Measure, Integration & Real Analysis · Problem 8C.17
Real and Complex Measures — Sheldon Axler
Mathematics — Measure Theory · extending-orthonormal-family-to-basis
Problem
Prove that each orthonormal family in a Hilbert space can be extended to an orthonormal basis of the Hilbert space. Specifically, suppose {ej}j∈Ω is an orthonormal family in a Hilbert space V. Prove that there exists a set Γ containing Ω and an orthonormal basis {fk}k∈Γ of V such that fj=ej for every j∈Ω.