Hilbert Space - Orthogonal Complements and Projections

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⊥ 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 in H can then be written uniquely as x = v + w, with v in V and w in V⊥. Therefore, H is the internal Hilbert direct sum of V and V⊥.

The linear operator PV : HH 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 P2 = P. Specifically,

Theorem. The orthogonal projection PV is a self-adjoint linear operator on H of norm ≤ 1 with the property P2V = PV. Moreover, any self-adjoint linear operator E such that E2 = E is of the form PV, where V is the range of E. For every x in H, PV(x) is the unique element v of V, which minimizes the distance ||xv||.

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

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

By restricting the codomain to the Hilbert space V, the orthogonal projection PV gives rise to a projection mapping π: HV; it is the adjoint of the inclusion mapping

meaning that

for all xV and yH.

The operator norm of a projection P onto a non-zero closed subspace is equal to one:

Every closed subspace V of a Hilbert space is therefore the image of an operator P of norm one such that P2 = P. The property of possessing appropriate projection operators characterizes Hilbert spaces:

  • A Banach space of dimension higher than 2 is (isometrically) a Hilbert space if and only if, for every closed subspace V, there is an operator PV of norm one whose image is V such that

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:

  • A Banach space X is topologically and linearly isomorphic to a Hilbert space if and only if, to every closed subspace V, there is a closed subspace W such that X is equal to the internal direct sum VW.

The orthogonal complement satisfies some more elementary results. It is a monotone function in the sense that if UV, then 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 . 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: . If the Vi are in addition closed, then .

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