Real and complex analysis
This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.
This text is part of the Walter Rudin Student Series in Advanced Mathematics.
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Positive Borel Measures
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a-algebra assume Banach algebra Banach space Blaschke product Borel measure Borel sets boundary bounded linear functional Cauchy CC(X Chap compact set compact subset completes the proof complex function complex measurable complex numbers constant contains continuous function convergence theorem converges uniformly convex countable define Definition dense differentiable disjoint entire function Exercise exists finite follows formula Fourier transform harmonic function Hausdorff space Hence Hilbert space holds holomorphic functions If/e Iffe implies inequality Lebesgue measure Lebesgue point Lemma mapping Math maximal measurable function measure space metric space Note obtain one-to-one open set Poisson integral polynomials positive measure Proof Let proof of Theorem properties Prove radius real number region satisfies sequence shows simply connected subspace Suppose/is supremum that/is Theorem Let Theorem Suppose union vector space zero