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. KEY TOPICS: 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 MARKET: For all readers interested in real analysis. |
Contents
Structure of Point Sets | 3 |
Sequences of Real Numbers | 47 |
Limits and Continuity | 115 |
Copyright | |
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a₁ b₁ c₁ calculus Cauchy sequence compact continuous function contradiction converges uniformly Corollary countable denoted differentiable diverges E₁ E₂ example exercise Exercise exists a positive finite number Fourier series function f ƒ is continuous given graph hypothesis improper integral inequality infinite least upper bound Lebesgue integrable Lemma Let f let ƒ lim f(x limit point Math mathematical induction measurable function measurable set measurable subset measure zero monotone increasing nonempty nonnegative nx dx one-to-one open intervals open set orthogonal P₁ pointwise polynomial positive integer power series previous theorem prove that ƒ prove that lim rational numbers real numbers Riemann integrable satisfies Section sequence f sequence of real series converges series of ƒ statement Suppose ƒ tautology true uniform convergence upper bound property Weierstrass Weierstrass M-test x₁