Characterizations of Banach Spaces via Convex and other Locally Lipschitz Functions

. Various properties of Banach spaces, including the reflexivity and the Schur property of a space, are characterized in terms of properties of corresponding classes of locally Lipschitz functions on those spaces. AMS Classification. 46A55, 46B20, 52A41. 1 Research partially supported by NSERC and the Shrum Endowment. 2 Research partially supported by grants AV CR 101-95-02 and GA CR 201-94-0069. 3 Research partially supported by a Walla Walla College Faculty Development Grant. 2 Borwein, Fabian & Vanderwerff 1. Introduction: "Sequentially Reflexive" spaces and motivating results. We will work with real Banach spaces and let BX and SX denote the closed unit ball and unit sphere of the Banach space X respectively. As in [2], we will say a Banach space is sequentially reflexive if Mackey and norm convergence coincide sequentially in its dual space. The following result was proved in [22]; see also [3, Theorem 5]. Theorem 1.1. For a Banach space X, the following are equivalent. ..