every Hilbert space has an orthonormal basis

Theorem - Every Hilbert spaceMathworldPlanetmath H≠{0} has an orthonormal basisMathworldPlanetmath.

Proof : As could be expected, the proof makes use of Zorn’s Lemma. Let π’ͺ be the set of all orthonormal sets of H. It is clear that π’ͺ is non-empty since the set {x} is in π’ͺ, where x is an element of H such that βˆ₯xβˆ₯=1.

The elements of π’ͺ can be ordered by inclusion, and each chain π’ž in π’ͺ has an upper bound, given by the union of all elements of π’ž. Thus, Zorn’s Lemma assures the existence of a maximal element B in π’ͺ. We claim that B is an orthonormal basis of H.

It is clear that B is an orthonormal set, as it belongs to π’ͺ. It remains to see that the linear span of B is dense in H.

Let span⁒BΒ― denote the closure of the span of B. Suppose span⁒BΒ―β‰ H. By the orthogonal decomposition theorem we know that


Thus, we conclude that (span⁒BΒ―)βŸ‚β‰ {0}, i.e. there are elements which are orthogonalMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/OrthogonalVectors) to span⁒BΒ―. This contradicts the maximality of B since, by picking an element y∈(span⁒BΒ―)βŸ‚ with βˆ₯yβˆ₯=1, Bβˆͺ{y} would belong belong to π’ͺ and would be greater than B.

Hence, span⁒BΒ―=H, and this finishes the proof. β–‘

Title every Hilbert space has an orthonormal basis
Canonical name EveryHilbertSpaceHasAnOrthonormalBasis
Date of creation 2013-03-22 17:56:05
Last modified on 2013-03-22 17:56:05
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Theorem
Classification msc 46C05