Provides avenues for applying functional analysis to the practical study of natural sciences as well as mathematics. Contains worked problems on Hilbert space theory and on Banach spaces and emphasizes concepts, principles, methods and major applications of functional analysis.
Most books on analysis could be subtitled "One damn theorem after another: written by mathematicians for mathematicians". This book is different. Though rigorous and concise, it takes the time to explain what theorems really mean and why concepts are worth understanding. It shows that functional analysis is a generalization and extension of many concepts from undergraduate algebra and calculus. As such, it is powerful, beautiful, and above all, useful.
The first half of the book covers the basic theory of metric spaces, normed/Banach spaces and inner-product/Hilbert spaces. Applications include approximation theory and numerical integration; differential and integral equations; and the Legendre, Hermite, Laguerre and Chebyshev polynomials. The second half of the book is devoted to spectral theory, the final chapter discussing operators in quantum mechanics. Although integration theory is not formally covered, the book does show its relationship to functional analysis.
The book provides numerous examples, counter-examples and exercises. The exercises really are do-able - mostly short but instructive - and answers are provided for odd-numbered questions.