Subdifferentials Whose Graphs Are Not Norm$\times$Weak* Closed

http://dx.doi.org/10.4153/CMB-2003-051-5
Canad. Math. Bull. 46(2003), 538-545

  Published:2003-12-01
 Printed: Dec 2003

In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a proper lower semicontinuous convex function on a separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed.