Convex Functions, Monotone Operators and Differentiability by Robert R. Phelps
By Robert R. Phelps
The enhanced and multiplied moment version 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 Theaffirmative resolution by way of Preiss of the many years outdated query of no matter if a Banachspace with an similar Gateaux differentiable norm is a susceptible Asplund house. The startlingly uncomplicated evidence by means of Simons of Rockafellar's basic maximal monotonicity theorem for subdifferentials of convex capabilities. The intriguing new edition of the precious Borwein-Preiss soft variational precept because of Godefroy, Deville and Zizler. the fabric is obtainable to scholars who've had a direction in sensible research; certainly, the 1st version has been utilized in a variety of graduate seminars. beginning with convex capabilities at the line, it results in interconnected themes in convexity, differentiability and subdifferentiability of convex services in Banach areas, prevalent continuity of monotone operators, geometry of Banach areas and the Radon-Nikodym estate, convex research, variational ideas and perturbed optimization. whereas a lot of this is often classical, streamlined proofs stumbled on extra lately are given normally. there are lots of routines, lots of which shape a vital part of the exposition.
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Extra info for Convex Functions, Monotone Operators and Differentiability
18. We next look at some additional properties which distinguish subdifferentials within the class of monotone operators. proving some basic results along the way. 20 Definition. A set-valued map monotone provided T:E -> whenever n:::: 2 and xo. XI. X2 .... xn £ E. xn 2 E>< is said to be n-cucljcallu = xo. and x k " £ T(x k ). 2. 3 ..... n. We say that T is cyclically monotone if it is n-cyclically monotone for every n. Clearly. a 2-cyclically monotone operator is monotone. 21 Examples. (a) The linear map in R2 defined b~ T(X1, X2) = (X2' -X1) is positive, hence monotone, but it is not 3-cyclically monotone : Look at the points (I, I), (0, 1) and (1, 0).
More generally, any separable reflexive Banach space. We will see later that separab Ie Asp lund spaces are character ized by the fact that their duals are separable. A more general fact, one of the most beautiful and fundamental results in this area, is that E is an Asplund space if and only if every separable subspace of E has a separable dual. Half of this latter result is a corollary to the next theorem. 14 Theorem. A Banach space E is an Asplund space if every separable closed subspace F of E is an Asp lund space.
14 Theorem. A Banach space E is an Asplund space if every separable closed subspace F of E is an Asp lund space. Proof. Q1 dense in D; we will produce a separable subspace F of E such that F n D ;z: 0 and the points of Frechet different iabi I ity of f I F are not dense in F n D. 25) GnU) to be the open set of all which there exists & > 0 satisfying sup lIy II = 1 f(x + &~) + f(x & &~) - 2f(x) < 1 In. x s D for 24 Since D is a Baire space and since G = nGn(f), we conclude that Gm(f) is not dense in D, for some m > 0, that is, there exists a nonempt~ open subset U C D\Gm(f).