INDEX 463
empty set, 3epsilon net, 227equality of mixed partial derivatives, 100equicontinuous, 228equivalence class, 8equivalence relation, 8equivalent norms, 65equivalent parametrizations, 130essentially bounded, 232extended complex plane, 374
separated sets, 374extending nonnegative functionals, 173extreme values theorem, 55
Fatou’s lemma, 168finite Borel measure
regular, 196finite dimensional normed linear space
compactness, 64complete, 64extreme values, 64
finite intersection propertycompact sets, 51
first mean value theorem integrals, 136fixed point property, 208fixed points contraction mapping, 224Fourier series, 314
convergence, 324uniform convergence, 322
Fourier transform, 274L1, 287L2, 289continuous, 287convolution, 288, 294of functions in G, 282convolution, 292general definition, 284in G ∗, 284inverse, 274, 284, 287polynomial growth, 286
fractional linear transformationmaps upper half plane to unit ball,
418fractional linear transformations, 416
maps unit ball to unit ball, 419Frechet derivative, 94Fresnel integrals, 401Fubini’s theorem, 177
function, 5functions of a linear operator, 434fundamental matrix, 410fundamental theorem of algebra, 14, 26,
56fundamental theorem of algebra
plausibility argument, 27fundamental theorem of calculus, 377
Radon measures, 235
G delta, 198Gamma function, 239gamma function, 137
analytic, 369existence and convergence, 271factorial, 271properties, 271
gauge function, 315general spherical coordinates, 254Gram Schmidt process, 18graph of a linear map, 308greatest common divisor, 21
description, 22Green’s theorem, 265
Jordan curve of finite length, 457
Hahndecomposition, 182, 343
Hahn Banach theorem, 316complex version, 317
Hahn Jordan decomposition, 343Hamel basis, 11Hardy’s inequality, 240Hausdorff
maximal principle, 9, 313Heine Borel theorem, 50higher order terms, 368Hilbert Space, 78Hilbert space, 310Holder’s inequality, 229
imaginary part, 348implicit function theorem, 104, 107, 108indicator function, 144infinitely differentiable function, 257initial value problem, 226inner product axioms, 16inner product space, 310inner regular, 192