Duality theorems for convex functions

where F* is the space of linear functionals on F. The conjugate function is proper convex on F*, and is always lower semi-continuous. If ƒ itself is l.s.c, then ƒ coincides with the conjugate/** of/* (where F** is identified with F). These facts and definitions have obvious analogs for concave functions, with " inP replacing "sup" in (1). Suppose ƒ is l.s.c. proper convex on F and g is u.s.c. proper concave on F. If ri (dom/) (~\ ri (dom g) ^ 0 ,