INDEX 451

embedding into its double dual space,313

empty set, 3epsilon net, 223equality of mixed partial derivatives, 98equicontinuous, 224equivalence class, 8equivalence relation, 8equivalent norms, 63equivalent parametrizations, 127essentially bounded, 228extended complex plane, 367, 368

separated sets, 368extending nonnegative functionals, 169extreme values theorem, 53

Fatou’s lemma, 164finite Borel measure

regular, 192finite dimensional normed linear space

compactness, 62complete, 62extreme values, 62

finite intersection propertycompact sets, 49

first mean value theorem integrals, 133fixed point property, 204fixed points contraction mapping, 220Fourier series, 308

convergence, 318uniform convergence, 316

Fourier transform, 268L1, 280L2, 283continuous, 281convolution, 282, 288of functions in G, 275convolution, 286general definition, 278in G ∗, 278inverse, 268, 278, 281polynomial growth, 280

fractional linear transformationmaps upper half plane to unit ball,

408fractional linear transformations, 406

maps unit ball to unit ball, 409Frechet derivative, 92

Fresnel integrals, 390Fubini’s theorem, 173function, 5functions of a linear operator, 424fundamental matrix, 399fundamental theorem of algebra, 14, 26,

54fundamental theorem of algebra

plausibility argument, 27fundamental theorem of calculus, 370

Radon measures, 231

G delta, 194Gamma function, 235gamma function, 134

analytic, 363existence and convergence, 265factorial, 265properties, 265

gauge function, 309general spherical coordinates, 250Gram Schmidt process, 19graph of a linear map, 302greatest common divisor, 22

description, 22Green’s theorem, 261

Jordan curve of finite length, 445

Hahndecomposition, 178, 337

Hahn Banach theorem, 310complex version, 311

Hahn Jordan decomposition, 337Hamel basis, 12Hardy’s inequality, 236Hausdorff

maximal principle, 9, 307Heine Borel theorem, 48higher order terms, 361Hilbert Space, 77Hilbert space, 304Holder’s inequality, 225

imaginary part, 342implicit function theorem, 102, 105, 106indicator function, 140infinitely differentiable function, 253initial value problem, 222inner product axioms, 17