@MISC{Farah908orthonormalbases, author = {Ilijas Farah}, title = {ORTHONORMAL BASES OF HILBERT SPACES}, year = {908} }

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Abstract

Assume H is a Hilbert space and K is a dense linear (not necessarily closed) subspace. The question whether K necessarily contains an orthonormal basis for H even when H is nonseparable was mentioned by Bruce Blackadar in an informal conversation during the Canadian Mathematical Society meeting in Ottawa in December 2008 and this note provides a negative answer. Note that the Gram– Schmidt process gives a positive answer when H is separable. I will use ℵ1 to denote both the first uncountable ordinal and the first uncountable cardinal and I will use c = 2 ℵ0 to denote both the cardinality of the continuum and the least ordinal of this cardinality. All bases are orthonormal. For cardinals λ < θ consider ℓ 2 (λ) as a subspace of ℓ 2 (θ) consisting of vectors supported on the first λ coordinates. Let pλ denote the projection of ℓ 2 (θ) to ℓ 2 (λ). Lemma 1. Assume λ < θ are infinite cardinals such that θ is regular and xγ, for γ < θ, is an orthonormal family in ℓ 2 (θ). Then there is γ0 < θ such that xγ is orthogonal to ℓ 2 (λ) for all γ ≥ γ0. Proof. For α ≤ θ let X(α) denote the closed linear span of xγ for γ < α. Let eξ,