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 Frenchman Jean-Paul ... Sartre I remember now was his last name had a dialectical mind good as a machine for cybernetics, immense in its way, he could peel a nuance like an onion, but he had no sense of evil, the anguish of God, and the possible existence of Satan.”
—Norman Mailer (b. 1923)
“The light that radiates from the great novels time can never dim, for human existence is perpetually being forgotten by man and thus the novelists’ discoveries, however old they may be, will never cease to astonish.”
—Milan Kundera (b. 1929)
“As humans have a prior right to existence over dogs by virtue of being more highly evolved and having a superior consciousness, so women have a prior right to existence over men. The elimination of any male is, therefore, a righteous and good act, an act highly beneficial to women as well as an act of mercy.”
—Valerie Solanas (b. 1940)