454 INDEX

Picarditerates, 223

piecewise definition of curves, 124piecewise smooth

differentiable, 131Plancherel theorem, 283, 284point of density, 233polar form complex number, 12poles, 403Polish space, 44polynomial, 274

degree, addition, multiply, equal,leading term, monic, division,divides, greatest commondivisor, 20

divides, 22greatest common divisor, 22greatest common divisor,

uniqueness, 22irreducible, 22irreducible factorization, 23relatively prime, 22

polynomialscanceling, 23factoring, 14factorization, 24

positive linear functional, 186positive part, 165precompact, 236primitive, 349principal part

Laurent series, 378product measure, 173product space

norm, 302projection in Hilbert space, 305

Radon measure, 189Radon Nikodym

theorem, 180Radon Nikodym derivative, 180Radon Nikodym theorem

Von Neumann, 317random variable

distribution measure, 144, 192real and imaginary parts, 165real part, 342reflexive Banach Space, 314

region, 258regularity

inner and outer, 151removable singularity, 384residue, 378, 385resolvent, 419resolvent set, 419, 439retract, 204

fixed point property, 204Riemann sphere, 367Riesz map, 307Riesz representation theorem

C (X), 334L1, finite measures, 326Lp, σ finite case, 327Lp, finite measures, 324Hilbert space, 306positive linear functionals, 188

ring, 371root test, 33Rouche’s theorem, 370, 405Runge, 418

Sard’s lemma, 246Schroder Bernstein theorem, 6Schwartz class, 286Schwarz lemma, 401, 410second mean value theorem, 134sectorial, 430, 431semigroup, 437

analytic, 437separable metric space

Lindeloff property, 44separated, 70separation theorem, 316sequence

Cauchy, 297subsequence, 39

sequentially compact, 45set notation, 3sigma algebra, 137sigma finite, 172signed measure, 178

Hahn decomposition, 178signed measures

Hahn decomposition, 178simple closed curve

orientation, 127