Orthogonal Projection

Let $H$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-hilbert-space-2026a}{complex Hilbert space}, with \reftext{def:inner-product-norm-2026a}{induced norm} $\lVert\cdot\rVert$, which is a \reftext{def:complex-normed-space-2026a}{norm} on $H$ by claim 2 of \ref{lem:inner-product-norm-is-norm-2026a}. Let $P$ be a \reftext{def:bounded-linear-operator-2026a}{bounded linear operator} on $H$, and let the product of operators be as in \reftext{def:operator-operations-2026a}{that definition}.

The operator $P$ is an \textbf{orthogonal projection} if it is \reftext{def:self-adjoint-operator-2026a}{self-adjoint} and satisfies