Introduction to Real Analysis

Assuming minimal background on the part of students, this text gradually develops the principles of basic real analysis and presents the background necessary to understand applications used in such disciplines as statistics, operations research, and engineering. The text presents the first elementary exposition of the gauge integral and offers a clear and thorough introduction to real numbers, developing topics in n-dimensions, and functions of several variables. Detailed treatments of Lagrange multipliers and the Kuhn-Tucker Theorem are also presented. The text concludes with coverage of important topics in abstract analysis, including the Stone-Weierstrass Theorem and the Banach Contraction Principle.

Preliminaries. Real Numbers. Sequences. Infinite Series. Euclidean Spaces. Limits of Functions. Continuity and Uniform Continuity. Sequences of Functions. The Riemann Integral Reviewed. The Gauge Integral. The Gauge Integral Over Unbounded Intervals. Convergence Theorems. Multiple Integrals. Convolution and Approximation. Metric Spaces. Topology in Metric Spaces. Continuity. Complete Metric Spaces. Contraction Mappings. The Baire Category Theorem. Compactness. Connectedness. The Stone-Weierstrass Theorem. Differentiation of Vector-valued Functions. Mapping Theorems. Bibliography. Index.