Geodesic
A Sublinear Analog of the Banach--Mazur Theorem in Separable Convex Cones with Norm
Matematičeskie zametki, Tome 104 (2018) no. 1, pp. 118-130.

Voir la notice de l'article provenant de la source Math-Net.Ru

A special class of separable normed cones, which includes convex cones in normed spaces and in spaces with an asymmetric norm, is distinguished on the basis of the functional separability of elements. It is shown that, generally, separable normed cones admit no linear injective isometric embedding in any normed space. An analog of the Banach–Mazur theorem on a sublinear injective embedding of a separable normed cone in the cone of real nonnegative continuous functions on the interval $[0;1]$ with the ordinary sup-norm is obtained. This result is used to prove the existence of a countable total set of bounded linear functionals for a special class of separable normed cones.
Keywords: separable normed cone, space with asymmetric norm, Hahn–Banach theorem, Banach–Mazur theorem, sublinear injective isometric embedding, total set of bounded linear functionals. 
@article{MZM_2018_104_1_a10,
     author = {F. S. Stonyakin},
     title = {A {Sublinear} {Analog} of the {Banach--Mazur} {Theorem} in {Separable} {Convex} {Cones} with {Norm}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {118--130},
     publisher = {mathdoc},
     volume = {104},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_104_1_a10/}
}
TY  - JOUR
AU  - F. S. Stonyakin
TI  - A Sublinear Analog of the Banach--Mazur Theorem in Separable Convex Cones with Norm
JO  - Matematičeskie zametki
PY  - 2018
SP  - 118
EP  - 130
VL  - 104
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2018_104_1_a10/
LA  - ru
ID  - MZM_2018_104_1_a10
ER  - 
%0 Journal Article
%A F. S. Stonyakin
%T A Sublinear Analog of the Banach--Mazur Theorem in Separable Convex Cones with Norm
%J Matematičeskie zametki
%D 2018
%P 118-130
%V 104
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2018_104_1_a10/
%G ru
%F MZM_2018_104_1_a10
F. S. Stonyakin. A Sublinear Analog of the Banach--Mazur Theorem in Separable Convex Cones with Norm. Matematičeskie zametki, Tome 104 (2018) no. 1, pp. 118-130. http://geodesic.mathdoc.fr/item/MZM_2018_104_1_a10/

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

[2] K. Keimel, W. Roth, Ordered Cones and Approximation, Lecture Notes in Math., 1517, Springer-Verlag, Berlin, 1992 | DOI | MR | Zbl

[3] W. Roth, “Hahn–Banach type theorems for locally convex cones”, J. Austral. Math. Soc. Ser. A, 68:1 (2000), 104–125 | DOI | MR | Zbl

[4] P. Selinger, “Towards a semantics for higher-order quantum computation”, Proceedings of the 2nd International Workshop on Quantum Programming Languages, TUCS General Publication, 33, Turku Centre for Computer Science, Turku, 2004, 127–143

[5] I. V. Orlov, “Teoremy ob obratnoi i neyavnoi funktsiyakh v klasse subgladkikh otobrazhenii”, Matem. zametki, 99:4 (2016), 631–634 | DOI | MR | Zbl

[6] L. M. García-Raffi, S. Romaguera, E. A. Sánchez-Pérez, O. Valero, “Metrizability of the unit ball of the dual of a quasi-normed cone”, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 7:2 (2004), 483–492 | MR | Zbl

[7] S. Romaguera, E. A. Sánchez-Pérez, O. Valero, “A characterization of generalized monotone normed cones”, Acta Math. Sin. (Engl. Ser.), 23:6 (2007), 1067–1074 | DOI | MR | Zbl

[8] M. G. Krein, A. A. Nudelman, Problema momentov Markova i ekstremalnye zadachi. Idei i problemy P. L. Chebysheva i A. A. Markova i ikh dalneishee razvitie, Nauka, M., 1973 | MR | Zbl

[9] E. P. Dolzhenko, E. A. Sevastyanov, “Approksimatsiya so znakochuvstvitelnym vesom (ustoichivost, prilozheniya k teorii uzhei i khausdorfovym approksimatsiyam)”, Izv. RAN. Ser. matem., 63:3 (1999), 77–118 | DOI | MR | Zbl

[10] P. A. Borodin, “Teorema Banakha–Mazura dlya prostranstv s nesimmetrichnoi normoi i ee prilozheniya v vypuklom analize”, Matem. zametki, 69:3 (2001), 329–337 | DOI | MR | Zbl

[11] A. R. Alimov, “Teorema Banakha–Mazura dlya prostranstv s nesimmetrichnym rasstoyaniem”, UMN, 58:2 (350) (2003), 159–160 | DOI | MR | Zbl

[12] G. E. Ivanov, M. S. Lopushanski, “O korrektnosti zadach approksimatsii i optimizatsii dlya slabo vypuklykh mnozhestv i funktsii”, Fundament. i prikl. matem., 18:5 (2013), 89–118 | MR

[13] G. E. Ivanov, “On well posed best approximation problems for a nonsymmetric seminorm”, J. Convex Anal., 20:2 (2013), 501–529 | MR | Zbl

[14] S. Cobzaş, Functional Analysis in Asymmetric Normed Spaces, Birkhauser Verlag, Basel, 2013 | MR | Zbl

[15] 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

[16] L. A. Lyusternik, V. I. Sobolev, Kratkii kurs funktsionalnogo analiza, Vysshaya shkola, M., 1982 | MR | Zbl

[17] F. S. Stonyakin, “Applications of anticompact sets to analogs of Denjoy–Young–Saks and Lebesgue theorems”, Eurasian Math. J., 6:1 (2015), 115–122 | MR

[18] F. S. Stonyakin, “Analogi teoremy Shaudera s ispolzovaniem antikompaktov”, Matem. zametki, 99:6 (2016), 950–953 | DOI | MR | Zbl

[19] F. S. Stonyakin, “Antikompakty i ikh prilozheniya k analogam teorem Lyapunova i Lebega v prostranstvakh Freshe”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 53, RUDN, M., 2014, 155–176