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 Eurasian Math. J., 2016, Volume 7, Number 3, Pages 89–99 (Mi emj234)

An analogue of the Hahn–Banach theorem for functionals on abstract convex cones

F. S. Stonyakin

Department of algebra and functional analysis, Crimea Federal University, 4 V. Vernadsky Ave, Simferopol, Russia

Abstract: We prove an analogue of the Hahn–Banach theorem on the extension of a linear functional with a convex estimate for each abstract convex cone with the cancellation law. Also we consider the special class of the so-called strict convex normed cones $(SCNC)$. For such structures we obtain an appropriate analogue of the Hahn–Banach separation theorem. On the base of this result we prove that each $(SCNC)$ is sublinearly, injectively and isometrically embedded in some Banach space.

Keywords and phrases: abstract convex cone, cancellation law, convex functional, Hahn–Banach theorem, convex normed come, Lemma on a support functional, strict convex normed cone, sublinear injective isometric embedding.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation MK-2915.2015.1 This work was supported by grant of the President of Russian Federation for young candidates of sciences, project no. MK-2915.2015.1.

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MSC: 46A22, 46A20, 46B10
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Citation: F. S. Stonyakin, “An analogue of the Hahn–Banach theorem for functionals on abstract convex cones”, Eurasian Math. J., 7:3 (2016), 89–99

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\Bibitem{Sto16} \by F.~S.~Stonyakin \paper An analogue of the Hahn--Banach theorem for functionals on abstract convex cones \jour Eurasian Math. J. \yr 2016 \vol 7 \issue 3 \pages 89--99 \mathnet{http://mi.mathnet.ru/emj234} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3581185} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000391008000007} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. F. S. Stonyakin, “Subdifferential calculus in abstract convex cones”, 2017 Constructive Nonsmooth Analysis and Related Topics, CNSA 2017, Dedicated to the Memory of V.F. Demyanov, ed. L. Polyakova, IEEE, 316–319
2. I. V. Orlov, “Embedding of a Uniquely Divisible Abelian Semigroup In a Convex Cone”, Math. Notes, 102:3 (2017), 361–368
3. F. S. Stonyakin, “A Sublinear Analog of the Banach–Mazur Theorem in Separable Convex Cones with Norm”, Math. Notes, 104:1 (2018), 111–120
4. F. S. Stonyakin, “On Sublinear Analogs of Weak Topologies in Normed Cones”, Math. Notes, 103:5 (2018), 859–864
5. I. V. Orlov, “Generalized Hamel basis and basis extension in convex cones and uniquely divisible semigroups”, Eurasian Math. J., 9:1 (2018), 69–82
6. F. S. Stonyakin, “Hahn–Banach type theorems on functional separation for convex ordered normed cones”, Eurasian Math. J., 10:1 (2019), 59–79
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