We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that Ï(x1+ x2) â Ï(x1) + Ï(x2). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i. e., there exists a linear operator A: X â Y such that Ax â Ï(x), x â X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces. © 2011 Springer Science+Business Media, Inc.
On linear selections of convex set-valued maps
Protasov, Vladimir
2011-01-01
Abstract
We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that Ï(x1+ x2) â Ï(x1) + Ï(x2). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i. e., there exists a linear operator A: X â Y such that Ax â Ï(x), x â X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces. © 2011 Springer Science+Business Media, Inc.Pubblicazioni consigliate
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