On subdifferentials of convex functions
Josef Kolomý · Czech digital mathematics library · 1993
Let -rY be a real normed linear space, X* and X* * its dual and bidual, respective-ly^,) the pairing between X and X*, S x (0) and 5^(0) the unit sphere in X and X*, respectively.By R we denote the set of all real numbers, while x denotes the image of an element x e X under the canonical mapping in X* *.If £ is a subspace of X, denote by E 1 its annihilator in X*.Let F and G be topological spaces, 2 G the family of all subsets of G, T: F -2 G a mapping, D(T) -{u e F: T(u) ¥> 0} its domain, G(T) -{(u, v)z FX G: ve T(u) for some u* D(T)} its graph in the space F X G.We shall say that T: F -* 2 G is upper semicontinuous at a 0 e F, if for each open subset W of G such that T(u 0 ) c W there exists an open neighborhood U of w 0 such that T(u) c W for every w e [/.Suppose now that X is a normed linear space.By the symbols o(X, X*) and o(X* 9 X), we mean the weak and the weak* topology on X and X*, respectively.Recall that T: X -2 X is said to be (i) monotone, if for every u,v^D(T) and every u* e T(u), v* e T(v) there is (v* -u*, v -w) ^ 0; (ii) maximal monotone, if T is monotone and for a given element (u 0 , «* fl )ax X* such that * -i& v -w 0 ) ^ 0 for every (v, v*) e G(7), we have that (w 0 , w*) e G(T).Let M c AT be an open nonvoid convex subset of a normed linear space X, f: M -+ R a convex continuous function.The multivalued mapping Af3«-df(u) defined by df(u) -{w* e X*, (u*, v -u) * f(v) -f(u) for every v e M} is called the subdifferential mapping (or subdifferential) of/on M. Note that w* e df(u 0 ), where w 0 € Af, if and only if the graph of the affine function h(v) -f(u 0 ) + (u*, v -u 0 ) is a supporting hyperplane to the epigraph of / at the point (u 0 ,f(u 0 )).