Let \( H \) denote a Hilbert space. If \( C \) is a closed subspace of \( H \), then
\[ H = C \oplus C^\perp, \]meaning that
\[ C \cap C^\perp = \{0\}, \quad C + C^\perp = H, \]and \( C^\perp \) is closed in \( H \).
Remarks
- In the proof, the Hilbert projection theorem is used to show that \( C + C^\perp = H \).
- This proposition is used for the proof of the Riesz representation theorem .