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
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