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:
“It is because everything is relative
That we shall never see in that sphere of pure wisdom and
Entertainment much more than groping shadows of an incomplete
Former existence so close it burns like the mouth that
Closes down over all your effort like the moment
Of death”
—John Ashbery (b. 1927)
“Every moment of one’s existence one is growing into more or retreating into less. One is always living a little more or dying a little bit.”
—Norman Mailer (b. 1923)
“During a walk or in a book or in the middle of an embrace, suddenly I awake to a stark amazement at everything. The bare fact of existence paralyzes me... To be alive is so incredible that all I can do is to lie still and merely breathe—like an infant on its back in a cot. It is impossible to be interested in anything in particular while overhead the sun shines or underneath my feet grows a single blade of grass.”
—W.N.P. Barbellion (1889–1919)