INDEX 455

simple curve, 72, 113, 127simple functions, 140simply connected, 368smooth functions

dense, 253Sobolev Space, 291

embedding theorem, 291equivalent norms, 291interpolation inequality, 293trace, 293

spectral mapping theorem, 426spectral radius, 422spectrum, 419spt, 186square root property, 410

simply connected, 410star center, 355star shaped, 355stereographic projection, 368Stirling’s formula, 135, 264Stone Weierstrass theorem, 85sup

changing order, 28support

function, 255

Tietze extension theorem, 61total variation, 320totally bounded, 46totally ordered, 9, 198translation invariance, 152, 197translations, 406

uniform boundedness, 299

uniform boundedness theorem, 299uniform continuity, 53uniform continuity and compactness, 54uniform convergence

infinite sums, 58uniform norm, 55, 220uniformly bounded, 224uniformly integrable, 175, 182union, 3

variation of constants formula, 400variational inequality, 305vector measures, 319Vitali

convergence theorem, 175Vitali convergence theorem, 177Vitali covering, 200

weak convergence, 317Weierstrass

approximation estimate, 79M test, 16, 59Stone Weierstrass theorem, 81

well ordered sets, 10

Young’s inequality, 336

zeromultiplicity, 35

zeros of analytic functioncounting them, 370

zeta functiondefinition, 364