922 INDEX

trace, 591trajectories, 807translation invariant, 258triangle inequality, 106, 358triangulated, 155triangulation, 155trivial, 100

uniform boundedness, 536uniform boundedness theorem, 536uniform continuity, 82uniform continuity and compactness, 82uniform convergence, 85uniform convergence and continuity, 85uniform convexity, 546, 572uniformly bounded, 690uniformly convex, 546uniformly integrable, 292, 372uniqueness of limits, 183upcrossing, 741, 787upper semicontinuity

set valued map, 167upper semicontinuous, 97upper semicontinuous composition, 168Urysohn’s lemma, 505, 506

variance, 759variational inequality, 577vector

measure, 621vector measures, 679vector space

dimension, 102vector space axioms, 99vector valued function

limit theorems, 183vectors, 99version, 807Vitali

convergence theorem, 292Vitali

convergence theorem, 673Vitali convergence theorem, 293Vitali cover, 263Vitali covering theorem, 114volume of unit ball, 452

weak and weak* topologies, 554weak convergence, 573

measures, 770weak convergence of measures, 772weak derivative, 444weak topology, 555weak∗ measurable, 653weak* topology, 555

metric space, 557weakly measurable, 647wedge product, 475

algebraic properties, 476Weierstrass

Stone Weierstrass theorem, 143Weierstrass approximation

estimate, 130well ordered sets, 70

Young’s inequality, 644