Introduction

The main topics to be studied in this chapter are orthogonal and orthonormal systems in a vector space with inner product, as well as various related concepts. These topics are sometimes, but not always, discussed in a basic course in linear algebra. Of central importance is the subject of infinite orthonormal systems which we present at the end of this chapter. These results will be applied in the next chapter on Fourier series. The first four sections of this chapter are a condensed review of some concepts and basic ideas (with proofs) from linear algebra. We use these facts in developing the different topics of this book. The reader will hopefully find in these sections a helpful synopsis

and review of his knowledge of the area.

Linear and Inner Product Spaces

The basic algebraic structure which we use is the linear space(often called vector space) over a field of scalars. Our “field of scalars” will always be either the real numbers R or the complex numbers <D. Elements of a linear space are called vectors. Formally, a non-empty set Vis called a linear space over a field Fif it satisfies the ollowing conditions:

1. Vector Addition: There exists an operation, generally denoted by"+", such that for any two vectors u, veV﹜ the “sum” u + v is also a vector in V.