Exercise 12.3: Direct Sum Decomposition of Hilbert Space
Suppose that $\mathbb{G}$ is a closed linear subspace. For some $f$, let $g$ to be the projection onto $\mathbb{G}$. Define $g_\parallel := g$ and $g_\perp := f - g$. Then $f = g_\parallel + g_\perp$. We claim that $g_\perp \in \mathbb{G}^\perp$. For suppose not: $\lra{f - g, v} > 0$ for some $v \in \mathbb{G}$. Then $v \in \mathbb{G}$ is positively correlated with the remainder, so we should be able to add a little to reduce the norm. In particular, we can show that $ \norm{f - (g + t v)}^2 < \norm{f - g}^2 $ for a sufficiently small $t > 0$, which contradicts the definition of $g$. Moreover, it is true that the decomposition $f = g_\parallel + g_\perp$ is unique. This proves that $\Hb = \mathbb{G} \oplus \mathbb{G}^\perp$ for every closed linear subspace $\mathbb{G}$.