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Contemporary Mathematics. Fundamental Directions, 2014, Volume 53, Pages 155–176 (Mi cmfd265)  

This article is cited in 4 scientific papers (total in 4 papers)

Anti-compacts and their applications to analogs of Lyapunov and Lebesgue theorems in Frechét spaces

F. S. Stonyakin
Full-text PDF (244 kB) Citations (4)
References:
Abstract: We introduce anti-compact sets (anti-compacts) in Frechét spaces. We thoroughly investigate the properties of anti-compacts and the scale of Banach spaces generated by anti-compacts. Special attention is paid to systems of anti-compact ellipsoids in Hilbert spaces. The existence of a system of anti-compacts is proved for any separable Frechét space $E$. Using the constructed theory, we obtain analogs of the Lyapunov theorem on the convexity and compactness of the range of vector measures in the class of separable Frechét spaces: We prove the convexity and compactness of the range of vector measure in a space $E_{\overline C}$ generated by an anti-compact $\overline C$. Also, the nondifferentiability problem with respect to the upper limit is investigated for the Pettis integral. We obtain differentiability conditions for the indefinite Pettis integrals in terms of the new weak integral boundedness and the $\sigma$-compact measurability. We prove an analog of the Lebesgue theorem on the differentiability of the indefinite Pettis integral for any strongly measurable integrand.
English version:
Journal of Mathematical Sciences, 2016, Volume 218, Issue 4, Pages 526–548
DOI: https://doi.org/10.1007/s10958-016-3041-5
Document Type: Article
UDC: 517.98
Language: Russian
Citation: F. S. Stonyakin, “Anti-compacts and their applications to analogs of Lyapunov and Lebesgue theorems in Frechét spaces”, Proceedings of the Crimean autumn mathematical school-symposium, CMFD, 53, PFUR, M., 2014, 155–176; Journal of Mathematical Sciences, 218:4 (2016), 526–548
Citation in format AMSBIB
\Bibitem{Sto14}
\by F.~S.~Stonyakin
\paper Anti-compacts and their applications to analogs of Lyapunov and Lebesgue theorems in Frech\'et spaces
\inbook Proceedings of the Crimean autumn mathematical school-symposium
\serial CMFD
\yr 2014
\vol 53
\pages 155--176
\publ PFUR
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd265}
\transl
\jour Journal of Mathematical Sciences
\yr 2016
\vol 218
\issue 4
\pages 526--548
\crossref{https://doi.org/10.1007/s10958-016-3041-5}
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  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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