INDEX 2743

inverse function theorem, 130, 132, 734, 750higher order derivatives, 735

inverses and determinants, 75inversions, 1738invertible maps, 728

different spaces, 730isodiametric inequality, 998, 1002isogonal, 1618, 1737isolated singularity, 1636isometric, 682Ito isometry, 2142, 2231Ito representation theorem, 2306

James map, 450Jensen’s formula, 1801Jensens inequality, 417, 1948Jordan curve theorem, 776, 781, 804Jordan separation theorem, 778

Kakutani fixed point theorem, 846, 1565Kolmogorov Centsov theorem, 2053Kolmogorov extension theorem, 393, 396

Polish Space, 1862Kolmogorov zero one law, 1874Kolmogorov’s inequality, 1876, 1877Kuratowski theorem, 1548

Lagrange multipliers, 132, 133Laplace expansion, 74Laplace transform, 1116Laurent series, 1683law, 2006Lebesgue

set, 937Lebesgue decomposition, 597, 1095Lebesgue measurable function

approximation with Borel measurable,263

Lebesgue measure, 339approximation with Borel sets, 263properties, 263

Lebesgue point, 935Leray Schauder alternative, 500Leray Schauder degree, 793

properties, 794Levy’s theorem, 2176, 2225

limiit point, 137limit

continuity, 710infinite limits, 707

limit of a function, 101, 707limit of a sequence, 138

well defined, 138limit point, 165, 707limit points, 101limits

combinations of functions, 708independent, 1875

limits and continuity, 710Lindeloff property, 143linear combination, 61, 72, 178linear independence, 182linearly dependent, 61, 179linearly independent, 61, 179linearly independent set

enlarging to a basis, 182Lions, 1445Liouville theorem, 1631Lipschitz, 38, 100, 109

functions, 946Lipschitz boundary, 1024Lipschitz continuous, 158Lipschitz function

integral of its derivative, 949Lipschitz manifold, 1257Lipschitz maps

extension, 955, 960, 1009, 1251little o notation, 711little Picard theorem, 1850local martingale, 2120local martingale

stochastic integral, 2256local submartingale, 2120localization

stochastically square integrable, 2255localization

elementary functions, 2251general case, 2253

localizing sequence, 2120locally bounded, 884locally compact, 149