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Linear Isometries of Spaces of Absolutely Continuous Functions

Published online by Cambridge University Press:  20 November 2018

V. D. Pathak
V. D. Pathak*
Affiliation:
Maharaja Sayajirao University of Baroda, Gujarat, India
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Let X be an arbitrary compact subset of the real line R which has at least two points. For each finite complex valued function f on X we denote by V(f; X) (and call it the weak variation of f on X) the least upper bound of the numbers ∑i|f(bi) – f(ai)| where {[ai, bi]} is any sequence of non-overlapping intervals whose end points belong to X. A function f is said to be of bounded variation (BV) on X if V(f; X) < ∞. A function f is said to be absolutely continuous (AC) on X, if given any ∈ > 0 there exists an n > 0 such that for every sequence of non-overlapping intervals {[au bi]} whose end points belong to X, the inequality

implies that

([7], p. 221, 223).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 2 , 01 April 1982 , pp. 298 - 306
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Cambern, M., Isometries of certain Banach algebras, Studia Mathematica, T. 25 (1965).Google Scholar
2. Dunford, N. and Schwartz, J., Linear operations (Part I) (New York, 1958).Google Scholar
3. Hewitt, E. and Stromberg, K., Real and abstract analysis (New York, 1969).Google Scholar
4. Hoffmann, K., Banach spaces of analytic functions (New York, 1962).Google Scholar
5. K., de Leeuw, Banach spaces of Lipschitz functions, Stud-Math. 21 (1961/62), 55–56.Google Scholar
6. Rao, N. V. and Roy, A. K., Linear isometries of some function spaces, Pacific J. Math. 38 (1971), 177–192.Google Scholar
7. Saks, S., The theory of integrals (New York, 1937).Google Scholar