A SHORT PROOF OF THE SEPARABLE REDUCTION THEOREM
Jean‐Paul Penot · Demonstratio Mathematica · 2010
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 Frechet subdifferentials.It relies on elementary results in convex analysis and avoids certain technicalities.The separable reduction theorem is an important result of nonsmooth analysis.It enables to pass from fuzzy sum rules in spaces with smooth norms to fuzzy sum rules in general Asplund spaces.Asplund spaces form the appropriate setting for such approximate rules and for extremal principles ([3], [5], [4], [6], [7], [8], [9], [13], [14], [15]).The proof of that result is rather sophisticated and long (see [1, pp.243-258], [14, pp.183-221]).It is the purpose of this note to present a short proof.While the core of the proof uses arguments similar to the ones in the original proofs, the simplification stems from elementary results about convexification.Such results of independent interest are gathered in Section 1.The characterization of nonemptiness of the Frechet subdifferential of a function is recalled in Section 2. The proof of the separable reduction theorem is the object of Section 3. The last section is devoted to known consequences of the theorem which may motivate the study and give an idea of its usefulness. Useful facts from convex analysisWe assume the reader has a basic knowledge of convex analysis.In particular, we assume some familiarity with the Fenchel-Moreau subdifferential: for a function / : X -> RQO := 111 {+00} on a normed vector space X, finite