INDEX 453

matrix of linear transformation, 64maximal chain, 9maximal function

Radon measures, 231maximum modulus theorem, 376mean value inequality, 78, 119measurability

limit of simple functions, 140measurable, 146

function, 138measurable complex functions

simple functions, 168measurable function

a.e. Borel, 144measurable sets, 146measure, 141

Borel, 150counting, 141finite, 141metric space, 150probability, 141properties, 141

measuresabsolutely continuous, 319decreasing sequences of sets, 141increasing sequences of sets, 141

measures from outer measures, 147Mellen transform, 394meromorphic, 385, 403

on extended complex plane, 405meromorphic functions

a field, 404meromorphic on extended complex

planerational function, 415

metric spacecompact sets, 46complete, 42completely separable, 43completion, 74open set, 38separable, 43

Minkowski functional, 316Minkowski inequality

integrals, 233Mittag Leffler, 391mixed partial derivatives, 97modular group, 417

mollifier, 253monotone convergence theorem, 162Montel’s theorem, 411multi-index, 274

negative part, 165nested interval lemma, 31, 32normal family of functions, 412nowhere differentiable function, 60nowhere differentiable functions, 314numerical range, 439

open ball, 38open set, 38

open cover, 44open mapping theorem, 300, 373open sets, 38

countable basis, 43intersection, 40union, 40

operatorsectorial, 430

operator norm, 65, 68order of a pole, 385order of a zero, 361ordered

partial, 9totally ordered, 9

orientation, 127simple closed curve, 128

oriented curve, 127orthonormal, 18orthonormal set, 307

maximal, 307outer measure

measurable, 146outer measure on R, 145outer regular, 189

parallelogram identity, 17partial derivatives, 94partial fractions, 25

unique, 26partial order, 9, 307partially ordered set, 9, 198partition of unity, 187periodic functions, 416

Jacobi, 416pi systems, 142