Elementary Real Analysis de H. G. Eggleston

Elementary Real Analysis

H. G. Eggleston

en Limba Engleză Paperback – 17 dec 2008

This textbook covers all the theoretical aspects of real variable analysis which undergraduates reading mathematics are likely to require during the first two or three years of their course. It is based on lecture courses which the author has given in the universities of Wales, Cambridge and London. The subject is presented rigorously and without padding. Definitions are stated explicitly and the whole development of the subject is logical and self-contained. Complex numbers are used but the complex variable calculus is not. 'Applied analysis', such as differential equations and Fourier series, is not dealt with. A large number of examples is included, with hints for the solution of many of them. These will be of particular value to students working on their own.

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Cuprins

Preface; Notation and Conventions; Preliminaries; 1. Enumerability and sequences; 2. Bounds for sets of numbers; 3. Bounds for functions and sequences; 4. Limits of the sequences; 5. Monotonic sequences; two important examples ; irrational powers of positive real numbers; 6. Upper and lower limits of real sequences: the general principle of convergence; 7. Convergence of series; absolute convergence; 8. Conditional convergence; 9. Rearrangement and multiplication of absolutely convergent series; 10. Double series; 11. Power series; 12. Point set theory; 13. The Bolzano-Weierstrass, Cantor and Heine-Borel theorems; 14. Functions defined over real or complex numbers; 15. Functions of a single real variable; limits and continuity; 16. Monotonic functions; functions of bounded variation; 17. Differentiation; mean-value theorems; 18. The nth mean-value theorem: Taylor's theorem; 19. Convex and concave functions; 20. The elementary transcendental functions; 21. Inequalities; 22 The Riemann integral; 23. Integration and differentiation; 24. The Riemann-Stiltjes integral; 25. Improper integrals; convergence of integrals; 26. Further tests for the convergence of series; 27. Uniform convergence; 28. Functions of two real variables. Continuity and differentiability; Hints on the solution of exercises and answers to exercises; Appendix; Index.