orthogonal decomposition theorem

Theorem - Let X be an Hilbert spaceMathworldPlanetmath and AX a closed subspace. Then the orthogonal complementMathworldPlanetmathPlanetmath (http://planetmath.org/Complimentary) of A, denoted A, is a topological complement of A. That means A is closed and


Proof :

  • A is closed :

    This follows easily from the continuity of the inner productMathworldPlanetmath. If a sequenceMathworldPlanetmath (xn) of elements in A converges to an element x0X, then

    x0,a=limnxn,a=limnxn,a=0for everyaA

    which implies that x0A.

  • X=AA :

    Since X is completePlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Complete) and A is closed, A is a subspacePlanetmathPlanetmath of X. Therefore, for every xX, there exists a best approximation of x in A, which we denote by a0A, that satisfies x-a0A (see this entry (http://planetmath.org/BestApproximationInInnerProductSpaces)).

    This allows one to write x as a sum of elements in A and A


    which proves that


    Moreover, it is easy to see that


    since if yAA then y,y=0, which means y=0.

    We conclude that X=AA.

Title orthogonal decomposition theorem
Canonical name OrthogonalDecompositionTheorem
Date of creation 2013-03-22 17:32:34
Last modified on 2013-03-22 17:32:34
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Theorem
Classification msc 46A99
Synonym closed subspaces of Hilbert spaces are complemented