2748 INDEX

Schauder fixed pointapproximate fixed point, 499, 1552

Schauder fixed point theorem, 500, 504measurability, 1553degree theory, 796

Schottky’s theorem, 1759Schroder Bernstein theorem, 33Schwartz class, 1301Schwarz formula, 1627Schwarz reflection principle, 1650Schwarz’s lemma, 1743second derivative, 721sections of open sets, 723sectorial, 1703self adjoint, 90, 560semigroup, 566

adjoint, 590contraction

bounded, 577generator, 576growth estimate, 576Hille Yosida theorem, 580strongly continuous, 576

seminorms, 479separability of C(H), 1992separable metric space

Lindeloff property, 143separated, 168separation theorem, 476sequence, 138

Cauchy, 140subsequence, 139

sequential compactness, 37, 636L1, 635L1, 635

sequential compactness in L1, 635sequential weak* compactness, 463sequentially compact set, 113set valued functions

measurability, 1533, 2391set valued map

locally bounded, 884sets, 29sgn, 66

uniqueness, 68

Shannon sampling theorem, 1118sigma finite, 261sign of a permutation, 68Simon, 1443simple function, 238, 643simple functions, 227singular values, 754Skorokhod’s theorem, 1996Sobolev Space

embedding theorem, 1117equivalent norms, 1117

Sobolev space, 1265Sobolev spaces, 1117space of continuous martingales, 2110

Hilbert space, 2135span, 61, 72, 178spectral radius, 1700spectral theory

Banach space, 468compact operator, 468

Sperner’s lemma, 212Steiner symetrization, 1000step functions

approximation result, 2233stereographic projection, 1599, 1760Stirling’s formula, 1811stochastic integral, 2229

stochastically square integrable, 2255stochastic integral , 2256

definition, 2243linear transformation, 2261main result, 2243quadratic variation, 2258

stochastic process, 2049stochastically square integrable, 2255Stokes theorem, 1072Stone’s theorem, 423stong law of large numbers, 1977stong topology, 487stopped martingale, 2118, 2120stopped process, 2107stopped submartingale, 2098, 2120stopping time, 1954, 1960, 2072strict convexity, 477strictly convex, 832