Introduction to real analysis
This text is a single variable real analysis text, designed for the one-year course at the junior, senior, or beginning graduate level. It provides a rigorous and comprehensive treatment of the theoretical concepts of analysis. The book contains most of the topics covered in a text of this nature, but it also includes many topics not normally encountered in comparable texts. These include the Riemann-Stieltjes integral, the Lebesgue integral, Fourier series, the Weiestrass approximation theorem, and an introduction to normal linear spaces. The Real Number System; Sequence Of Real Numbers; Structure Of Point Sets; Limits And Continuity; Differentiation; The Riemann And Riemann-Stieltjes Integral; Series of Real Numbers; Sequences And Series Of Functions; Orthogonal Functions And Fourier Series; Lebesgue Measure And Integration; Logic and Proofs; Propositions and Connectives For all readers interested in real analysis.
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The Real Number System
Sequences of Real Numbers
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approximation Cauchy sequence chapter closed and bounded consequence continuous function continuous real-valued function converges uniformly Corollary DEFINITION Let denoted differentiable diverges example exercises Exercise exists a positive finite number following theorem Fourier series function on a,b function/on G a,b If/is improper integral inequality infinite least upper bound Lebesgue integrable Lemma Let f Let/be lim f(x limit point Math mean value theorem measurable function measurable set measurable subset measure zero Miscellaneous Exercise monotone increasing non-empty non-negative normed linear space Nt(p open intervals open set open subset polynomial positive integer power series previous theorem Prove that lim Prove Theorem proves the result rational numbers rb rb real-valued function defined Riemann integrable Riemann-Stieltjes integral satisfying series converges Since/is subsequential limits that/is THEOREM Let uniform convergence uniformly continuous upper bound property Weierstrass Weierstrass M-test