## On subdifferentials of convex functions.(English)Zbl 0928.49017

The paper deals with subdifferentials of continuous convex functions $$f$$ defined on a normed space $$X$$. Conditions are given ensuring that subgradients of $$f$$ belong to the complementary space of a closed subspace $$E\subseteq X$$. Furthermore, the continuity and related properties of the duality mappings $$J: X\to 2^{X^*}$$ and $$J: X^*\to 2^{X^{**}}$$ of a Banach space $$X$$ and its dual space $$X^*$$ are investigated.

### MSC:

 49J52 Nonsmooth analysis 46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics 47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics 58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds

### Keywords:

subdifferentials; convex functions; normed space; dual space
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