vi CONTENTS Chapter 4. Normed Linear Spaces 101 4.1. Definitions and Basic Properties 102 4.2. Bounded Linear Operators 106 4.3. Fundamental Theorems about Linear Operators 109 4.4. Extending Linear Functionals 112 4.5. Generalized Limits and the Dual of (F ) 114 4.6. Adjoint Operators and Isometries of Normed Linear Spaces 116 4.7. Concrete Facts about Isometries of Normed Linear Spaces 119 4.8. Locally Compact Groups 123 4.9. Hilbert Spaces 126 4.10. Convergence and Selfadjoint Operators 134 4.11. Independent Projects 136 Chapter 5. Differentiation 141 5.1. Review of Differentiation in One Variable 142 5.2. Differential Calculus in Rn 149 5.3. The Derivative as a Matrix of Partial Derivatives 154 5.4. The Mean Value Theorem 158 5.5. Higher-Order Partial Derivatives and Taylor’s Theorem 160 5.6. Hypersurfaces and Tangent Hyperplanes in Rn 164 5.7. Max-Min Problems 166 5.8. Lagrange Multipliers 170 5.9. The Implicit and Inverse Function Theorems 175 5.10. Independent Projects 182 Chapter 6. Integration 191 6.1. Measures 192 6.2. Lebesgue Measure 194 6.3. Measurable Functions 205 6.4. The Integral 208 6.5. Lp Spaces 216 6.6. Fubini’s Theorem 220 6.7. Change of Variables in Integration 224 6.8. Independent Projects 227 Chapter 7. Fourier Analysis on Locally Compact Abelian Groups 237 7.1. Fourier Analysis on the Circle 238 7.2. Fourier Analysis on Locally Compact Abelian Groups 246 7.3. The Determination of ˆ 249 7.4. The Fourier Transform on (R, +) 251 7.5. Fourier Inversion on (R, +) 254 7.6. Fourier Analysis on p-adic Fields 259 7.7. Independent Projects 263
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