Functional Analysis and Its Applications

, Volume 45, Issue 1, pp 46–55 | Cite as

On linear selections of convex set-valued maps

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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 φ(x 1 + x 2) ⊂ φ(x 1) + φ(x 2). 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 → Y such that Ax ∈ φ(x), x ∈ 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.

Key words

set-valued map linear selection subadditivity Lipschitz function stability of linear operators 
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Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia

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