Give an example of a sequence of nonzero vectors phinn1infin

Give an example of a sequence of non-zero vectors (phi_n)_n=1^infinity in a Hilbert space H that is not an orthonormal basis but satisfies ||x||^2 = sigma_n = 1^infinity |(x, phi_n)|^2 for all x H.

Solution

Let X be an inner product space and let M be a subset. Then M is called total if span(M) = X.

  A total orthonormal set in an inner product space is called an orthonormal basis.

define an orthonormal basis as a maximal orthonormal set, e.g., an orthonormal set which is not properly contained in any other orthonormal set. The two definitions are equivalent (Hunter and Nachtergaele’s theorem).

Every Hilbert space contains a total orthonormal set. (Furthermore, all total orthonormal sets in a Hilbert space H {0} have the same cardinality, which is known as the Hilbert dimension).

The proof requires the axiom of choice or Zorn’s lemma.

Proof. Let M be the set of all orthonormal subsets of H. Pick any nonzero element x. Then one such orthonormal subset is {y} where y = x/||x||, so M . Set inclusion defines a partial ordering on M, and every chain C M has an upper bound, namely the union of all subsets of X which are elements of C. By Zorn’s lemma, M has a maximal element F. We claim F is total in H. If not, then there is a nonzero z H such that z F, but then we could have normalized and added z to F to create a larger orthonormal set, which is impossible by the maximality of F.