From Euclidean to Hilbert Spaces

5The Geometric Structure of Hilbert Spaces

Among the infinite-dimensional vector spaces, Hilbert spaces are the closest to the Euclidean spaces ⁿ presented in Chapter 1 with respect to their geometric structure, which is the focus of the present chapter.

Infinite-dimensional Banach spaces do not share this characteristic, with structural properties that can be far more complicated than those of Hilbert spaces.

The rich geometric structure of Hilbert spaces makes it possible to extend the discrete Fourier transform (DFT) to spaces in infinite dimensions, using the concepts of series and the continuous Fourier transform.

Suggested reading for those wishing to go further into the subjects discussed in this chapter and in Chapter 6 includes Berberian (1961), Abbati and Cirelli (1997), Saxe (2000), Debnath and Mikusinski (2005) and Moretti (2013).

The first step in analyzing the geometric structure of Hilbert spaces is to consider the concept of orthogonal complement.

5.1. The orthogonal complement in a Hilbert space and its properties

The set of all vectors which are orthogonal to the vectors of a subset in a Hilbert space is of crucial importance in understanding the geometric properties of these spaces.

The definition and properties of this set are given below.

DEFINITION 5.1.– Let be a Hilbert space and M ⊆ any subset. The orthogonal complement of M in is:

that is M^⊥ contains ...