Functional analysis

Cover......Page 1
Title page......Page 2
Preface......Page 6
List of Symbols......Page 14
1.1 Basic definitions and properties......Page 16
1.2 Examples of normed spaces and related concepts......Page 21
1.3 Operators and functionals......Page 32
1.4 Convex functionals (real-valued convex functions on normed spaces)......Page 52
1.5 Topological properties of normed spaces......Page 58
1.6 Geometrical properties of normed spaces......Page 69
1.7 Some more exampIes......Page 70
2.1 Basic definition and properties......Page 97
2.2 Orthogonal complements and projection theorem......Page 109
2.3 Orthonormal systems and Fourier expansion......Page 115
2.4 Duality and reflexivity......Page 122
2.5 Operators in Hilbert space......Page 126
2.6 Bilinear forms and Lax-Milgram lemma......Page 139
2.7 Some more examples......Page 145
3.1 Extension form of the Hahn-Banach theorem and its consequences......Page 160
3.2 Geometric form of the Hahn-Banach theorem and its corol1aries......Page 168
3.3 Principle of uniform boundedness and its applications......Page 170
3.4 Open mapping and closed graph theorems......Page 174
3.5 Examples......Page 177
4.1 Weak topologies......Page 181
4.2 Weak convergence......Page 182
4.3 Reflexive Banach spaces......Page 184
4.4 Examples......Page 185
5.1 Gâteaux derivative......Page 189
5.2 Fréchet derivative......Page 193
5.3 Subdifferential......Page 195
5.4 Integration in normed spaces......Page 197
6.1 Ordered Banach spaces......Page 201
6.2 Banach lattice......Page 204
6.3 Ordered Hilbert spaces......Page 206
7.1 Banach contraction principle and its generalizations......Page 208
7.2 Schauder's fixed-point theorem......Page 212
7.3 Fixed-point theorems for ordered Banach spaces......Page 213
7.4 Applications of Banach contraction principle......Page 214
8.1 Definition and examples of boundary value problems......Page 227
8.2 Abstract equations......Page 233
8.3 Sobolev space......Page 235
8.4 Certain remarks concerning the solutions of BVPs......Page 242
9.1 Minimization of functionals......Page 247
9.2 Calculus of variation and linear programming......Page 251
10 Variational Inequalities......Page 256
10.1 Lions-Stampacchia theory......Page 257
10.2 Physical phenomena represented by variational inequalities......Page 260
11 The Finite-Element Method......Page 265
11.1 Approximate problem......Page 266
11.2 Internal approximation H¹( )......Page 269
11.3 Finite elements......Page 271
11.4 Application of the finite element method to solve boundary-value problem......Page 275
11.5 The effect of numerical integration......Page 277
11.7 Abstract error estimation for variational inequalities......Page 280
12 Optimal Control......Page 283
12.1 Illustration of the problem with the help of an example and formulation of the general problem......Page 284
12.2 Linear quadratic control problem......Page 286
Appendix A......Page 293
Appendix B......Page 296
Appendix C......Page 299
Appendix D......Page 303
Appendix E......Page 310
References......Page 312
Suggestions for further study......Page 319
Index......Page 320