Geodesic
Universal Spaces of Subdifferentials of Sublinear Operators Ranging in the Cone of Bounded Lower Semicontinuous Functions
Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 547-557

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We study Fréchet's problem of the universal space for the subdifferentials $\partial P$ of continuous sublinear operators $P\colon V\to BC(X)_{\sim}$ which are defined on separable Banach spaces $V$ and range in the cone $BC(X)_\sim$ of bounded lower semicontinuous functions on a normal topological space $X$. We prove that the space of linear compact operators $L^{\mathrm c}(\ell^2,C(\beta X))$ is universal in the topology of simple convergence. Here $\ell^2$ is a separable Hilbert space, and $\beta X$ is the Stone–Ĉech compactification of $X$. We show that the images of subdifferentials are also subdifferentials of sublinear operators.
Keywords: sublinear operator, subdifferential, topology of simple convergence, lower semicontinuous function, separable Banach space, continuous selection.
Mots-clés : Fréchet problem for universal spaces 
Yu. E. Linke. Universal Spaces of Subdifferentials of Sublinear Operators Ranging in the Cone of Bounded Lower Semicontinuous Functions. Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 547-557. http://geodesic.mathdoc.fr/item/MZM_2011_89_4_a6/
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     title = {Universal {Spaces} of {Subdifferentials} of {Sublinear} {Operators} {Ranging} in the {Cone} of {Bounded} {Lower} {Semicontinuous} {Functions}},
     journal = {Matemati\v{c}eskie zametki},
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[1] Yu. E. Linke, “Primeneniya teoremy Maikla i ee obrascheniya k sublineinym operatoram”, Matem. zametki, 52:1 (1992), 67–75 | MR | Zbl

[2] Yu. E. Linke, “O konuse ogranichennykh polunepreryvnykh snizu funktsii”, Matem. zametki, 77:6 (2005), 886–902 | MR | Zbl

[3] Yu. E. Linke, “Ob opornykh mnozhestvakh sublineinykh operatorov”, Dokl. AN SSSR, 207:3 (1972), 531–533 | MR | Zbl

[4] I. Gelfand, “Abstrakte Funktionen und lineare Operatoren”, Matem. sb., 4(46):2 (1938), 235–286 | Zbl

[5] R. G. Bartle, “On compactness in functional analysis”, Trans. Amer. Math. Soc., 79 (1955), 35–57 | MR | Zbl

[6] N. Danford, Dzh. Shvarts, Lineinye operatory, v. 1, Obschaya teoriya, IL, M., 1962 | MR | Zbl

[7] V. L. Klee, Jr., “Some topological properties of convex sets”, Trans. Amer. Math. Soc., 78 (1955), 30–45 | MR | Zbl

[8] C. Bessaga, A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, Monografie Matematyczne, 58, PWN, Warszawa, 1975 | MR | Zbl

[9] R. Engelking, Obschaya topologiya, Mir, M., 1986 | MR | Zbl

[10] I. M. Gelfand, A. N. Kolmogorov, “O koltsakh nepreryvnykh funktsii na topologicheskikh prostranstvakh”, Dokl. AN SSSR, 22:1 (1939), 11–15 | Zbl

[11] M. M. Dei, Normirovannye lineinye prostranstva, IL, M., 1961 | MR | Zbl

[12] L. Hörmander, “Sur la fonction d'appui des ensembles convexes dans une espace lokalement convexe”, Ark.Mat., 3:2 (1955), 181–186 | DOI | MR | Zbl