Universal Spaces of Subdifferentials of Sublinear Operators Ranging in the Cone of Bounded Lower Semicontinuous Functions

Yu. E. Linke

Matematičeskie zametki, Tome 89 (2011) no. 4, pp. 547-557

Résumé

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.

Détail

Keywords: sublinear operator, subdifferential, topology of simple convergence, lower semicontinuous function, Fréchet problem for universal spaces, separable Banach space, continuous selection.