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λ-Variation and Hausdorff Distance

✍ Scribed by Franciszek Prus-Wiśniowski


Book ID
102943410
Publisher
John Wiley and Sons
Year
2006
Tongue
English
Weight
646 KB
Volume
158
Category
Article
ISSN
0025-584X

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✦ Synopsis


Abstract

It is shown that points of varying monotonicity of a function play an important role in calculation of its λ‐variation. It is proven that for a given continuous function x: [0,1] → IR of bounded λ‐variation, the function A → λ‐var(x, A) is continuous on the metric space (ℋ︁, d~H~) of all closed subsets of [0, 1] with Hausdorff distance. Another function from (ℋ︁, d~H~) into ℛ is used to characterize the compactness in CΛBV~c~, i.e. in the Banach space of continuous functions that are continuous in λ‐variation. As a corollary the separability of CΛABV~c~ is proven.


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