Index

C1, 96Ck, 96C∞

c , 252Cm

c , 252Cc(Ω), 186Fσ , 152, 194Gδ , 152, 194Lp

continuity of translation, 255separable, 256

L1(Ω), 166Lp

compactness, 262X×Y

norm, 302∩, 3∪, 3ε net, 46G , 274L (X ,Y ), 69

Banach space, 69π systems, 142σ algebra, 137

epsilon net, 223

a.e., 153accumulation point, 38adjoint, 76adjoint linear map, 312algebra, 81almost everywhere, 153alternative norm, 221analytic functions

counting zeros, 362integral domain, 371uniform convergence, 371

analytic mappingspreservation of angles, 370

annulus, 376approximate identity, 253approximation

Gδ and Fσ , 194arc, 72

Jordan, 72arcwise connected, 72

connected, 72Arzela Ascoli theorem, 224

at most countable, 7axiom of choice, 6

Bairecategory, 298, 299

Banach fixed point theorem, 220Banach space, 55, 219Banach Steinhaus theorem, 299beta function, 364binomial theorem, 34border regions, 441Borel measure

regular, 192bounded continuous linear functions, 68bounded variation, 113branch cut, 392Bromwich integral, 394Brouwer fixed point theorem, 305Browder’s lemma, 217

Cantor function, 156Cantor set, 156Caratheodory’s procedure, 147Casorati Weierstrass theorem, 385Cauchy principal value, 398Cauchy Riemann equations, 342Cauchy Schwarz

inequality, 17Cauchy sequence, 297

bounded, 42convergence, 43convergence of subsequence, 41

chain, 9chain rule, 93change of variables

map not one to one, 249change of variables general case, 249characteristic function

of a measure, 335closed graph theorem, 303closed set, 39closed sets

intersection, 40limit points, 39union, 40

closure of a set, 41limit points, 41

cofactor identity, 99

449