450 INDEX

compact and sequentially compact, 48compact set, 45compact sets

intersection, 49compact support, 255compactness

equivalent conditions, 46complete, 220completely separable, 43completion of Borel measure, 194complex measure

Radon Nikodym, 322total variation, 322

complex numbersroots, 13

conjugateof product, 35

connected, 70open balls, 72

connected component, 71connected components, 71

equivalence class, 71equivalence relation, 71open sets, 72

connected setcontinuous function, 73continuous image, 71

connected setsintersection, 70intervals, 71real line, 71union, 70

conservative, 126continuity

uniform, 53continuous

not differentiable anywhere, 59continuous function, 50

maximum and minimum, 53continuous functions

dense, 252equivalent conditions, 50

continuous image of compact set, 53contour integral, 345contraction mapping

fixed points, 220new norm, 219

convergence

in measure, 182infinite series, 58pointwise, 55uniform, 55, 220

convexset, 304

convexfunctions, 235

convolution, 263, 288countable, 7countable basis, 43counting measure, 141counting zeros, 362, 370counting zeros and poles, 405curves

piecewise smooth, 131cycle, 379

around compact sets, 383

De Moivre’s theorem, 13derivative

chain rule, 93continuous, 92Frechet, 92matrix, 94well defined, 92

derivatives, 92differentiable, 92

continuous, 92differentiable function

measurable sets, 239sets of measure zero, 239

differential equationsPeano existence theorem, 236

dilations, 406direct sum, 427distance to a set, 44dominated convergence

generalization, 171dominated convergence theorem, 170double sums

changing order, 28dual space, 311duality maps, 317Dynkin’s lemma, 142

Eggoroff theorem, 174, 216elliptic functions, 417