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 a basis and thus an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same 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 if and only if it admits a countable orthonormal basis. (One can prove this last statement without using the axiom of choice).
Read more about this topic: Orthonormal Basis
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“Generality is, indeed, an indispensable ingredient of reality; for mere individual existence or actuality without any regularity whatever is a nullity. Chaos is pure nothing.”
—Charles Sanders Peirce (1839–1914)
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—David Hume (1711–1776)
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—George Bernard Shaw (1856–1950)