Hilbert Spaces

Abstract

A Hilbert space is the abstraction of the finite-dimensional Euclidean spaces of geometry. Its properties are very regular and contain few surprises, though the presence of an infinity of dimensions guarantees a certain amount of surprise. Historically, it was the properties of Hilbert spaces that guided mathematicians when they began to generalize. Some of the properties and results seen in this chapter and the next will be encountered in more general settings later in this book, or we shall see results that come close to these but fail to achieve the full power possible in the setting of Hilbert space.