L2-spaces: orthogonal sequences

Abstract

Abstract Among the LP-spaces, L2 is special. Like the Euclidean spaces ℝk, its norm comes from an inner product. This gives the notion of orthogonality which underlies Fourier theory in L2, and gives a rich geometrical theory. In this chapter we assume familiarity with rudimentary linear algebra. Where exactly the same arguments work in an arbitrary inner product space as in the special case of an L2-space we work in the general setting. Our results are motivated by the geometry of the most familiar inner product spaces, namely the real Euclidean spaces ℝk (k = 1, 2, 3), with the usual scalar product as the inner product. They subsume the corresponding results in Chapter 30.