# Download PDF by V. I. Smirnov, A. J. Lohwater: A Course of Higher Mathematics. Integration and Functional

By V. I. Smirnov, A. J. Lohwater

ISBN-10: 1483197476

ISBN-13: 9781483197470

Best mathematics_1 books

Read e-book online The Oxford handbook of generality in mathematics and the PDF

Generality is a key worth in medical discourses and practices. all through background, it has obtained various meanings and of makes use of. This selection of unique essays goals to inquire into this variety. via case experiences taken from the heritage of arithmetic, physics and the existence sciences, the ebook offers proof of alternative methods of figuring out the overall in a number of contexts.

Additional info for A Course of Higher Mathematics. Integration and Functional Analysis

Example text

Sufficiency. Since the points at which a function of bounded variation is continuous are distributed densely in [0,1], we can make use of these points only in forming the Riemann-Stieltjes sum. ^—1) vanish, so that (90) holds for any choice of continuous f(x). The theorem is proved. We have thus shown that the necessary and sufficient condition for integral (89) to give the same linear functional in O as (88) is that the difference h(x) — g(x) be equal to h(0) — g(0) at every point at which it is continuous, as also at x = 1.

There remain the discontinuities at which gr(f — 0) φ g(Ç + 0). If we take

Contained in [a, 6] and consisting of the left-hand end x = a and of all the points x that have rational abscissae. This set of points is dense in [a, 6], and we can extract a subsequence hnjt(x) which is convergent at all the points x^. We thus obtain a limit function h(x), as yet defined only at the points a and x^. e. we put h (x) = sup h (xk). Xk