Calculus of sequential normal compactness in variational analysis. (Q1399373)

Variational analysis, and in particular the theory of generalized differentiation, is based on various kinds of limiting procedures, which in infinite-dimensional spaces require suitable compactness properties. In the present paper, the authors study systematically sequential normal compactness and related properties. A subset \(\Omega\) of a Banach space is said to be sequentially normally compact (SNC) at a point \(\overline x\in\Omega\) if for any sequences \(x_k@>\Omega>> \overline x\), \(\varepsilon_k\downarrow 0\) and \(x^*_k\in\widehat N_{\varepsilon_k}(x_k; \Omega)\) (\(\varepsilon_k\)-normals to \(\Omega\) at \(x_k\)) one has \[ x^*_k@> w^*>> 0\Rightarrow\| x^*_k\|\to 0\qquad\text{as }k\to\infty. \] Among others, it is shown that under a qualification condition, the intersection of finitely many SNC subsets of an Asplund space is again an SNC set. Moreover, the SNC property is studied for set-valued and extended-real-valued mappings.