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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

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