Convex Functions, Monotone Operators and Differentiability by Robert R. Phelps

By Robert R. Phelps

The enhanced and improved moment variation includes expositions of a few significant effects that have been received within the years because the 1st variation. Theaffirmative resolution by way of Preiss of the many years outdated query of even if a Banachspace with an similar Gateaux differentiable norm is a vulnerable Asplund area. The startlingly uncomplicated evidence via Simons of Rockafellar's primary maximal monotonicity theorem for subdifferentials of convex features. The interesting new edition of the priceless Borwein-Preiss gentle variational precept as a result of Godefroy, Deville and Zizler. the fabric is offered to scholars who've had a path in sensible research; certainly, the 1st version has been utilized in a variety of graduate seminars. beginning with convex services at the line, it results in interconnected themes in convexity, differentiability and subdifferentiability of convex capabilities in Banach areas, prevalent continuity of monotone operators, geometry of Banach areas and the Radon-Nikodym estate, convex research, variational rules and perturbed optimization. whereas a lot of this can be classical, streamlined proofs came across extra lately are given commonly. there are many routines, lots of which shape a vital part of the exposition.

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Then for any x E dom(r) we have Y E E. Proof. Note, first, that because of the decreasing size Of the sets as E -+ 0+, the limit on the right exists (although it possibly equals Given E > 0, choose any x" E <\f(x), so that for t > 0 and y E E ~ t- I [f(x + ty) - f(x) o£f(x) :too) . + EJ . Let t = F ; then this becomes ~ F-l[f(X + Fy) - f(x)] + F Since the inequality (#) holds for all x" E o£f(x£ ), it follows that Note that if d+f(x)(y) = -00, then this actually yields (,,), so to prove the reverse inequality, we can assume that d+f(x)(y) > -00 .

Thus, if y £ B(O; o)non) and y" £ TCy ), then 201lY"1I = sup{(y .. s. 20} ~ Ily"II'lIyll ... I ~ olly"ll ... I, so IIY"II~l/o. Note that the foregoing result does not require that OCT) be convex. There are trivial examples which show that 0 can be an absorbing point of on) but not an interior point (for instance, let T be the restriction of the subdifferential of the norm to the se t A, defined above). Even if OCT) ~ convex and T is maximal monotone, O(T) can have empty interior, as shown by the following example.

41 Definition. A Banach space E is said to be weakly comDactly Qenerated (WCG) provided there exists a weak I~ compact subset K of E whose linear span is dense in E. Since the closed convex hull of a weakl~ compact subset of a Banach space is weakl~ compact, one can alwa~s assume that K is convex. For background, see J. Diestel's lecture notes [Di]. 42 Examples. (a) If E is separable or reflexive, then it is WCG. In the first case, let {x n} be dense in E and let K = {n-1xn/IIXnll} U {O}: this is actuall~ compact.

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