By V. I. Smirnov, A. J. Lohwater
ISBN-10: 1483197476
ISBN-13: 9781483197470
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Additional resources for A Course of Higher Mathematics. Integration and Functional Analysis
Example text
We have already investigated a selection principle for sets of continuous functions [IV; 15 and 16]. We now prove a theorem which gives us a selection principle for functions of bounded variation. THEOREM (Hetty). e. all the g(x) are bounded in absolute value and their variations over [a, b] are also bounded by some number. Now, from any infinite sequence gn(x) of functions belonging to the set 8, we can select a subsequence gnje(x) which tends to some function of bounded variation g(x) at every point of [a, 6].
Gt(ß) - 0Î(«) · (49) We use reductio ad absurdum. Suppose that the reverse inequality holds: γ [ V&g) + g(ß) - g(a)] > g*(ß) - gf(a). (50) 8] FUNCTIONS OP BOUNDED VAKIATION 29 We choose a subdivision δ of the interval [α, β] such t h a t the sum td is so close t o the variation Vßa(g) t h a t inequality (50) still remains in force when this total variation is replaced b y the sum in question. Thus we have, for some subdivision a = x0 < xx < . . : (51) On the other hand, we can evidently write so t h a t inequality (51) can be written as At least one of the terms on the left-hand side must be greater t h a n the corresponding term on the right.
With this choice of e m , (84) can be written as J § | Φ [O? xm (1 - o;) n - m ] | < ηφ. e. „(*)= 9n (*) : m=0 for 20[O^xm(l-x)"-m) n-i ^0[C^xm(l-x)n^m] for 0< x < —, n nf1] 1 2 —< x < —, 2 3 ^ x < — — < n n n - 1 (86) < £ < 1, 771 = 0 0„(1)= ^ Φ [ 0 ΐ ? α ; ' π ( 1 - ζ ) π - ' π ] . m=0 The total variation of gn(x) is evidently equal to the sum of the absolute values of the jumps of gn(x) at the points of subdivision and at the ends of the interval. By (85), we have V^(gn) < ηφ. Similarly, it follows at once from (85) and the definition of the function gn(x) that | gn(x) \ < η φ .