Banach Spaces of Type (NI) and Monotone Operators on Non-reflexive Spaces

A convex set C ⊆ X∗∗ × X∗ admits the variant Banach-Dieudonné property (VBDP) if the weak∗-strong closure C w×‖·‖ is the smallest set containing C that is closed to all limits of its bounded and weak∗×‖ · ‖ convergent nets. We show in particular, that all convex sets in X∗∗×X∗ admit the VBDP when E∗ := X∗×X∗∗ is weakly-compactly generated (WCG) and hence if E is either a dual separable or a reflexive Banach space. This allows us to answer some outstanding problems regarding the embedding of various representative functions, including what we call the Penot function (but which seems first to have been clearly identified by Svaiter) , for nonreflexive spaces [15, 7]. In particular, we show that the conjugate of the Fitzpatrick representative function F̂T ∗ : X∗ × X∗∗ → R and the Penot representative function P̂T ∗ : X∗ × X∗∗ → R are themselves representative functions when T ⊆ X × X∗ is maximal monotone and E is as above. It follows that in such spaces all maximal monotone operators are well-behaved and the classical sum rule holds. Notice: We no longer trust Theorem 5 and so the results herein should be taken as conjectured not proven.