A sequence of norm-one elements of a Hilbert space that are mutually orthogonal is said to form an orthonormal sequence. If, additionally, such a sequence spans the entire space, it is said to be complete. As it turns out, if in a Hilbert space there is a complete orthonormal sequence, this space is indistinguishable from the space of square summable sequences. In particular, perhaps contrary to our misleading intuition saying that there are many more square integrable functions than there are square summable sequences, the space of the former is as large as (in fact much the same as) the space of the latter. We will see one important consequence of this stunning result in the next chapter.

Chapter 6: In this chapter, we explore the role of orthonormal (orthogonal and normalized) vectors in an inner-product space. Matrix representations of linear transformations with respect to orthonormal bases are of particular importance. They are associated with the notion of an adjoint transformation. We give a brief introduction to Fourier series that highlights the orthogonality properties of sine and cosine functions. In the final section of the chapter, we discuss orthogonal polynomials and the remarkable numerical integration rules associated with them.