466 INDEX

poles, 413Polish space, 46polynomial, 280

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

divides, 21greatest common divisor, 21greatest common divisor,

uniqueness, 21irreducible, 21irreducible factorization, 22relatively prime, 21

polynomialscanceling, 23factoring, 14factorization, 23

positive linear functional, 190positive part, 169precompact, 240primitive, 355principal part

Laurent series, 387product measure, 177product space

norm, 308projection in Hilbert space, 311

Radon measure, 192Radon Nikodym

theorem, 184Radon Nikodym derivative, 184Radon Nikodym theorem

Von Neumann, 323random variable

distribution measure, 148, 196real and imaginary parts, 169real part, 348reflexive Banach Space, 320region, 262regularity

inner and outer, 156removable singularity, 393residue, 394resolvent, 429resolvent identity, 430resolvent set, 429, 450

retract, 208fixed point property, 208

Riemann sphere, 374Riesz map, 313Riesz representation theorem

C (X), 340L1, finite measures, 332Lp, σ finite case, 333Lp, finite measures, 330Hilbert space, 312positive linear functionals, 192

ring, 380root test, 34Rouche’s theorem, 377, 415

Sard’s lemma, 250Schroder Bernstein theorem, 6Schwartz class, 293Schwarz lemma, 379, 419second mean value theorem, 136sectorial, 442semigroup, 448

analytic, 448separable metric space

Lindeloff property, 46separated, 72separation theorem, 322sequence

Cauchy, 303subsequence, 41

sequentially compact, 47set notation, 3sigma algebra, 141sigma finite, 176signed measure, 182

Hahn decomposition, 182signed measures

Hahn decomposition, 182simple closed curve

definition, 130orientation, 131

simple curve, 74, 115, 130simple functions, 144simply connected, 374smooth functions

dense, 257Sobolev Space, 298

embedding theorem, 298