Orthogonality, Orthonormality, Inner Products, and Completeness

If the chosen basis functions satisfy the rule that the integral of the conjugate product of any two different members of the family equals zero,

$$\int_{-\infty}^{\infty} \Psi_{k}^{*}(x) \Psi_{j}(x) dx = 0 \; \; (k \ne j)$$ (9)

then this family of functions is called orthogonal.

The above integral is called an inner product, and it is often denoted by putting the two functions inside angle brackets (conjugation of one of them is implied:)

$$< \Psi_{k}(x), \Psi_{j}(x)> \; \; \equiv \; \; \int_{-\infty}^{\infty} \Psi_{k}^{*}(x) \Psi_{j}(x) dx$$ (10)

If it is also true that the inner product of any member of this family of functions with itself is equal to 1,

$$< \Psi_{k}(x), \Psi_{j}(x)> \; \; = \; \; \int_{-\infty}^{\infty} \Psi_{k}^{*}(x) \Psi_{j}(x) dx = 1 \; \; (k = j)$$ (11)

then these functions are said to be orthonormal. If they form a complete basis, then all of the coefficients a_k that are needed to represent some arbitrary function f(x) exactly in terms of the chosen family of orthonormal basis functions $\Psi_{k}(x)$ can be obtained just by taking the inner products of the original function f(x) with each of the basis functions $\Psi_{k}(x)$:

$$a_{k} \; \; = \; \; < \Psi_{k}(x), f(x) > \; \; = \; \; \int_{-\infty}^{\infty} \Psi_{k}^{*}(x) f(x) dx$$ (12)

One example of such a representation is the Fourier Transform, which we will examine later.