Introductory Real Analysis (Dover Books on Mathematics)

This volume in Richard Silverman's exceptional series of translations of Russian works in the mathematical science is a comprehensive | elementary introduction to real and functional analysis by two faculty members from Moscow University. It is self-contained | evenly paced | eminently readable | and readily accessible to those with adequate preparation in advanced calculus.The first four chapters present basic concepts and introductory principles in set theory | metric spaces | topological spaces | and linear spaces. The next two chapters consider linear functionals and linear operators | with detailed discussions of continuous linear functionals | the conjugate space | the weak topology and weak convergence | generalized functions | basic concepts of linear operators | inverse and adjoint operators | and completely continuous operators. The final four chapters cover measure | integration | differentiation | and more on integration. Special attention is here given to the Lebesque integral | Fubini's theorem | and the Stieltjes integral. Each individual section — there are 37 in all — is equipped with a problem set | making a total of some 350 problems | all carefully selected and matched.With these problems and the clear exposition | this book is useful for self-study or for the classroom — it is basic one-year course in real analysis. Dr. Silverman is a former member of the Institute of Mathematical Sciences of New York University and the Lincoln Library of M.I.T. Along with his translation | he has revised the text with numerous pedagogical and mathematical improvements and restyled the language so that it is even more readable.