Suppose X is a Hilbert space over a field k, with inner product ( | ). A (not necessarily finite or even countable) set B = {bj} of elements of X is called an orthonormal basis for X if the following are true:

Orthonormal bases are most commonly encountered when dealing with function spaces which happen to be Hilbert spaces. Under these circumstances we can approximate any function in the space by a finite sum of basis elements (by the density property above). If we can choose a special orthonormal basis for our function space whose elements have some nice property, then we may be able to use that property to prove things about arbitrary elements of the space.

For instance, harmonic analysis or Fourier analysis begins by considering the space L2(T), which is roughly the set of real- or complex-valued functions on the unit circle T, whose squares are Lebesgue integrable. (See Hilbert space for details.) This space has an orthonormal basis {x → (2π)-1/2 einx | n ∈ Z} (in the complex-valued case) or {x → π-1/2 sin(nx); x → π-1/2 cos(nx) | n ∈ N} (in the real-valued case). Approximation in this basis is precisely the Fourier transform.

Other orthonormal bases are used for approximating functions by polynomials, for instance the Chebyshev polynomials and Legendre polynomials.

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