By J. Prolla

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**Extra resources for Approximation Theory and Functional Analysis**

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I) Let f:X ~ R be convex; then D(f *) :#: <1> iff there exists a nonvertical hyperplane H in X* x R such that Epi f is contained in one of the half-spaces determined by H. ii) Epi C is convex and closed. 8. Let f:X ~R be convex, xe IntDCf) and F(y)=f~(x,y),ye X; then F* = I Clf(x)" f(x+ty) -f(x) Hint. -1 X* F t ( Xj t ,vX'"e. 14. 9. Let f:X ~B. be proper and convex; then x* e af(x) iff x e aC(x*). 10. A function f:X ~R is affine (Le. the function x linear) iff ± f are convex. ~f(x) -f(O) is 11.

If f:X ~R is continuous and convex, then its subdifferential map is norm-to-weak* upper semi continuous. 39 Sl1BDIFFERENTIABILITY AND DUALITY MAPPINGS Hint. , Phelps [1]. 20. Let X, Y be two vector topological spaces and DeY; we introduce the following notations: -1 -1 T + (D) = {xe X; TxcD} T - (D) ={x eX; Txn D*- cp}. Prove that the following statements are equivalent: a) T is upper-semicontinuous; b) for any open set V c Y, T:1(V) is open in X; -1 c) for any closed set W c Y , T - (W) is closed in X; d) if Dc Y, then T::::1(:6):2 T=I(D) .

J n E N. By hypothesis there is B > 0 with Ilxn + y nil S; 2(1 - B) and this is a contradiction. We can prove ii) and iii) analogously. 4. REMARK. llyll S; l. 5 NOTATIONS. £ (2-

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