Problems in Real and Complex Analysis

In the pages that follow there are: A. A revised and enlarged version of Problems in analysis (PIA) . (All typographical, stylistic, and mathematical errors in PIA and known to the writer have been corrected.) B. A new section COMPLEX ANALYSIS containing problems distributed among many of the principal topics in the theory of functions of a complex variable. C. A total of 878 problems and their solutions. D. An enlarged Index/Glossary and an enlarged Symbol List. Notational and terminological conventions are to be found for the most part under Conventions at the beginnings of the chapters. Spe cial items not included in Conventions are completely explained in the Index/Glossary. The audience to which the current book is addressed differs little from the audience for PIA. The background of the reader is assumed to include a knowledge of the basic principles and theorems in real and complex analysis as those subjects are currently viewed. The aim of the problems is to sharpen and deepenthe understanding of the mechanisms that underlie modern analysis. I thank Springer-Verlag for its interest in and support of this project. State University of New York at Buffalo B. R. G. v Contents The symbol alb under Pages below indicates that the Problems for the section begin on page a and the corresponding Solutions begin on page b. Thus 3/139 on the line for Set Algebra indicates that the Problems in Set Algebra begin on page 3 and the corresponding Solutions begin on page 139.

1 Set Algebra and Function Lattices. - 2 Topology, Limits, and Continuity. - 3 Real- and Complex-valued Functions. - 4 Measure and Topology. - 5 Measure Theory. - 6 Topological Vector Spaces. - 7 Elementary Theory. - 8 Functions Holomorphic in a Disc. - 9 Functions Holomorphic in a Region. - 10 Entire Functions. - 11 Analytic Continuation. - 12 Singularities. - 13 Harmonic Functions. - 14 Families of Functions. - 15 Convexity Theorems. - Symbol List. - Glossary/Index.