This is our book on the theory of Hilbert spaces, its methods and usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. There are many fine books on Hilbert spaces and our intention is not to generate another book to stick on the pile or to be used to level a desk. So from the onset, we have sought to synthesize the book with special goals in mind.

The needs and concerns of researchers in engineering differ from those of the pure sciences. It is difficult to put the finger on what distinguishes the engineering approach that we take. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a foundational concept that links deeply with other areas of mathematics or is part of a long-standing human intellectual endeavor — not that engineering, in comparison, concerns less intellectual pursuits. As an example, Carleson in 1966 proved a conjecture by Luzin in 1915 concerning the almost-everywhere convergence of Fourier series of continuous functions. Carleson's theorem, as it is called, has its roots in the questions Fourier asked himself, in French presumably, about the nature of convergence of the series named after him. As a result it is important for mathematics, but less clear for engineers.