Orthonormal Basis - Basic Formula

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.