Hilbert Spaces and L²

Abstract

Hilbert spaces are a special class of Banach spaces. Hilbert spaces are simpler than Banach spaces owing to an additional structure called an inner product. These spaces play a significant role in functional analysis and have found widespread use in applied mathematics. We shall see at the end of this section that the Lebesgue space L² (and its complex relative H²) is a Hilbert space. In this and the next section, we introduce some basic definitions and facts concerning Hilbert spaces of immediate interest to our discussion of the space L². Further details and proofs of the results presented in these sections can be found in most books on functional analysis, e.g., [25].