quality:correction
Orthogonal projections

When the vector space W {\displaystyle W} has an inner product and is complete (is a Hilbert space) the concept of orthogonality can be used. An orthogonal projection is a projection for which the range U {\displaystyle U} and the kernel V {\displaystyle V} are orthogonal subspaces. Thus, for every x {\displaystyle \mathbf {x} } and y {\displaystyle \mathbf {y} } in W {\displaystyle W}, ⟨ P x, (y − P y) ⟩ = ⟨ (x − P x), P y ⟩ = 0 {\displaystyle \langle P\mathbf {x},(\mathbf {y} -P\mathbf {y})\rangle =\langle (\mathbf {x} -P\mathbf {x}),P\mathbf {y} \rangle =0}. Equivalently: ⟨ x, P y ⟩ = ⟨ P x, P y ⟩ = ⟨ P x, y ⟩. {\displaystyle \langle \mathbf {x},P\mathbf {y} \rangle =\langle P\mathbf {x},P\mathbf {y} \rangle =\langle P\mathbf {x},\mathbf {y} \rangle.}

A projection is orthogonal if and only if it is self-adjoint. Using the self-adjoint and idempotent properties of P {\displaystyle P}, for any x {\displaystyle \mathbf {x} } and y {\displaystyle \mathbf {y} } in W {\displaystyle W} we have P x ∈ U {\displaystyle P\mathbf {x} \in U}, y − P y ∈ V {\displaystyle \mathbf {y} -P\mathbf {y} \in V}, and ⟨ P x, y − P y ⟩ = ⟨ x, (P − P 2) y ⟩ = 0 {\displaystyle \langle P\mathbf {x},\mathbf {y} -P\mathbf {y} \rangle =\langle \mathbf {x},\left(P-P^{2}\right)\mathbf {y} \rangle =0} where ⟨ ⋅, ⋅ ⟩ {\displaystyle \langle \cdot,\cdot \rangle } is the inner product associated with W {\displaystyle W}. Therefore, P {\displaystyle P} and I − P {\displaystyle I-P} are orthogonal projections. The other direction, namely that if P {\displaystyle P} is orthogonal then it is self-adjoint, follows from the implication from ⟨ (x − P x), P y ⟩ = ⟨ P x, (y − P y) ⟩ = 0 {\displaystyle \langle (\mathbf {x} -P\mathbf {x}),P\mathbf {y} \rangle =\langle P\mathbf {x},(\mathbf {y} -P\mathbf {y})\rangle =0} to ⟨ x, P y ⟩ = ⟨ P x, P y ⟩ = ⟨ P x, y ⟩ = ⟨ x, P ∗ y ⟩ {\displaystyle \langle \mathbf {x},P\mathbf {y} \rangle =\langle P\mathbf {x},P\mathbf {y} \rangle =\langle P\mathbf {x},\mathbf {y} \rangle =\langle \mathbf {x},P^{*}\mathbf {y} \rangle } for every x {\displaystyle x} and y {\displaystyle y} in W {\displaystyle W}; thus P = P ∗ {\displaystyle P=P^{*}}.

The existence of an orthogonal projection onto a closed subspace follows from the Hilbert projection theorem.