Let $G \subset E$ a closed subspace of Banach space $E$. A subspace $L \subset E$ is is said to complement $G$ if $L$ is closed, $G \cap L = \{0\}$ and $G+L = E$. Prove that every closed subset of a Hilbert space has a complement.
Comments: I was able to show the overall result for a finite-dimensional subspace using the kernels of functional $\phi_k : G \longrightarrow \mathbb{R} $, $\phi_k(\sum\alpha_iv_i) = \alpha_k$, where $\{v_1, . . . , v_n\}$ is basis of $G$. I think the idea must be the same to generalize, but I can not.
Hint:
Every closed subspace $G$ of a Hilbert space $H$ has an orthogonal complement $G^\perp$. The Riesz projection theorem says $$H = G \oplus G^\perp$$