Basic Formula
If B is an orthogonal basis of H, then every element x of H may be written as
When B is orthonormal, we have instead
and the norm of x can be given by
Even if B is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of x, and the formula is usually known as Parseval's identity. See also Generalized Fourier series.
If B is an orthonormal basis of H, then H is isomorphic to ℓ 2(B) in the following sense: there exists a bijective linear map Φ : H → ℓ 2(B) such that
for all x and y in H.
Read more about this topic: Orthonormal Basis
Famous quotes containing the words basic and/or formula:
“I fly in dreams, I know it is my privilege, I do not recall a single situation in dreams when I was unable to fly. To execute every sort of curve and angle with a light impulse, a flying mathematics—that is so distinct a happiness that it has permanently suffused my basic sense of happiness.”
—Friedrich Nietzsche (1844–1900)
“Hidden away amongst Aschenbach’s writing was a passage directly asserting that nearly all the great things that exist owe their existence to a defiant despite: it is despite grief and anguish, despite poverty, loneliness, bodily weakness, vice and passion and a thousand inhibitions, that they have come into being at all. But this was more than an observation, it was an experience, it was positively the formula of his life and his fame, the key to his work.”
—Thomas Mann (18751955)