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A short proof of the separable reduction theorem

Penot J. P.

Demonstratio Mathematica

2010

Vol. 43, nr 3

653-663

We present a simple proof of the separable reduction theorem, a crucial result of nonsmooth analysis which allows to extend to Asplund spaces the results known for separable spaces dealing with Fréchet subdifferentials. It relies on elementary results in convex analysis and avoids certain technicalities.

φ-regular functions in Asplund spaces

Ngai H. V., Thera M.

Control and Cybernetics

2007

Vol. 36, no 3

755-774

We introduce in the context of Asplund spaces, a new class of (φ-regular functions. This new concept generalizes the one of prox-regularity introduced by Poliquin & Rockafellar (2000) in Rn and extended to Banach spaces by Bernard & Thibault (2004). In particular, the class of φ-regular functions includes all lower semi-continuous convex functions, all lower-C2 functions, and convexly C1,0-composite functions as well. Geometrical and subdifferential characterizations for this new class of functions are investigated.