Hilbert Spaces

Abstract

Inner products are generalizations of the dot product in $\mathbb {R}^n$. They extend the concept of distance to include orthogonality, with results like Pythagoras’ theorem and the Cauchy-Schwarz inequality. Norms that are induced by inner products are characterized by the parallelogram law and the polarization identity. Complete inner product spaces, called Hilbert spaces, have various special properties, including the least distance theorem for closed convex sets, the Riesz representation theorem on dual spaces, and adjoint operators. Applications are made to least squares approximation and Inverse Problems, with examples from statistics, image reconstruction, tomography, Tikhonov regularization, and Wiener deconvolution. The chapter closes with a section on orthogonal bases, a generalization of Fourier series and the Parseval identity, from which follows that every separable Hilbert space is isomorphic to $\ell ^2$. Further applications include time-frequency and wavelet bases, solving infinite dimensional linear equations, Gaussian quadrature, and the JPEG image format.