INDEX 359

dual basis, 344dyadics, 113Dynkin’s lemma, 184

alternative conditions, 199

Eggoroff theorem, 213eigenvalues, 47Einstein summation convention, 335elementary matrices, 18equality of mixed partial derivatives, 158equi-integrable, 215, 216equicontinuous, 78equivalence class, 55equivalence relation, 54essential singularity, 298exchange theorem, 93extensions of real functions, 299extreme values theorem, 74

F sigmaset, 191

Faddeyev, 231Fatou’s lemma, 206Fick’s law, 276finding the inverse, 31finite measure

regularity, 236fixed point property, 226, 326Frechet derivative, 144Fubini’s theorem, 223

general product measures, 222function, 52functions

measurable, 179fundamental theorem

algebra, 297fundamental theorem of calculus

Radon measures, 239

G delta, 191gamma function, 162Gateaux derivative, 147, 149

continuous, 151Gauss Jordan method for inverses, 31general spherical coordinates, 255gradient

contravariant components, 351covariant components, 351

Gram Schmidt process, 101

Gram Schmidt process., 100Green’s theorem, 287, 294

area, 288

Hahndecomposition, 218

Hamel basis, 109harmonic, 286Hausdorff

maximal principle, 60Hausdorff max. Principle

Zorn’s lemma, 62Hausdorff maximal principle, 62Hermitian

diagonalization, 14non-defective, 14

Hermitian matrix, 14Hessian matrix, 171higher order derivative

multilinear form, 152higher order derivatives, 152Holder’s inequality, 98homeomorphism, 73

implicit function theorem, 164, 167, 168index

lowering, 341raising, 341

indicator functionapproximation, 237

inner regular, 191compact sets, 191

integralcontinuous function, 125decreasing function, 201functions in L1, 208linear, 208

integral over a measurable set, 212integrals

iterated, 127interior point, 63intermediate value theorem, 81invariance of domain, 229, 316inverse, 23, 31inverse

left right, 32product of matrices, 24row reduced echelon form, 33