Agreement with the concrete L-two projection

Assume the Axiom of Choice, and let $H$ be complex $L^2(\mu )$ or a closed linear subspace of it with $M\subseteq H$ a closed linear subspace (Closed l two subspaces have orthogonal projections). That published theorem constructs a linear contraction $P_M:H\to H$ with $P_{M}f\in M$ and $f-P_{M}f\perp M$ for every $f$, together with $H=M\oplus M^{\perp }$.

The Hilbert orthogonal projection of The Hilbert orthogonal projection onto a closed subspace is characterised by the same two properties: its value at $f$ is the unique $M$-component of the unique orthogonal decomposition $f=P_{M}f+(f-P_{M}f)$ with $f-P_{M}f\in M^{\perp }$. Since the concrete construction supplies a vector of $M$ whose residual is orthogonal to $M$, and the orthogonal decomposition is unique, the concrete map and the Hilbert projection agree wherever both are defined. The concrete theorem is used only to identify the two constructions: it is not a supplier for the existence, linearity, contractivity or self-adjointness of the Hilbert projection, which are proved on this page from the abstract decomposition.

The choice cost of the identification is the concrete theorem's: it assumes AC, and AC implies the Axiom of Countable Choice, under which the abstract projection exists (The Axiom of Choice). No stronger principle is claimed, and the identification is orientation only, not a load-bearing prerequisite of any theorem on this page.