Direct Sums of Hilbert Spaces

Recall, if $M \leq \mathcal{H}$, then every vector $h \in \mathcal{H}$ can be written uniquely as $x + y$ where $x \in M$ and $y \in M^\perp$.
Thus, if $h = x + y$, $k = a + b$, $x,a \in M$, $y,b \in M^\perp$, then

$$\left\langle{ h,k }\right\rangle = \left\langle{ x,a }\right\rangle + \left\langle{ y,b }\right\rangle$$

Goal: Given two Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, find a Hilbert space $\mathcal{L}$ such that $\mathcal{H}, \mathcal{K}\leq \mathcal{L}$ and $\mathcal{H}^\perp = \mathcal{K}$. Denote $\mathcal{L}$ by $\mathcal{H}\oplus \mathcal{K}$.

Construction: Let $\mathcal{H}\oplus \mathcal{K}$ be the set $H \times K$ with coordinate-wise addition and scalar multiplication. Define

$$\left\langle{ (h_1,k_1), (h_2,k_2) }\right\rangle := \left\langle... ...}\right\rangle _\mathcal{H}+ \left\langle{ k_1, k_2 }\right\rangle _\mathcal{K}$$

It is easy to show

• This is an inner product
• $\mathcal{H}\oplus \mathcal{K}$ is complete.
• $\mathcal{H}\cong \{ (h,0) \in \mathcal{H}\oplus \mathcal{K}\}$
• $\mathcal{K}\cong \{ (0,k) \in \mathcal{K}\oplus \mathcal{K}\}$
• $\mathcal{H}^\perp = \mathcal{K}$

Similarly, if $\mathcal{H}_n$, $n \in \mathbb{N}$ is a Hilbert space, let

$$\mathcal{H}:= \{ (h_n)_{n=1}^\infty : h_n \in \mathcal{H}_n, \sum_{n=1}^\infty \Vert h_n\Vert^2 < \infty \}$$

and define, for $g = (g_n)$, $h = (h_n)$ in $\mathcal{H}$,

$$\left\langle{ g,h }\right\rangle = \sum_{n=1}^\infty \left\langle{ g_n, h_n }\right\rangle _{\mathcal{H}_n}$$

This is well-defined, as

$$\sum_{n=1}^\infty \vert\left\langle{ g_n,h_n }\right\rangle \vert \leq \sum_{n=1}^\infty \Vert g_n \Vert \cdot \Vert h_n \Vert$$

$$\leq \left( \sum_{n=1}^\infty \Vert g_n\Vert^2 \right)^\frac12 \left( \sum_{n=1}^\infty \Vert h_n\Vert^2 \right)^\frac12 < \infty$$

This also shows $\left\langle{ g,g }\right\rangle = 0$ implies $g = 0$ for $g \in \mathcal{H}$. Showing that $\mathcal{H}$ is a Hilbert space is routine. $\mathcal{H}$ is denoted $$\bigoplus_{n=1}^\infty \mathcal{H}_n$$.

Finally, given $\mathcal{H}_i, i \in I$, a set of Hilbert space, let $\mathcal{H}$ be the set of functions

$$f: I \rightarrow \bigcup_{i \in I} \mathcal{H}_i$$

such that $f(i) \in \mathcal{H}_i$ for all $i$ and $\sum \{ \Vert f(i)\Vert^2: i \in I \} < \infty$.
Define, for $f,g \in \mathcal{H}$,

$$\left\langle{ f,g }\right\rangle = \sum \{ \left\langle{ f(i), g(i) }\right\rangle _{\mathcal{H}_i} : i \in I \}$$

$\mathcal{H}$ is a Hilbert space and is denoted $\sum \{ H_i : i \in H \}$ or $$\bigoplus_{i \in I} \mathcal{H}_i$$.