Problems in Analysis

Description

These problems and solutions are offered to students of mathematics who have learned real analysis, measure theory, elementary topology and some theory of topological vector spaces. The current widely used texts in these subjects provide the background for the understanding of the problems and the finding of their solutions. In the bibliography the reader will find listed a number of books from which the necessary working vocabulary and techniques can be acquired. Thus it is assumed that terms such as topological space, u-ring, metric, measurable, homeomorphism, etc., and groups of symbols such as AnB, x EX, f: IR 3 X 1-+ X 2 - 1, etc., are familiar to the reader. They are used without introductory definition or explanation. Nevertheless, the index provides definitions of some terms and symbols that might prove puzzling. Most terms and symbols peculiar to the book are explained in the various introductory paragraphs titled Conventions. Occasionally definitions and symbols are introduced and explained within statements of problems or solutions. Although some solutions are complete, others are designed to be sketchy and thereby to give their readers an opportunity to exercise their skill and imagination. Numbers written in boldface inside square brackets refer to the bib­ liography. I should like to thank Professor P. R. Halmos for the opportunity to discuss with him a variety of technical, stylistic, and mathematical questions that arose in the writing of this book. Buffalo, NY B.R.G. Front Matter....Pages i-vii Front Matter....Pages 1-1 Set Algebra....Pages 3-4 Topology....Pages 5-7 Limits....Pages 8-9 Continuous Functions....Pages 10-14 Functions from ℝ n to ℝ m ....Pages 15-19 Measure and Topology....Pages 20-22 General Measure Theory....Pages 23-27 Measures in ℝ n ....Pages 28-30 Lebesgue Measure in ℝ n ....Pages 31-34 Lebesgue Measurable Functions....Pages 35-37 L 1 ( X, μ )....Pages 38-42 L 2 ( X, μ ) or ℌ (Hilbert Space)....Pages 43-46 L p ( X , μ), 1 ≦ p ≦ ∞....Pages 47-48 Topological Vector Spaces....Pages 49-54 Miscellaneous Problems....Pages 55-64 Front Matter....Pages 65-65 Set Algebra....Pages 67-68 Topology....Pages 69-75 Limits....Pages 76-78 Continuous Functions....Pages 79-87 Functions from ℝ n to ℝ m ....Pages 88-100 Front Matter....Pages 65-65 Measure and Topology....Pages 101-108 General Measure Theory....Pages 109-117 Measures in ℝ n ....Pages 118-125 Lebesgue Measure in ℝ n ....Pages 126-132 Lebesgue Measurable Functions....Pages 133-138 L 1 ( X, μ )....Pages 139-148 L 2 ( X, μ ) or ℌ (Hilbert Space)....Pages 149-158 L p ( X, μ ), 1 ≦ p ≦ ∞....Pages 159-164 Topological Vector Spaces....Pages 165-177 Miscellaneous Problems....Pages 178-202 Back Matter....Pages 203-206