The present volume is a text designed for a first course in analysis. Although it is logically self-contained, it presupposes the mathematical maturity acquired by students who will ordinarily have had two years of calculus. When used in this context, most of the first part can be omitted, or reviewed extremely rapidly, or left to the students to read by themselves. The course can proceed immediately into Part Two after covering Chapters o and 1. However, the techniques of Part One are precisely those which are not emphasized in elementary calculus courses, since they are regarded as too sophisticated. The context of a third-year course is the first time that they are given proper emphasis, and thus it is important that Part One be thoroughly mastered. Emphasis has shifted from computational aspects of calculus to theoretical aspects: proofs for theorems concerning continuous 2 functions; sketching curves like x e-X, x log x, xlix which are usually regarded as too difficult for the more elementary courses; and other similar matters.

Inhalt

0 Sets and Mappings.- 1 Real Numbers.- 2 Limits and Continuous Functions.- 3 Differentiation.- 4 Elementary Functions.- 5 The Elementary Real Integral.- 6 Normed Vector Spaces.- 7 Limits.- 8 Compactness.- 9 Series.- 10 The Integral in One Variable.- 11 Approximation with Convolutions.- 12 Fourier Series.- 13 Improper Integrals.- 14 The Fourier Integral.- 15 Functions on n-Space.- 16 Derivatives in Vector Spaces.- 17 Inverse Mapping Theorem.- 18 Ordinary Differential Equations.- 19 Multiple Integrals.- 20 Differential Forms.