Generalized Hamel basis and basis extension in convex cones and uniquely divisible semigroups

I. V. Orlov

Eurasian mathematical journal, Tome 9 (2018) no. 1, pp. 69-82.

Résumé

In the work, a concept of sublinear independence in an arbitrary convex cone is introduced and the corresponding generalization of Hamel basis is studied. Applying these results to the cones generated by uniquely divisible semigroups ((UD)-semigroups) allows us to extend obtained results for the class of (UD)-semigroups. Some applications are considered.

Bibliographie

[1] T.A. Abreu, T.A. Corbacho, V. Tarieladze, Uniform type conoids, Preprint, 2006 | MR

[2] E.G. Bachtigareeva, M.L. Goldman, “Associated norms and optimal embeddings for some class of two weight integral quasinorms”, Fundamental and Appl. Math., 19:5 (2014), 3–33 (in Russian) | MR

[3] E.G. Bachtigareeva, “Optimal Banach function space for given cone of increasing functions in weighted $L_p$-space”, Eurasian Math. J., 6:1 (2015), 6–25 | MR

[4] R.G. Bartle, D.R. Shelbort, Introduction to Real Analysis, Wiley, N.Y., 2011 | MR

[5] R. Becker, Convex cones in Analysis, Hermann, 2006 | MR

[6] G. Birkhoff, “Integration of functions with values in a Banach space”, Trans. Amer. Math. Soc., 38 (1935), 357–378 | MR

[7] H. Cartan, S. Eilenberg, Homological algebra, Princeton Univer. Press, New York, 1999 | MR

[8] P.M. Cohn, Universal algebra, Reidel, 2011 | MR

[9] E. Corbacho, D. Dikranjan, V. Tarieladze, “Absorption adjunctable semigroups”, Research and Exposition in Math., 24 (2000), 77–103 | MR

[10] S. Feigelstock, “Divisible is injective”, Soochow J. Math., 32:2 (2006), 241–243 | MR

[11] A. Fraenkel, I. Bar-Hillel, Foundations of set theory, North Holland, 1973 | MR

[12] B. Fuchssteiner, W. Lusky, Convex cones, North Holland Math. Stud., 56, North Holland, Amsterdam, 1981 | MR

[13] G. Godini, “A framework for best simultaneous approximation, normed almost linear spaces”, Approx. Theory, 43 (1985), 338–358 | DOI | MR

[14] M.L. Goldman, P.P. Zabreiko, “Optimal Banach function spaces, generated by the cone of non-negative increasing functions”, Proceedings of Inst. Math. of Nat. Acad. of Sc. of Belarus, 22:1 (2014), 24–34 (in Russian)

[15] G. Hamel, “Eine Basis aller Zahlen und die unstetigen Lösungen der Functionalgleichung: $f(x + y) = f(x) + f(y)$”, Math. Ann., 60 (1905), 459–462 | DOI | MR

[16] A. Hamel, A. Löhne, Minimal set theorems, Reports of the Institute of Optimization and Stochastics No 11, Martin-Luther-Universität, Halle-Vittenberg, 2008

[17] K. Keimel, W. Roth, Ordered cones and approximation, Lecture Notes in Math., 1517, Springer, Berlin, 1992 | DOI | MR

[18] A. Löhne, On convex functions with values in semi-linear spaces, , 2004 http://ito.mathematik.uni-halle.de/l̃oehne/pdf/cf_web.pdf

[19] I.V. Orlov, “Embedding of a uniquely divisible Abelian semigroup in a convex cone”, Math. Notes, 102:3–4 (2017), 361–368 | DOI | MR

[20] I.V. Orlov, “Introduction to Sublinear Analysis”, Journal of Mathematical Sciences, 218:4 (2016), 430–502 | DOI | MR

[21] I.V. Orlov, “Inverse and implicit function theorems in the class of subsmooth maps”, Math. Notes, 99:3 (2016), 619–622 | DOI | MR

[22] E.S. Polovinkin, M.V. Balashov, Elements of the convex and strongly convex analysis, Fizmatlit, M., 2004 (in Russian) | MR

[23] J.H. Rädström, “An embedding theorem for space of convex sets”, Proc. Amer. Math. Soc., 3 (1952), 165–169 | DOI | MR

[24] V.D. Stepanov, “On optimal Banach spaces containing a weight cone of monotone or quasiconcave function”, Math. Notes, 98:6 (2015), 907–922 | MR

[25] F.S. Stonyakin, “An analogue of the Hahn–Banach theorem for functionals on abstract convex cones”, Eurasian Math. J., 7:3 (2016), 89–99 | MR

[26] R. Urbanski, “A generalization of the Minkovski–Rädström–Hörmander theorem”, Bull. L'Acad. Pol. Sci., 24:9 (1976), 709–715 | MR

[27] R.E. Worth, “Boundaries of semilinear spaces and semialgebras”, Trans. Amer. Math. Soc., 148 (1970), 99–119 | DOI | MR