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Relatively weakly compact sets in the Denjoy space. (English) Zbl 1045.26502

By the “Denjoy space” the author means the space \(H\) of all Henstock-Kurzweil integrable functions on a compact interval \(I :=[a,b]\) of \(\mathbb{R}\), equipped with the seminorm \(| f| := \sup\{|\int_a^xf|:x\in I\}\). The author finds necessary and sufficient conditions for a subset of \(H\) (and of its completion) to be relatively weakly compact. The conditions involve (i) boundedness, (ii) equicontinuity on a dense subset of \(I\) and (iii) an “asymptotic \(ACG^*\)-condition”.

MSC:

26A39 Denjoy and Perron integrals, other special integrals
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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