Real AnalysisCambridge University Press, 15. aug. 2000 - 284 sider This is a course in real analysis directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something to specialists and non-specialists. The course consists of three major topics: metric and normed linear spaces, function spaces, and Lebesgue measure and integration on the line. In an informal style, the author gives motivation and overview of new ideas, while supplying full details and proofs. He includes historical commentary, recommends articles for specialists and non-specialists, and provides exercises and suggestions for further study. This text for a first graduate course in real analysis was written to accommodate the heterogeneous audiences found at the masters level: students interested in pure and applied mathematics, statistics, education, engineering, and economics. |
Indhold
Countable and Uncountable Sets | |
Metrics and Norms Metric Spaces | |
The Relative Metric Notes andRemarks 5 Continuity | |
Connectedness Connected Sets | |
Notes andRemarks 7 Completeness | |
Category | |
Sequences of Functions | |
Fourier Series | |
Lebesgue Measure The Problem of Measure Lebesgue Outer Measure Riemann Integrability | |
Measurable Functions Measurable Functions Extended RealValued Functions | |
Notes andRemarks 19 Additional Topics Convergence inMeasure The L p Spaces | |
The LebesgueIntegral | |
Notes andRemarks 20 Differentiation Lebesgues Differentiation Theorem | |
Notes andRemarks 11 The SpaceofContinuous Functions The WeierstrassTheorem Trigonometric Polynomials Infinitely Differentiable Functions Eq... | |
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algebra bounded variation Cantor function Cauchy sequence Chapter closed sets complete consider contains continuous function converges in measure converges pointwise converges uniformly Corollary define definition denote dense derived numbers differentiable discontinuous easy equicontinuous equivalent example Exercise exists f is continuous fact finite follows forall Fourier series function f Given hence Hint homeomorphic infinite inthe isometry lattice Lebesgue integral Lebesgue measurable Lebesgue’s Lemma Let f limit linear Lipschitz measurable functions measurable sets measure zero monotone nonempty nonnegative normed vector space notation ofthe onetoone open interval open sets pairwise disjoint particular partition PROOF Prove real numbers realvalued functions Recall Riemann integrable Riemann–Stieltjes Riesz satisfies Show that f showthat simple functions step functions subalgebra subintervals subsequence subset subspace suchthat thatis thatthe totally bounded trig polynomial uncountable uniform convergence uniformly continuous Weierstrass theorem wewill