Nonreflexive Banach Ssd Spaces

In this paper, we unify the theory of SSD spaces, part of the theory of strongly representable multifunctions, and the theory of the equivalence of various classes of maximally monotone multifunctions. 0 Introduction In this paper, we unify three different lines of investigation: the theory of SSD spaces as expounded in [11] and [13], part of the theory of strongly representable multifunctions as expounded in [15] and [4], and the equivalence of various classes of maximally monotone multifunctions, as expounded in [5]. The purely algebraic concepts of SSD space and q–positive set are introduced in Definition 1.2. These were originally defined in [11], and the development of the theory was continued in [13]. Apart from the fact that we write " P " instead of " pos " , we use the notation of the latter of these references. We show in Lemma 1.9 how certain proper convex functions f on an SSD space lead to a q–positive set, P(f). In Definition 1.10, we define the intrinsic conjugate, f @ , of a proper convex function on an SSD space, and we end Section 1 by proving in Lemma 1.11 a simple, but useful, property of intrinsic conjugates. In Definition 2.1, we introduce the concept of a Banach SSD space, which is an SSD space with a Banach space structure satisfying the compatibility conditions (2.1.1) and (2.1.2). A proper convex function on a Banach SSD space may be a VZ function, which is introduced in Definition 2.5. Our main result on VZ functions, established in Theorem 2.9(c,d), is that if f is a lower semicontinuous VZ function then P(f) is maximally q– positive, f @ is also a VZ function, and P f @ = P(f). Lemma 2.7(b) is an important stepping–stone to Theorem 2.9. In Definition 2.12 and Lemma 2.13, we introduce and discuss the properties of various convex functions on a Banach SSD space and its dual, and show in Theorem 2.15(c) that if f is a lower semicontinuous VZ function on a Banach SSD then there is a whole family of VZ functions h associated with f such that P(h) = P(f). If E is a nonzero Banach space then it is shown in Examples 1.4, 2.3, and 2.4 that E × E * is a Banach SSD space under various different norms. We show in Section 3 how the definitions and results of Section 2 specialize …