Hilbert Spaces

This chapter begins with vector spaces and inner (scalar) products. The proof of the all-important Schwarz inequality is given. Then an intuitive idea of measures, in particular the Lebsegue measure, is described as also the Lebesgue integral. The chapter then deals with function spaces \(l^2\) —the space of Fourier coefficients—and the \(L^2\) space of square-integrable functions—both of which are Hilbert spaces. Then the parallelogram law, Gram- Schmidt orthonormalisation procedure, Bessels’s inequality, and complete orthonormal sets are discussed in depth.