Index(−∞,∞], 180Ck, 152C1, 150, 151C1 and differentiability, 150C∞

c , 250Cm

c , 250Fσ , 191Gδ , 191L1

approximation, 237L1

complex vector space, 208L1(Ω), 208L1

loc, 251∆, 337ε net, 70Fn, 91∇

2, 337π systems, 184σ algebra, 179

accumulation point, 63adjugate, 43algebra, 128analytic functions

real valued, 300zeros, 299

annulus, 298approximate identity, 250arcwise connected, 82

connected, 82, 131area measure

on manifold, 267area of a parallelogram, 329Ascolil Arzela theorem

general form, 79at most countable, 53axiom of choice, 49, 53

Hausdorff maximal principle, 62well ordering, 62

axiom of extension, 49axiom of specification, 49axiom of unions, 49

Baire theorem, 110Banach space, 98basis, 93basis, 339Bernstein polynomial, 120

approximation of derivative, 120

Binet Cauchy formula, 39block matrix, 9block multiplication, 9, 11Borel

measure, 190Borel Cantelli lemma, 183Borel measure

metric space, 190Borel sets, 235Borsuk, 315Borsuk Ulam theorem, 317Borsuk’s theorem, 313bounded linear maps

continuity, 114bounded set, 102box product, 332Brouwer fixed point theorem, 226, 312

Cantor function, 199Cantor set, 198Caratheodory’s procedure, 187Casorati Weierstrass theorem, 299Cauchy

integral formula, 296integral theorem, 296

Cauchy Schwarz inequality, 96Cauchy sequence, 65Cayley Hamilton theorem, 47chain, 60chain rule, 146change of variables

linear map, 242linear maps, 243map not one to one, 250

change of variables general case, 250characteristic polynomial, 47Clairaut’s theorem, 156closed disk, 102closed set, 64closed sets

limit points, 64closure of a set, 66, 67cofactor, 41cofactor identity, 159, 306column rank, 44compact set, 68compactness

closed interval, 101equivalent conditions, 70

357