orthonormal basis

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

• B is a basis for X in the analyst's sense, that is, the finite linear combinations α₁b₁ + ... + α_kb_k (where α_i are scalars from k, and b_i are any elements of B) are dense in X. Compare and oppose Hamel basis.
• Each b_j is normalized, that is, (b_j | b_j) = 1 for every j.
• The b_j are orthogonal, that is, (b_j | b_k) = 0 if j ≠ k.

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 e^inx | 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.