**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 P_{V} : *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*2 = *P*. Specifically,

**Theorem**. The orthogonal projection P_{V}is a self-adjoint linear operator on*H*of norm ≤ 1 with the property P2_{V}= P_{V}. Moreover, any self-adjoint linear operator*E*such that*E*2 =*E*is of the form P_{V}, where*V*is the range of*E*. For every*x*in*H*, P_{V}(*x*) is the unique element*v*of*V*, which minimizes the distance ||*x*−*v*||.

This provides the geometrical interpretation of *P _{V}*(

*x*): it is the best approximation to

*x*by elements of

*V*.

Projections *P _{U}* and

*P*are called mutually orthogonal if

_{V}*P*

_{U}

*P*

_{V}= 0. This is equivalent to

*U*and

*V*being orthogonal as subspaces of

*H*. The sum of the two projections

*P*

_{U}and

*P*

_{V}is a projection only if

*U*and

*V*are orthogonal to each other, and in that case

*P*

_{U}+

*P*

_{V}=

*P*

_{U+V}. The composite

*P*

_{U}

*P*

_{V}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*

_{U}

*P*

_{V}=

*P*

_{U∩V}.

By restricting the codomain to the Hilbert space *V*, the orthogonal projection *P*_{V} gives rise to a projection mapping π: *H* → *V*; it is the adjoint of the inclusion mapping

meaning that

for all *x* ∈ *V* and *y* ∈ *H*.

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 *P*2 = *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*P*_{V}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*V*⊕*W*.

The orthogonal complement satisfies some more elementary results. It is a monotone function in the sense that if *U* ⊂ *V*, 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 *V*_{i} are in addition closed, then .

Read more about this topic: Hilbert Space

### Famous quotes containing the word projections:

“Western man represents himself, on the political or psychological stage, in a spectacular world-theater. Our personality is innately cinematic, light-charged *projections* flickering on the screen of Western consciousness.”

—Camille Paglia (b. 1947)