Hilbert space
Orthogonal complements and projections

If S is a subset of a Hilbert space H, the set of vectors orthogonal to S is defined by S ⊥ = { x ∈ H ∣ ⟨ x, s ⟩ = 0 for all s ∈ S }. {\displaystyle S^{\perp }=\left\{x\in H\mid \langle x,s\rangle =0\ {\text{ for all }}s\in S\right\}\,.}

The set S is a closed subspace of H (can be proved easily using the linearity and continuity of the inner product) and so forms itself a Hilbert space. If V is a closed subspace of H, then V is called the orthogonal complement of V. In fact, every x ∈ H can then be written uniquely as x = v + w, with v ∈ V and w ∈ V. Therefore, H is the internal Hilbert direct sum of V and V.

The linear operator P : H → H that maps x to v is called the orthogonal projection onto V. There is a natural one-to-one correspondence between the set of all closed subspaces of H and the set of all bounded self-adjoint operators P such that P = P. Specifically,

Theorem — The orthogonal projection P is a self-adjoint linear operator on H of norm ≤ 1 with the property P = P. Moreover, any self-adjoint linear operator E such that E = E is of the form P, where V is the range of E. For every x in H, P (x) is the unique element v of V that minimizes the distance ‖ x − v ‖.

This provides the geometrical interpretation of P (x): it is the best approximation to x by elements of V.

Projections P and P are called mutually orthogonal if P P = 0. This is equivalent to U and V being orthogonal as subspaces of H. The sum of the two projections P and P is a projection only if U and V are orthogonal to each other, and in that case P + P = P. The composite P P is generally not a projection; in fact, the composite is a projection if and only if the two projections commute, and in that case P P = P.

By restricting the codomain to the Hilbert space V, the orthogonal projection P gives rise to a projection mapping π : H → V; it is the adjoint of the inclusion mapping i: V → H, {\displaystyle i:V\to H\,,} meaning that ⟨ i x, y ⟩ H = ⟨ x, π y ⟩ V {\displaystyle \left\langle ix,y\right\rangle _{H}=\left\langle x,\pi y\right\rangle _{V}} for all x ∈ V and y ∈ H.

The operator norm of the orthogonal projection P onto a nonzero closed subspace V is equal to 1: ‖ P V ‖ = sup x ∈ H, x ≠ 0 ‖ P V x ‖ ‖ x ‖ = 1. {\displaystyle \|P_{V}\|=\sup _{x\in H,x\neq 0}{\frac {\|P_{V}x\|}{\|x\|}}=1\,.}

Every closed subspace V of a Hilbert space is therefore the image of an operator P of norm one such that P = P. The property of possessing appropriate projection operators characterizes Hilbert spaces:

While this result characterizes the metric structure of a Hilbert space, the structure of a Hilbert space as a topological vector space can itself be characterized in terms of the presence of complementary subspaces:

The orthogonal complement satisfies some more elementary results. It is a monotone function in the sense that if U ⊂ V, then V ⊆ U with equality holding if and only if V is contained in the closure of U. This result is a special case of the Hahn–Banach theorem. The closure of a subspace can be completely characterized in terms of the orthogonal complement: if V is a subspace of H, then the closure of V is equal to V. The orthogonal complement is thus a Galois connection on the partial order of subspaces of a Hilbert space. In general, the orthogonal complement of a sum of subspaces is the intersection of the orthogonal complements: (∑ i V i) ⊥ = ⋂ i V i ⊥. {\displaystyle {\biggl (}\sum _{i}V_{i}{\biggr)}^{\perp }=\bigcap _{i}V_{i}^{\perp }\,.}

If the V are in addition closed, then ∑ i V i ⊥ | ¯ = (⋂ i V i) ⊥. {\displaystyle {\overline {\sum _{i}V_{i}^{\perp }{\vphantom {\Big |}}}}={\biggl (}\bigcap _{i}V_{i}{\biggr)}^{\perp }\,.}
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