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Duality in the space of regulated functions and the play operator. (English) Zbl 1055.46023
Let $$G(a,b;X)$$ be the space of all regulated functions defined on the interval $$[a,b]\subset \mathbb R$$ with values in the Hilbert space $$X$$. A regulated function is a function having two-sided limits at each point in $$[a,b]$$. With respect to the uniform norm $$\| f\| _{[a,b]},\; G(a,b;X)$$ is a Banach space that contains $$C(a,b;X)$$, the space of continuous functions. For a regulated function $$f:[a,b]\to X$$ and a partition $$d:a=t_ 0<...<t_ m=b$$ of the interval $$[a,b]$$, the essential variation of $$f$$ on $$[a,b]$$ is defined by $$\overline{\mathcal V}_ d(f) = \sum_{i=1}^ m | f(t_ i-)-f(t_{i-1}+)| + \sum_{i=0}^ m | f(t_ i+)-f(t_ i-)| ,$$ and the total essential variation by $$\overline{Var}_{[a,b]}=\sup_ d \overline{\mathcal V}_ d(f).$$ The space of all functions with finite total essential variation, denoted by $$\overline{BV}(a,b;X)$$, contains the space $$BV(a,b;X)$$ of all functions with bounded variation.
In Theorem 2.6, the authors give a representation for the dual of the Banach space $$G(a,b;X)$$ in terms of the Young integral and of some functions in $$BV(a,b;X)$$. Using this representation, they establish some relationships between weak and wbo-convergence (wbo comes from “weak bounded oscillation”). The play operator, considered by P. Krejči and Ph. Laurençot [J. Conv. Anal. 9, 159–183 (2002; Zbl 1001.35118)] is used in an essential manner throughout the paper. This is a natural extension of the classical play operator from the mathematical theory of hysteresis.

##### MSC:
 46E40 Spaces of vector- and operator-valued functions 26A45 Functions of bounded variation, generalizations 26A39 Denjoy and Perron integrals, other special integrals 26E20 Calculus of functions taking values in infinite-dimensional spaces 34C55 Hysteresis for ordinary differential equations
##### Keywords:
regulated functions; functions with bounded variation
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