920 INDEX

inverse, 30product space

norm, 538product topology, 504products of Borel sets, 259, 322progressively measurable, 817progressively measurable

integral of, 819progressively measurable version, 821projection in Hilbert space, 577Prokhorov’s theorem, 770

quadratic variationconvergence in probability, 904fantastic properties, 887

Rademacher’s theorem, 441Radon measure, 253, 263, 362, 363Radon Nikodym

Radon measures, 343theorem, 302

Radon Nikodym derivative, 302Radon Nikodym property, 680Radon Nikodym theorem

Radon Measures, 343random variable, 715

distribution measure, 256random vector, 715

independent, 736rank, 35rank of a matrix, 50rank theorem, 221rational function, 130real and imaginary parts, 284recognizing a martingale

stopping times, 859refinement of a cover, 705reflexive Banach Space, 546reflexive Banach space, 631

weak compactness, 560regular, 253regular measure space, 362regular topological space, 502regular values, 410resolvent, 605retract, 329

Banach space, 712closed and convex set, 712

fixed point property, 329reverse sub-martingale, 802Reynolds

transport formula, 494Riemann integral, Lebesgue integral, 294Riesz F., 374Riesz map, 579Riesz representation theorem, 684

Hilbert space, 579locally compact Hausdorff space, 516

Riesz Representation theoremC (X), 642

Riesz representation theorem Lp

σ finite case, 630, 687finite measures, 626

Riesz representation theorem for L1

finite measures, 629right inverse, 37–39right polar factorization, 21row equivalent, 33row operations, 24row operations

linear relationships, 31row rank, 50row reduced echelon form, 32row reduced echelon form

existence, 33uniqueness, 34

Russell’s paradox, 59

saddle point, 147Sard’s theorem, 332, 412scalars, 99Schaefer fixed point theorem, 165Schauder fixed point

approximate fixed point, 164Schauder fixed point theorem, 165, 166Schroder Bernstein theorem, 60Schur’s theorem, 19Schwartz class, 386second derivative, 193second derivative test, 219, 220sections of open sets, 196self adjoint, 595semigroup, 600

adjoint, 614contraction

bounded, 603