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