Vector Spaces with an Inner Product

Abstract

This chapter begins with an optional section on symmetry of plane figures, which shows how some natural geometrical questions lead to the problem of studying linear transformations that preserve length. The concept of length in a general vector space over the real numbers is introduced in the next section, where it is shown how length is related to an inner product. The language of orthonormal bases and orthogonal transformations is developed with some examples from geometry and analysis. Beside the fact that the real numbers form a field, we shall use heavily in this chapter the theory of inequalities and absolute value, and the fact that every real number a ≥ 0 has a unique nonnegative square root $\sqrt a$.