An Introduction to Nonlinear Analysis Martin Schechter Index More Information

Cambridge University Press 0521843979 - An Introduction to Nonlinear Analysis Martin Schechter Index More information

Index

(f,g), 315 brachistochrone, 145, 151 (u, v), 1 Brouwer degree, 171 (u, v)H ,2 B(X, Y ), 320 calculus of variations, 145 co(M), 179 Carathéodory function, 20, 338 i(A), 330 Cauchy sequence, 314, 341 Jx, 328 Cauchy–Schwarz inequality, 316 K(X), 320 characteristic function, 336 K(X, Y ), 320 closed graph theorem, 322 L1, 334 closed operator, 322 Lp, 335 closed set, 314, 342 N(A), 322 closure, 343 R(A), 321 compact map, 179 W m,p(Q), 306 compact operator, 320 X, 317 compact set, 177, 325, 342 α(A), 330 compact support, 90 β(A), 330 complete metric space, 341 dim V , 326 completeness, 314 2, 175 conjugate operator, 321 co(M), 179 conjugate space, 317 Φ(X, Y ), 330 continuous, 342 , 327 contraction, 137 contraction mapping principle, a.e., 331 66 absolutely continuous, 338 convex, 125 adjoint operator, 321 almost everywhere, 331 convex hull, 179 Arzelà–AscoliTheorem, 343 convex set, 318 cover, 344, 346 Banach fixed point theorem, 66 critical point theory, 1 Banach space, 314 cuboid, 331 Banach–Steinhaus theorem, 322 cycloid, 148 bang–bang control, 215 basis, 327 Dancer–Fuˇc´ık spectrum, 245 Beppo-Levi theorem, 334 degree, 171 bounded functional, 317 dense subset, 177, 325 bounded inverse theorem, 323 derivative, 5 bounded linear functional, 317 differentiable, 6 bounded operator, 319 dimension, 326 bounded set, 325, 342 domain, 319 box, 331 dual space, 317

355

356 Index

Egoroff theorem, 337 locally finite, 344, 346 eigenelement, 321 lower semi-continuous, 126 eigenfunctions, 264 Lusin theorem, 337 eigenspace, 321 measurable function, 335 eigenvalue, 321 measurable set, 336 eigenvalues, 72, 264 measure, 331 eigenvector, 321 measure zero, 331 equicontinuous, 343 metric, 341 equivalent norms, 327 metric space, 341 Euler equations, 151 minimizing sequence, 24 Euler’s equation, 149 extrema, 1 negative part of a function, 332 Newton’s equation of motion, Fatou lemma, 334 149 fixed point, 173 nonexpansive, 137 fixed point property, 173 nonexpansive operator, 137 Fourier series, 17 norm, 314 Fréchet Derivative, 269 norm of an operator, 319 Fubini theorem, 336 normed vector space, 314 functional, 3, 316 null space, 322 geometric Hahn–Banach Theorem, open cover, 344, 346 318 open covering, 342 Hahn–Banach theorem, 317 open set, 342 Hamilton’s principle, 148 operator, 319 Hilbert cube, 175 Palais–Smale condition, 40 Hilbert space, 316 Palais–Smale sequence, 39, 47 homeomorphic, 172 para-compact, 344, 346 homeomorphism, 172 parallelepiped, 331 homotopic, 172 partial derivative, 133 homotopy, 172 Peano’s theorem, 201 Hölderinequality, 336 pendulum, 149 iff=if, and only if, 132 Picard’s theorem, 66 index, 330 point-wise convergence, 342 infinite dimensional, 326 positive part of a function, 332 inner product, 315 potential energy, 148 integrable, 333 PS condition, 40 integrable function, 332 PS sequence, 40, 47 integral, 334 pseudo-gradients, 345 interior point, 342 range, 321 inverse operator, 323 refinement, 344, 346 iso-perimetric, 207 renamed subsequence, 9 jumping nonlinearities, 245 retraction, 180 Riesz representation theorem, kinetic energy, 148 317 l.s.c., 126 saddle point, 124 Lagrange multiplier rule, 226 Sard’s theorem, 184 Lagrangian, 149 scalar product, 315 Lebesgue dominated convergence Schauder’s fixed point theorem, theorem, 335 178 Leray–Schauder degree, 200 self-adjoint operators, 323 linear operator, 319 semi-Fredholm operators, 330 linear space, 314 separable, 177, 325 linear span, 325 simple harmonic motion, 150 linearly independent vectors, 326 step function, 331

Index 357

strictly convex, 125 u.s.c., 125 strong convergence, 328 uniformly continuous, 342 subspace, 314 upper semi-continuous, 125 summable, 333 support, 191 vector space, 314 vectors, 313 test functions, 90 transformation, 319 weak derivative, 11 triangle inequality, 316 weakly closed, 127