916 INDEX
Hahn decomposition, 645Hahn Jordan decomposition, 645Hamel basis, 121Hardy’s inequality, 372Haursdorff measures, 445Hausdorff
maximal principle, 68Hausdorff and Lebesgue measure, 452Hausdorff dimension, 452Hausdorff maximal principle, 507, 540Hausdorff measure
set of measure 0, 453Hausdorff measures, 446Hausdorff space, 502Hermitian
diagonalization, 20non-defective, 20
Hermitian matrix, 20Hessian matrix, 219higher order derivative
multilinear form, 193higher order derivatives, 193
implicit function theorem, 212inverse function theorem, 212
Hilbert Schmidtoperator, 592
Hilbert Schmidt theorem, 586, 665Hilbert space, 575hitting this before that, 853Holder inequality
backwards, 546Holder space
not separable, 567Holder spaces, 567Holder’s inequality, 106, 357homeomorphism, 81
implicit function theorem, 208higher order derivatives, 212
increasing functionexistence of the derivative, 437
independent, 313independent events, 720independent random vectors, 721independent sigma algebras, 721indicator function
approximation, 315inner product space, 575
inner regular, 253, 717compact sets, 253
inner regularity, 257Integral
Riemann and Lebesgue, 531integral
continuous function, 138decreasing function, 279functions in L1, 285linear, 285
integral over a measurable set, 290integrals
iterated, 141integration
with respect to a martingale, 883integration by parts, 323integration with respect to martingales
Ito isometry, 888interior point, 71intermediate value theorem, 89invariance of domain, 230, 424inverse, 29, 37inverse
left right, 38product of matrices, 30row reduced echelon form, 39
inverse function theorem, 211, 232higher order derivatives, 212
inverse image, 59inverses and determinants, 48invertible, 29invertible maps, 205
different spaces, 206isodiametric inequality, 449, 451isometric, 682iterated integrals, 141Ito isometry, 888
James map, 545Jensen’s inequality, 741Jensens inequality, 784Jordan curve theorem, 430Jordan separation theorem, 431
Kakutani fixed point theorem, 169Kantorovitch, 635Kolmogorov Centsov theorem, 810, 813Kolmogorov extension theorem, 527, 719