Hilbert spaces and Banach algebras are needed for further study in areas such as dynamical systems,

partial differential equations, and abstract Fourier analysis. 

The book assumes the reader has an undergraduate background in basic real analysis, including elementary set theory, limits, continuity, differentiation, and Riemann integration.   

An acquaintance with the properties of complex numbers is also assumed. Linear spaces, metric spaces, general topology, and integration are developed in detail in the appendices. The book contains numerous exercises to reinforce the material in the book, including exercises of independent interest. Hints are given for the more difficult exercises and solutions to odd-numbered exercises may be found in an appendix. The book should be of use to advanced undergraduates and first-year graduate students, as well as for self-study.