462 INDEX

compact set, 47compact sets

intersection, 51compact support, 259compactness

equivalent conditions, 48complete, 224completely separable, 45completion of Borel measure, 198complex measure

Radon Nikodym, 328total variation, 328

complex numbersroots, 13

conjugateof product, 35

connected, 72open balls, 74

connected component, 73connected components, 73

equivalence class, 73equivalence relation, 73open sets, 74

connected setcontinuous function, 75continuous image, 73

connected setsintersection, 72intervals, 73real line, 73union, 72

conservative, 128continuity

uniform, 55continuous

not differentiable anywhere, 61continuous function, 52

maximum and minimum, 55continuous functions

dense, 256equivalent conditions, 52

continuous image of compact set, 55contour integral, 351contraction mapping

fixed points, 224new norm, 223

convergencein measure, 186

infinite series, 60pointwise, 57uniform, 57, 224

convexset, 310

convexfunctions, 239

convolution, 267, 295countable, 7countable basis, 45counting measure, 145counting zeros, 377counting zeros and poles, 415counting zeros of analytic function, 377cycle, 387

around compact sets, 391

De Moivre’s theorem, 12density of G in Lp, 281derivative

chain rule, 95continuous, 94Frechet, 94matrix, 96well defined, 94

derivatives, 94differentiable, 94

continuous, 94differentiable function

measurable sets, 243sets of measure zero, 243

differential equationsPeano existence theorem, 240

dilations, 416direct sum, 438distance to a set, 46dominated convergence

generalization, 175dominated convergence theorem, 174double sums

changing order, 28dual space, 317duality maps, 323Dynkin’s lemma, 146

Eggoroff theorem, 178, 220elliptic functions, 426embedding into its double dual space,

319