Index(−∞,∞], 238Ck, 193C1, 192C1 and differentiability, 192C∞

c , 366Cm

c , 366Fσ , 253Gδ , 534Gδ , 253L1

approximation, 315weak compactness, 631

Lp

definition, 359density of continuous functions, 363density of simple functions, 362density of smooth functions, 368norm, 359separability, 363

Lp

reflexive, 631weak compactness, 631

L1

complex vector space, 286L1(Ω), 285L∞, 361Lp

compactness, 371continuity of translation, 365

Lp(Ω), 357Lp (Ω;X), 671L1

loc, 366X×Y

norm, 538ε net, 78Fn, 99D∗, 443G , 375L (X ,Y ), 535

Banach space, 535π systems, 243σ algebra, 237

a-priori estimates, 167absolutely continuous

existence of derivative, 346function, 345function and measure, 345integral of derivative, 349

integral of the derivative, 346Lipschitz, 347measure, 345

accumulation point, 71adapted, 786, 817adjoint linear map, 544adjugate, 49Alexander subbasis theorem, 507algebra, 141

Cartesian product, 524measure on algebra, 524recognizing one, 523

algebra of sets, 523approximate identity, 366apriori estimates, 167arcwise connected, 90

connected, 90, 144area measure

on manifold, 399arithmetic mean, 233Arzela Ascoli theorem

Banach space, 690Ascolil Arzela theorem

general form, 87at most countable, 61axiom of choice, 57, 61axiom of extension, 57axiom of specification, 57axiom of unions, 57

backwards Holder inequality, 373backwards Minkowski inequality, 374Baire

category, 533, 534Baire theorem, 122Banach

space, 533Banach Alaoglu theorem, 556Banach space, 106, 205, 359, 533, 711Banach Steinhaus theorem, 536barycenter, 157basis, 101basis of a topology, 501Bernstein polynomial

approximation of derivative, 133Besicovitch

covering theorem, 119, 317Besicovitch covering theorem, 263Bessel’s inequality, 585, 619

912