Editing Orthonormal basis (section)

==Existence==

Using [[Zorn's lemma]] and the [[Gram–Schmidt process]] (or more simply well-ordering and transfinite recursion), one can show that ''every'' Hilbert space admits an orthonormal basis;<ref> [https://books.google.com/books?id=-m3jBwAAQBAJ Linear Functional Analysis] Authors: Rynne, Bryan, Youngson, M.A. page 79</ref> furthermore, any two orthonormal bases of the same space have the same [[Cardinal number|cardinality]] (this can be proven in a manner akin to that of the proof of the usual [[dimension theorem for vector spaces]], with separate cases depending on whether  the larger basis candidate is countable or not). A Hilbert space is [[Separable metric space|separable]] if and only if it admits a [[countable]] orthonormal basis. (One can prove this last statement without using the [[axiom of choice]]. However, one would have to use the [[axiom of countable choice]].)