Let \( H \) denote a Hilbert space . For every vector \( x \in H \) and every convex, closed set \( C \subseteq H \), there exists a unique vector \( m \in C \) such that

\[ \|x - m\| = \inf_{c \in C} \|x - c\|. \]

Additionally, if \( C \) is a vector subspace of \( H \), then \( m \) is the unique vector in \( C \) such that \( x - m \) is orthogonal to \( C \).

Remarks
  • The Hilbert projection theorem implies, that if \(C\) is closed subspace, then \[ H = C \oplus C^\perp \] (see (0x68e89b9a) ).