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**Example text**

37) and the total variation 01 I on I as Vr(I) = sup Vr(p) . 38) The function I is said to be of bounded variation on I as Vr(I) is finite. We recall that monotonic functions and differentiable functions with bounded derivatives on I are of bounded variation on I. In the first case, Vr(I) = I/(b) - l(a)1 and, in the second, Vr(I) if If' (t)1 ~ c(b - a) 1. Furthermore, if pi is a refinement of p, then ~ c for all t Vr(p') ~ and, if a ~ c Vr([a, bD E Vr(p) ~ b and I is of bounded variation on [a, b], = Vr([a, cD + Vr([c, bD .

Definition. s. R-S integrable on [a, b] with respect to X. s. s. integral on [a, b] of f with respect to X, and is denoted by b fa f(t) dX(t) . 20) Dermition. s. R-S integrable on [a, b] with respect to f. s. s. integral on [a, b] of X with respect to f, and is denoted by b fX(t)df(t) . 20, 21) are elements of L2 (D). 2. 16. , all Cauchy sequences involved in the definitions have one and the same limit in L 2 (D). s. integrable on [a, b] with respect to X, (f) if and only if there is an element in L2 (D), denoted by [f(t)dX(t) , ([ X(t)df(t») 42 2.

If Xl> X 2, ... , Xn in S are mutually orthogonal, we obtain the Pythagorean law «Xl + . + X n), (Xl + . + Xn» = (Xl' Xl) + . . + (Xn> Xn) . 57) Cauchy, Bunyakowsky, and Schwarz Inequality. : 0. 58) Norm. Let (X, X) 112 = Ilxll . ; 0 ; Ilxll = 0 ¢> X ) =0. 58) Ilx + YI1 2 = EX2 + 2E{XY} + Ey2 ~ IIXW + 211XIIIIYII + IIYI1 2 = {IIXII + II YI1}2 . v. X in S, II xl I has all the norm properties when the equivalence classes are taken into account. 61) when Xl> X 2, . . , Xn are mutually orthogonal.