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
Famous quotes containing the word existence:
“The individual who has to justify his existence by his own efforts is in eternal bondage to himself.”
—Eric Hoffer (1902–1983)
“What is a novel if not a conviction of our fellow-men’s existence strong enough to take upon itself a form of imagined life clearer than reality and whose accumulated verisimilitude of selected episodes puts to shame the pride of documentary history?”
—Joseph Conrad (1857–1924)
“Did men but consider that the sun, moon, and stars, and every other object of the senses, are only so many sensations in their minds, which have no other existence but barely being perceived, doubtless they would never fall down and worship their own ideas; but rather address their homage to that eternal invisible Mind which produces and sustains all things.”
—George Berkeley (1685–1753)