Introduction to Functional Analysis With Applications A.H. Siddiqi Khalil Ahmad P. Manchanda Tunbridge Wells, UK Anamaya Publishers New Delhi
Contents Preface vii List of Symbols.: ' - ix 1. Normed and Banach Spaces 1 1.1 Basic Definitions and Properties 1 1.2 Examples of Normed Spaces and Related Concepts 6 Normed Spaces 6 Banach Spaces 13 Dimension of Normed Spaces 13 Open and Closed Spheres 13 Normed Subspaces 16 Completion of Normed Spaces 16 Isometry and Isomorphism 16 1.3 Operators and Functionals 17 Definitions and Examples 17 Properties of Linear Operators and Dual Space 25 Algebra of Operators 32 1.4 Convex Functionals (Real-Valued Convex Functions on Normed Spaces) 38 Convex Sets 38 Affine Operator 40 Convex Functionals 40 Lower Semicontinuous (lsc) and Upper Semicontinuous (use) Functionals 43 1.5 Topological Properties of Normed Spaces 44 Compactness in Normed Spaces 44 Separability and Connectedness for Normed Spaces 49 Equivalent Norms and Finite-Dimensional Spaces SO Reflexive Normed Spaces and Different Kinds of Topologies 53 Norme'd Spaces with Basis or Base 54 1.6 Geometrical Properties of Normed Spaces 55 1.7 Some More Examples 56 Exercises 75 Solutions to Exercises 77 2. Inner Product and Hilbert Spaces : 81 2.1 Basic Definitions and Properties 81 Definitions, Examples and Properties of Inner-product Space 81
xiv Contents Hilbert Space 86 Parallelogram Law and Characterization of Hilbert Space 87 2.2 Orthogonal Complements and Projection Theorem 93 Orthogonal Complements and Projections 93 Projection Theorem 95 2.3 Orthonormal Systems and Fourier Expansion 100 Definitions, Examples and Gram-Schmidt Orthogonalization - Process 100 Bessel's Inequality 104 2.4 Duality and Reflexivity 106 Riesz Representation Theorem 106 Reflexivity of Hilbert Spaces 109 2.5 Operators in Hilbert Space 110 Adjoint of a Bounded Linear Operator on a Hilbert Space 110 Self-Adjoint, Positive, Normal and Unitary Operators 115' Adjoint of an Unbounded Linear Operator 122 2.6 Bilinear Forms and Lax-Milgrarri Lemma 124 2.7 Projection on Convex Sets 131 2.8 Some More Examples 134 Exercises 144 Solutions to Exercises 146 3. Fundamental Theorems 149 3.1 Extension Form of the Hahn-Banach Theorem and Its Consequences 149 Consequences of the Extension Form of the Hahn-Banach Theorem 154 3.2 Geometric Form of the Hahn-Banach Theorem and Its Corollaries 157 3.3 Principle of Uniform Boundedness and Its Applications 159 Principle of Uniform Boundedness 159 Applications of the Principle of Uniform Boundedness in Fourier Analysis 160 3.4 Open Mapping and Closed Graph Theorems 162 Graph of a Linear Operator and Closedness Property 162 Open Mapping Theorem 164 Closed Graph Theorem 165 3.5 Examples 166 Exercises 168 Solutions to Exercises 169 4. Weak Topologies, Weak Convergence and Reflexive Spaces 170 4.1 Weak Topologies 170 4.2 Weak Convergence 171 4.3 Reflexive Banach Spaces 173 :
4.4 Weak Convergence in Hilbert Spaces 174 4.5 Examples 176 Exercises 178 Solutions to Exercises 178 Contents xv 5. Differentiation and Integration in Normed Spaces 180 5.1 Gateaux Derivative 180 5.2 Frechet Derivative 183 5.3 Subdifferential 186 5.4 Integration in Normed Spaces 188 Exercises 189 Solutions to Exercises 190 6. Fixed-Point Theorems and their Applications 191 6.1 Banach Contraction Principle and Its Generalizations 191 6.2 Schauder's Fixed-Point Theorem 195 6.3 Applications of Banach Contraction Principle 196 Application to Matrix Equation 196 " Application to Differential Equations 201 Exercises 203 Solutions to Exercises 206 7. Rudiments of Spectral Theory 210 7.1 Spectral Properties of Bounded Linear Operators 210 7.2 Compact Operators 211 7.3 Spectral Properties of Self-Adjoint and Compact Operators 215 7.4 Spectral Decomposition 216 Solvability of Operator Equations 219 Characterization of Solvability in Terms of Range and Null Spaces 222 Characterization of Lax-Milgram Lemma 223 Existence Theorem for Nonlinear Operators 224 7.5 Examples 225 Exercises 226 Solutions to Exercises 227 8. Boundary Value Problems 230 8.1 Definition and Examples of Boundary Value Problems 230 Definition 230 Examples of BVPs 231 8.2 Abstract Equations 236 8.3 Sobolev Space 238 Examples of Distribution 239 8.4 Certain Remarks Concerning the Solutions of BVPs 246
xvi Contents Exercises 249 Solutions to Exercises 249 9. Optimization 251 9.1 Minimization of Functionals 251 9.2 Calculus of Variation and Linear Programming -255 Calculus of Variation 255 Linear Programming 257 Exercises 257 Solutions to Exercises 258 10. Variational Inequalities 260 10.1 Lions-Stampacchia Theory 260 10.2 Physical Phenomena Represented by Variational Inequalities 265 Exercises 267 Solutions to Exercises 267 11. The Finite-Element Method 269 11.1 Approximate Problem 270 11.2 Internal Approximation of//'(q) 273 11.3 Finite Elements 275 11.4 Application of the Finite-Element Method to Solve Boundary Value Problems 279 Practical Method to Compute a(wj, w,) 281 11.5 Effect of Numerical Integration 281 11.6 Abstract Error Estimate for the Nonconforming Finite-Element Method 284 11.7 Abstract Error Estimation for Variational Inequalities 285 Exercises 286 Solutions to Exercises 287 12. Optimal Control 288 12.1 Problem Illustration with the Help of an Example and Formulation of General Problem 289 Formulation of the General Optimal Control Problem for a System Represented by Differential Equation 290 12.2 Linear Quadratic Control Problem 291 Exercises 297 Solutions to Exercises 297 13. Wavelets 298 13.1 Recapitulation of Some Basic Concepts 298 13.2 Continuous Wavelet Transform 300
13.3 Examples of Wavelets 301 13.4 Decay of Continuous Wavelet Transform 308 13.5 Multiresolution Analysis 310 Examples of Multiresolution Analysis 311 Important Properties of MRA 311 13.6 Decomposition and Reconstruction Algorithms 319 13.7 BestAf-TermApproximation 320 13.8 Wavelet and Function Spaces 322 Exercises 322 Contents xvii Appendix A 325 Appendix B 328 Appendix C 331 Appendix D 335 Appendix E 343 Appendix F 345 REFERENCES 351 INDEX 359