914 INDEX

continuity set, 777continuous function, 80, 503

maximum and minimum, 82continuous functions

compact support, 290equivalent conditions, 80

continuous image of compact set, 81continuous martingale

not of bounded variation, 863contraction map, 82

fixed point, 208fixed point theorem, 82

convergence in measure, 310convex, 711

set, 576sets, 145

convexfunctions, 372

convex combination, 120convex function

continuous, 741convex hull, 120, 155, 711convolution, 366, 388convolution of measures, 755coordinate map, 111coordinates, 155countable, 61covariance matrix, 759covariation, 871cowlicks, 425Cramer’s rule, 49cylindrical set, 725, 749

definition of Lp, 359definition of a Ck function, 194density of G in Lp, 376density of continuous functions in Lp, 363derivative

chain rule, 188continuity, 193continuity of Gateaux derivative, 193continuous, 187continuous Gateaux derivatives, 191Frechet, 186Gateaux, 189, 191generalized partial, 196higher order, 193matrix, 189

partial, 196second, 193well defined, 186

derivative of inverse, 230derivatives, 186determinant

definition, 41expansion along row, column, 47matrix inverse, 48permutation of rows, 42product, 45row, column operations, 43symmetric definition, 43transpose, 43

diameter of a set, 96differentiable, 186

continuous, 187continuous partials, 197

differentiable map of Lebesgue measurableset, 332

differential equationsdependence on data, 562global existence, 166

differential formsgeneralalities, 472

differentiationRadon measures, 341

differentiation almost everywheremonotone function, 267

dimension of a vector space, 102Dini derivates, 265directional derivative, 191distance, 71distribution, 443, 715distribution function, 297divergence theorem, 402dominated convergence

generalization, 290dominated convergence theorem, 289, 673Doob Dynkin lemma, 723Doob estimate, 793Doob’s sub-martingale estimate, 826dot product, 103dual space, 543duality maps, 572dyadics, 125Dynkin’s lemma, 243

alternative conditions, 269