Introduction to Functional Analysis With Applications

Transcription

1 Introduction to Functional Analysis With Applications A.H. Siddiqi Khalil Ahmad P. Manchanda Tunbridge Wells, UK Anamaya Publishers New Delhi

2 Contents Preface vii List of Symbols.: ' - ix 1. Normed and Banach Spaces Basic Definitions and Properties 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 Operators and Functionals 17 Definitions and Examples 17 Properties of Linear Operators and Dual Space 25 Algebra of Operators 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 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 Geometrical Properties of Normed Spaces Some More Examples 56 Exercises 75 Solutions to Exercises Inner Product and Hilbert Spaces : Basic Definitions and Properties 81 Definitions, Examples and Properties of Inner-product Space 81

3 xiv Contents Hilbert Space 86 Parallelogram Law and Characterization of Hilbert Space Orthogonal Complements and Projection Theorem 93 Orthogonal Complements and Projections 93 Projection Theorem Orthonormal Systems and Fourier Expansion 100 Definitions, Examples and Gram-Schmidt Orthogonalization - Process 100 Bessel's Inequality Duality and Reflexivity 106 Riesz Representation Theorem 106 Reflexivity of Hilbert Spaces 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 Bilinear Forms and Lax-Milgrarri Lemma Projection on Convex Sets Some More Examples 134 Exercises 144 Solutions to Exercises Fundamental Theorems Extension Form of the Hahn-Banach Theorem and Its Consequences 149 Consequences of the Extension Form of the Hahn-Banach Theorem Geometric Form of the Hahn-Banach Theorem and Its Corollaries Principle of Uniform Boundedness and Its Applications 159 Principle of Uniform Boundedness 159 Applications of the Principle of Uniform Boundedness in Fourier Analysis Open Mapping and Closed Graph Theorems 162 Graph of a Linear Operator and Closedness Property 162 Open Mapping Theorem 164 Closed Graph Theorem Examples 166 Exercises 168 Solutions to Exercises Weak Topologies, Weak Convergence and Reflexive Spaces Weak Topologies Weak Convergence Reflexive Banach Spaces 173 :

4 4.4 Weak Convergence in Hilbert Spaces Examples 176 Exercises 178 Solutions to Exercises 178 Contents xv 5. Differentiation and Integration in Normed Spaces Gateaux Derivative Frechet Derivative Subdifferential Integration in Normed Spaces 188 Exercises 189 Solutions to Exercises Fixed-Point Theorems and their Applications Banach Contraction Principle and Its Generalizations Schauder's Fixed-Point Theorem Applications of Banach Contraction Principle 196 Application to Matrix Equation 196 " Application to Differential Equations 201 Exercises 203 Solutions to Exercises Rudiments of Spectral Theory Spectral Properties of Bounded Linear Operators Compact Operators Spectral Properties of Self-Adjoint and Compact Operators 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 Examples 225 Exercises 226 Solutions to Exercises Boundary Value Problems Definition and Examples of Boundary Value Problems 230 Definition 230 Examples of BVPs Abstract Equations Sobolev Space 238 Examples of Distribution Certain Remarks Concerning the Solutions of BVPs 246

5 xvi Contents Exercises 249 Solutions to Exercises Optimization Minimization of Functionals Calculus of Variation and Linear Programming -255 Calculus of Variation 255 Linear Programming 257 Exercises 257 Solutions to Exercises Variational Inequalities Lions-Stampacchia Theory Physical Phenomena Represented by Variational Inequalities 265 Exercises 267 Solutions to Exercises The Finite-Element Method Approximate Problem Internal Approximation of//'(q) Finite Elements Application of the Finite-Element Method to Solve Boundary Value Problems 279 Practical Method to Compute a(wj, w,) Effect of Numerical Integration Abstract Error Estimate for the Nonconforming Finite-Element Method Abstract Error Estimation for Variational Inequalities 285 Exercises 286 Solutions to Exercises Optimal Control 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 Linear Quadratic Control Problem 291 Exercises 297 Solutions to Exercises Wavelets Recapitulation of Some Basic Concepts Continuous Wavelet Transform 300

6 13.3 Examples of Wavelets Decay of Continuous Wavelet Transform Multiresolution Analysis 310 Examples of Multiresolution Analysis 311 Important Properties of MRA Decomposition and Reconstruction Algorithms BestAf-TermApproximation 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