CONTENTS 5.3 General Measure 5.3.1 Signed measures 5. 4 Examples Using Measure Theory 5.4.1 Probability Spaces 5.4.21 195 5.5 Appendix- Proofs in Chapter 5 200 5.6 Bibilography for Chapter 5 6 Function Spaces 213 6. 1 The set of bounded continuous functions 6.1.1 Completeness 216 6. 1.2 Compactness 6. 1.3 Approximation arability of C(X) 6. 1.5 Fixed point theorems 6.2 Classical Banach spaces: L 229 6.2.1 Additional Topics in Lp(X 235 6.2.2 Hilbert Spaces(L2(X)) 37 6.3 Linear operators 241 6.4 Linear functionals 6.4.1 Dual spaces 248 6.4.2 Second Dual Space 6.5 Separation Results 6.5.1 Existence of equilibrium 260 6.6 timization of Nonlinear Operators 262 6.6.1 Variational methods on infinite dimensional vector spaces 262 6.6.2 Dynamic Programming 274 6.7 Appendix- Proofs for Chapter 6 284 6.8 Bibilography for Chapter 6 297 7 Topological Spaces 7.1 Continuous Functions and Homeomorphisms 7.2 Separation Axioms 7.3 Convergence and Completeness 305CONTENTS 5 5.3 General Measure . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.3.1 Signed Measures . . . . . . . . . . . . . . . . . . . . . 185 5.4 Examples Using Measure Theory . . . . . . . . . . . . . . . . 194 5.4.1 Probability Spaces . . . . . . . . . . . . . . . . . . . . 194 5.4.2 L1 .................... . . . . . . . . . 195 5.5 Appendix - Proofs in Chapter 5 . . . . . . . . . . . . . . . . . 200 5.6 Bibilography for Chapter 5 . . . . . . . . . . . . . . . . . . . . 211 6 Function Spaces 213 6.1 The set of bounded continuous functions . . . . . . . . . . . . 216 6.1.1 Completeness . . . . . . . . . . . . . . . . . . . . . . . 216 6.1.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . 218 6.1.3 Approximation . . . . . . . . . . . . . . . . . . . . . . 221 6.1.4 Separability of C(X) . . . . . . . . . . . . . . . . . . . 227 6.1.5 Fixed point theorems . . . . . . . . . . . . . . . . . . . 227 6.2 Classical Banach spaces: Lp . . . . . . . . . . . . . . . . . . . 229 6.2.1 Additional Topics in Lp(X) . . . . . . . . . . . . . . . 235 6.2.2 Hilbert Spaces (L2(X)) . . . . . . . . . . . . . . . . . . 237 6.3 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . 241 6.4 Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . . 245 6.4.1 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . . 248 6.4.2 Second Dual Space . . . . . . . . . . . . . . . . . . . . 252 6.5 Separation Results . . . . . . . . . . . . . . . . . . . . . . . . 254 6.5.1 Existence of equilibrium . . . . . . . . . . . . . . . . . 260 6.6 Optimization of Nonlinear Operators . . . . . . . . . . . . . . 262 6.6.1 Variational methods on infinite dimensional vector spaces262 6.6.2 Dynamic Programming . . . . . . . . . . . . . . . . . . 274 6.7 Appendix - Proofs for Chapter 6 . . . . . . . . . . . . . . . . . 284 6.8 Bibilography for Chapter 6 . . . . . . . . . . . . . . . . . . . . 297 7 Topological Spaces 299 7.1 Continuous Functions and Homeomorphisms . . . . . . . . . . 302 7.2 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . . 303 7.3 Convergence and Completeness . . . . . . . . . . . . . . . . . 305