Full Calculus in Asplund Spaces
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This chapter is devoted to developing a comprehensive calculus for our basic generalized differential constructions: normals to sets, coderivatives of set-valued and single-valued mappings, and subgradients of extended-real-valued functions. A useful part of the generalized differential calculus has been presented in Chap. 1 in the setting of arbitrary Banach spaces. However, a number of important results therein impose differentiability assumptions on some mappings involved in compositions. In this chapter we don’t require any smoothness and/or convexity of sets and mappings under consideration developing a full calculus in the framework of Asplund spaces at the same level of perfection as in finite dimensions. The main impact to this development comes from the results of Chap. 2 on the extremal principle and variational properties of Fréchet-like constructions in Asplund spaces. In this way we obtain general calculus rules for our basic objects using a geometric approach, i.e., starting with calculus rules for normal cones and then deriving from them sum and chain rules as well as other results for coderivatives and subdifferentials. It happens that the calculus rules obtained involve sequential normal compactness (SNC) assumptions on sets and mappings that are automatic in finite dimensions and reveal one of the most principal differences between finite-dimensional and infinite-dimensional variational theories.
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