On the range of the subdifferential in non reflexive Banach spaces

Abstract

We present the following unbounded version for James's theorem on weak compactness in Banach spaces: let C be a closed, convex but not necessarily bounded subset in the Banach space E, and Λ be a non-void and τ(E⁎,E)-open subset of E⁎; i.e. Mackey open in the dual space, such thatsup⁡{z⁎(c):c∈C}<+∞ whenever z⁎∈Λ. If C is not σ(E⁎⁎,E⁎)-closed in E⁎⁎ there is a linear form z⁎∈Λ such that the sup⁡{z⁎(c):c∈C} is not attained.As a main application we have the following: if f:E→R∪{∞} is a proper function such that the range of the subdifferential ∂f(E) contains a nonvoid open subset for the Mackey topology on the dual space (E⁎,τ(E⁎,E)), then for each set c∈R the sublevel set f−1((−∞,c] is relatively weakly compact. If in addition the function f has a domain with non-empty norm interior, the Banach space E must be reflexive.Straightforward applications to robust representation of risk measures and monotone operators are also derived.