... (previous)... (next) $\S I.6$
