Orthogonal Basis and Orthogonalization

Lecture 6

In the realm of infinite dimensional vector spaces, a Hilbert space is the next best thing to an Euclidean space, i.e., a finite dimensional inner product space. The single most useful property of these spaces is that they permit the introduction of an orthonormal basis.

The first and most important way of specifying such a basis is to introduce a Hermitian matrix or operator. Its eigenvectors form an orthonormal basis. In fact, this is why a Hilbert space was invented in the first place: to accomodate the eigenvalue problem of a Hermitian operator,

$$Au = \lambda u \quad ,$$

arising, as we shall see, in boundary value problems, for example.

The second way of specifying such a basis is by means of the Gram-Schmidt orthogonalization process. From a given linearly independent set of vectors, one constructs by an iterative process a corresponding set of orthonormal vectors.

Let us, therefore, assume that we have acquired by one of these, or by some other method, the system of orthonormal elements

$$\{ u_1,u_2,\dots ,u_n,\dots\colon \langle u_i,u_j\rangle =\delta_{ij}\}$$

of the Hilbert space ${\cal H}$ .