Vol. 47, No. 1, 1973

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Subdifferentials of convex functions on Banach spaces

George William Luna

Vol. 47 (1973), No. 1, 161–175

This note was motivated by a paper of P. D. Taylor, which contains a simple proof of Rockafellar’s basic theorem that the subdifferential map ∂f of a lower semicontinuous proper convex function f on a Banach space is maximal monotone. Taylor based his proof on a theorem which can be considered as a sharpening (for the epigraph of a convex function) of a result (Lemma 1.1) concerning support points and functions of convex sets due to Brondsted-Rockafellar and Phelps. It is shown that Taylor’s theorem can be generalized somewhat, using related methods. (It is shown, by an example, that there is a limitation on the extent of generalization possible.) The theorem follows from a slightly technical result (Proposition 1.3) which admits a dual version (Proposition 2.2). As an application of Proposition 2.2, a short proof of Rockafellar’s theorem relating the graph of (∂f)1 to that of ∂f is given. The methods of this paper yield a generalization (Corollary 1.9) of one of the density results of Bishop-Phelps.

Mathematical Subject Classification
Primary: 46B05
Received: 26 May 1972
Revised: 18 October 1972
Published: 1 July 1973
George William Luna