Introduction to Real AnalysisThis textbook is designed for a one-year course in real analysis at the junior or senior level. An understanding of real analysis is necessary for the study of advanced topics in mathematics and the physical sciences, and is helpful to advanced students of engineering, economics, and the social sciences. Stoll, who teaches at the U. of South Carolina, presents examples and counterexamples to illustrate topics such as the structure of point sets, limits and continuity, differentiation, and orthogonal functions and Fourier series. The second edition includes a self-contained proof of Lebesgue's theorem and a new appendix on logic and proofs. Annotation copyrighted by Book News Inc., Portland, OR |
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The Real Number System | 1 |
Sequences of Real Numbers | 44 |
Structure of Point Sets | 83 |
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a₁ a₂ approximation b₁ bounded real-valued function c₁ Cauchy sequence closed and bounded consequence continuous function continuous real-valued function converges uniformly Corollary countable DEFINITION Let denoted differentiable diverges E₂ example exercises Exercise exists a positive finite number following theorem Fourier series function f ƒ is continuous given improper integral inequality infinite integral of ƒ least upper bound Lebesgue integrable Lemma Let f let ƒ lim f(x limit point Math mean value theorem measurable function measurable set measurable subset measure zero monotone increasing non-negative nx dx open intervals open set open subset orthogonal P₁ positive integer power series previous theorem prove that ƒ Prove that lim rational numbers real-valued function defined Riemann integrable Riemann-Stieltjes integral satisfies sequence f series converges subsequential limits Suppose f THEOREM Let uniform convergence upper bound property Weierstrass x₁