Duality mapping. Projection

Let y be a reflexive Banach space, and V be a space dual of V. We assume that the functionals u — u, u — u defined on V, V, respectively, are strictly convex. In this case the spaces V, V are called strictly convex. A value of a functional u G V on elements u G y is denoted by (u, u). An operator I y — y is called a duality mapping if the following conditions hold (Gajewski et al., 1974)  

We note that Ju is a linear and continuous functional on V meanwhile the operator I is not linear, in general. 

The existence of a unique duality mapping could easily be proved. Indeed, let 1 = u G y I u = 1 be a unit sphere in V. According to the Hahn-Banach theorem, for every fixed u G E there exists a unique element u G V such that u = 1, u, u) = 1 due to the strict convexity of V. Let us define 

Note that relations (1.91) and (1.92) mean linearity of the duality mapping I and its inverse I in Hilbert spaces due to the linearity of the scalar product. 

Let K c V he a convex closed set. We assume that y is a strictly convex reflexive Banach space. For given u G V an element Pu G K is called a projection of u onto the set K if 

---

Let K cV he a. convex closed subset of a reflexive Banach space V, I he a duality mapping, and P be a projection operator of V onto K. We are in a position to give a definition of a penalty operator. An operator (5 V V is called a penalty operator connected with the set K if the following conditions are fulfilled. Firstly, / is a monotonous bounded semicontinuous operator. Secondly, a kernel of / coincides with K, i.e. 

---