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:
“How old the world is! I walk between two eternities.... What is my fleeting existence in comparison with that decaying rock, that valley digging its channel ever deeper, that forest that is tottering and those great masses above my head about to fall? I see the marble of tombs crumbling into dust; and yet I don’t want to die!”
—Denis Diderot (1713–1784)
“If the heart beguiles itself in its choice [of a wife], and imagination will give excellencies which are not the portion of flesh and blood:Mwhen the dream is over, and we awake in the morning, it matters little whether ‘tis Rachael or Leah,—be the object what it will, as it must be on the earthly side ... of perfection,—it will fall short of the work of fancy, whose existence is in the clouds.”
—Laurence Sterne (1713–1768)
“The hour when you say, “What does my happiness matter? It is poverty and filth, and a wretched complacency. Yet my happiness should justify existence itself!””
—Friedrich Nietzsche (1844–1900)