Primer of Modern Analysis

This book discusses some of the first principles of modern analysis. I t can be used for courses at several levels, depending upon the background and ability of the students. It was written on the premise that today's good students have unexpected enthusiasm and nerve. When hard work is put to them, they work harder and ask for more. The honors course (at the University of Wisconsin) which inspired this book was, I think, more fun than the book itself. And better. But then there is acting in teaching, and a typewriter is a poor substitute for an audience. The spontaneous, creative disorder that characterizes an exciting course becomes silly in a book. To write, one must cut and dry. Yet, I hope enough of the spontaneity, enough of the spirit of that course, is left to enable those using the book to create exciting courses of their own. Exercises in this book are not designed for drill. They are designed to clarify the meanings of the theorems, to force an understanding of the proofs, and to call attention to points in a proof that might otherwise be overlooked. The exercises, therefore, are a real part of the theory, not a collection of side issues, and as such nearly all of them are to be done. Some drill is, of course, necessary, particularly in the calculation of integrals.

I.- 1 Applications.- 1. Tangent Lines.- 2. Derivatives.- 3. Maximum and Minimum Problems.- 4. Velocity and Acceleration.- 5. Area.- 2 Calculation of Derivatives.- 1. Limits.- 2. Limits and Derivatives.- 3. Derivatives of Sums, Products, and Quotients.- 4. Continuity.- 5. Trigonometric Functions.- 6. Composite Functions.- 7. Logarithms and Exponentials.- 3 Deeper Properties of Continuous Functions.- 1. Inverse Functions.- 2. Uniform Continuity.- 3. Maxima and Minima.- 4. The Mean-Value Theorem.- 5. Zero and Infinity.- 4 Riemann Integration.- 1. Area.- 2. Integrals.- 3. Elementary Functions.- 4. Change of Variable.- 5. Integration by Parts.- 6. Riemann Sums.- 7. Arc Length.- 8. Polar Coordinates.- 9. Volume.- 10. Improper Integrals.- 5 Taylor’s Formula.- 1. Taylor’s Formula.- 2. Equivalent Formulas.- 3. Local Maxima and Minima.- 6 Sequences and Series.- 1. Sequences and Series.- 2. Increasing Sequences and Positive Series.- 3. Cauchy Sequences.- 4. Sequences of Functions.- 5. Power Series.- 6. Analytic Functions.- 7. Examples.- 8. Weierstrass Approximation Theorem.- II.- 7 Metric Spaces.- 1. The spaceRn.- 2. Absolute Value in Rn.- 3. Metric Spaces.- 4. Function Spaces.- 5. Equivalent Metrics.- 6. Open and Closed Sets.- 7. Connected Spaces.- 8. Composite Functions and Subsequences.- 9. Compact Spaces.- 10. Equivalence of Absolute Values on Rn.- 11. Products.- 12. Stone-Weierstrass Approximation Theorem.- 8 Functions From R1to Rn.- 1. Lines, Half-lines, and Directions.- 2. Derivatives and Integrals.- 3. Tangent Lines, Velocity, and Acceleration.- 4. Geometric Models of R“.- 5. Missiles, Moons, and so on.- 6. Arc Length.- 9 Algebra and Geometry in Rn.- 1. Subspaces.- 2. Bases.- 3. Orthonormal Bases.- 4. Linear Transformations.- 5. Sums and Products.- 6. Null Space and Range.- 7. Matrices and Linear Equations.- 8. Continuity of Linear Transformations.- 9. Self-adjoint Transformations.- 10. Orthogonal Transformations.- 11. Determinants.- 10 Linear Approximation.- 1. Directional Derivatives and Partial Derivatives.- 2. The Differential.- 3. Existence of the Differential.- 4. Composite Functions.- 5. The Mean-Value Theorem.- 6. A Fixed-Point Theorem.- 7. The Inverse-Function Theorem.- 8. The Implicit-Function Theorem.- 11 Surfaces.- 1. Algebraic Curves.- 2. Manifolds.- 3. Tangent Spaces.- 4. Functions on Manifolds.- 5. Quadratic Forms and Quadric Surfaces.- 12 Higher Derivatives.- 1. Second Derivatives.- 2. Higher Derivatives.- 3. The Inverse-and Implicit-Function Theorems.- 4. Taylor’s Formula.- 5. Local Maxima and Minima.- III.- 13 Integration.- 1. Introduction.- 2. Lebesgue Measure.- 3. Outer Measures.- 4. Measurability in RRn.- 5. Measurable Functions.- 6. Definition of the Integral.- 7. Convergence Theorems.- 8. Integrable Functions.- 9. Product Measures.- 10. Functions Defined by Integrals.- 11. Convolution.- 12. Approximation Theorems.- 13. Multiple Series.- 14. Regular Values and Sard’s Theorem.- 14 Differentiation.- 1. Regular Borel Measures.- 2. Differentiability Theorems.- 3. Integration of Derivatives.- 4. Change of Variable.- 5. Differentiability of Lipschitz Functions.- 15 Surface Area.- 1. Area Measures.- 2. Parametric—SurfacesIntroductory Remarks.- 3. The Jacobian.- 4. Absolute Continuity.- 5. Variation.- 6. The Jacobian Formula for Surface Area.- 7. Examples.- 8. Polar Coordinates.- 16 The Brouwer Degree.- 1. Introduction.- 2. The Degree for C?Functions.- 3. The Degree for Continuous Functions.- 4. Some Applications of the Degree.- 5. Change of Variable Revisited.- 17 Extensions of Differentiable Functions.- 1. Introduction.- 2. Reflection Across Hyperplanes.- 3. Regularized Distance.- 4. Reflection Across Lipschitz Graphs.- 5. Reflection of Hölder Functions.- 6. Reflection of Sobolev Functions.- 7. Extension from Lipschitz Graph Domains.