Norm–to–weak Upper Semicontinuous Monotone Operators Are Generically Strongly Continuous

In any Banach space a monotone operator with a norm-to-weak upper semicontinuous multivalued selection on an open set D is singlevalued and normto-norm upper semicontinuous at the points of a dense Gδ subset of D. Monotone operators — and especially a special case of them, subdifferentials of convex functions — play an important role in various parts of nonlinear analysis. One of the often investigated problems is the question about generic continuity of monotone operators, which in the case of subdifferentials means generic Fréchet differentiability of convex functions. In general Banach spaces monotone operators are not always generically continuous. There are numerous characterizations of Asplund spaces, i.e. the spaces in which any monotone operators is generically continuous on the interior of its effective domain (see e.g. [Ph]). The aim of this note is to prove that monotone operator of a certain class are generically continuous in an arbitrary Banach space (Theorem 2). Our result can be deduced from a selection result by Ch. Stegall [St], based on hard topological techniques (Remark 2). We present a relatively simple alternative proof, self-contained in the sense that it uses monotone operator techniques only. Three main tools of this note are following: a slight modification of a result by D. Preiss and L. Zaj́ıček [Pr–Zaj] (Theorem P–Z), the observation that the image of a separable set by a norm-to-weak continuous mapping is separable, and the well known method of separable reduction which extends the result from the separable into the nonseparable case. Definitions and Notations Let us recall some definitions and notations. X always denotes a real Banach space, X∗ its continuous dual. Closed and open balls with centre c (in X or X∗) and radius r > 0 are denoted by B(c; r) and B(c; r), respectively. Received March 10, 1992. 1980 Mathematics Subject Classification (1991 Revision). Primary 47H04, 47H05; Secondary 54C60.