Introduction to Analysis

<p>Preface</p> <p>Part I. ONE-DIMENSIONAL THEORY</p> <p>&nbsp;</p> <p>1. The Real Number System</p> <p>1.1 Introduction</p> <p>1.2 Ordered field axioms</p> <p>1.3 Completeness Axiom</p> <p>1.4 Mathematical Induction</p> <p>1.5 Inverse functions and images</p> <p>1.6 Countable and uncountable sets</p> <p>&nbsp;</p> <p>2. Sequences in R</p> <p>2.1 Limits of sequences</p> <p>2.2 Limit theorems</p> <p>2.3 Bolzano-Weierstrass Theorem</p> <p>2.4 Cauchy sequences</p> <p>*2.5 Limits supremum and infimum</p> <p>&nbsp;</p> <p>3. Continuity on R</p> <p>3.1 Two-sided limits</p> <p>3.2 One-sided limits and limits at infinity</p> <p>3.3 Continuity</p> <p>3.4 Uniform continuity</p> <p>&nbsp;</p> <p>4. Differentiability on R</p> <p>4.1 The derivative</p> <p>4.2 Differentiability theorems</p> <p>4.3 The Mean Value Theorem</p> <p>4.4 Taylor's Theorem and l'Hôpital's Rule</p> <p>4.5 Inverse function theorems</p> <p>&nbsp;</p> <p>5 Integrability on R</p> <p>5.1 The Riemann integral</p> <p>5.2 Riemann sums</p> <p>5.3 The Fundamental Theorem of Calculus</p> <p>5.4 Improper Riemann integration</p> <p>*5.5 Functions of bounded variation</p> <p>*5.6 Convex functions</p> <p>&nbsp;</p> <p>6. Infinite Series of Real Numbers</p> <p>6.1 Introduction</p> <p>6.2 Series with nonnegative terms</p> <p>6.3 Absolute convergence</p> <p>6.4 Alternating series</p> <p>*6.5 Estimation of series</p> <p>*6.6 Additional tests</p> <p>&nbsp;</p> <p>7. Infinite Series of Functions</p> <p>7.1 Uniform convergence of sequences</p> <p>7.2 Uniform convergence of series</p> <p>7.3 Power series</p> <p>7.4 Analytic functions</p> <p>*7.5 Applications</p> <p>&nbsp;</p> <p>Part II. MULTIDIMENSIONAL THEORY</p> <p>&nbsp;</p> <p>8. Euclidean Spaces</p> <p>8.1 Algebraic structure</p> <p>8.2 Planes and linear transformations</p> <p>8.3 Topology of Rⁿ</p> <p>8.4 Interior, closure, boundary</p> <p>&nbsp;</p> <p>9. Convergence in Rⁿ</p> <p>9.1 Limits of sequences</p> <p>9.2 Heine-Borel Theorem</p> <p>9.3 Limits of functions</p> <p>9.4 Continuous functions</p> <p>*9.5 Compact sets</p> <p>*9.6 Applications</p> <p>&nbsp;</p> <p>10. Metric Spaces</p> <p>10.1 Introduction</p> <p>10.2 Limits of functions</p> <p>10.3 Interior, closure, boundary</p> <p>10.4 Compact sets</p> <p>10.5 Connected sets</p> <p>10.6 Continuous functions</p> <p>10.7 Stone-Weierstrass Theorem</p> <p>&nbsp;</p> <p>11. Differentiability on Rⁿ</p> <p>11.1 Partial derivatives and partial integrals</p> <p>11.2 The definition of differentiability</p> <p>11.3 Derivatives, differentials, and tangent planes</p> <p>11.4 The Chain Rule</p> <p>11.5 The Mean Value Theorem and Taylor's Formula</p> <p>11.6 The Inverse Function Theorem</p> <p>*11.7 Optimization</p> <p>&nbsp;</p> <p>12. Integration on Rⁿ</p> <p>12.1 Jordan regions</p> <p>12.2 Riemann integration on Jordan regions</p> <p>12.3 Iterated integrals</p> <p>12.4 Change of variables</p> <p>*12.5 Partitions of unity</p> <p>*12.6 The gamma function and volume</p> <p>&nbsp;</p> <p>13. Fundamental Theorems of Vector Calculus</p> <p>13.1 Curves</p> <p>13.2 Oriented curves</p> <p>13.3 Surfaces</p> <p>13.4 Oriented surfaces</p> <p>13.5 Theorems of Green and Gauss</p> <p>13.6 Stokes's Theorem</p> <p>&nbsp;</p> <p>*14. Fourier Series</p> <p>*14.1 Introduction</p> <p>*14.2 Summability of Fourier series</p> <p>*14.3 Growth of Fourier coefficients</p> <p>*14.4 Convergence of Fourier series</p> <p>*14.5 Uniqueness</p> <p>&nbsp;</p> <p>References</p> <p>Answers and Hints to Exercises</p> <p>Subject Index</p> <p>Symbol Index</p> <p>&nbsp;</p>*Enrichment section <br> <br>