Edgar Asplund

Bio: Edgar Asplund is an academic researcher from University of Washington. The author has contributed to research in topics: Banach space & Gâteaux derivative. The author has an hindex of 4, co-authored 4 publications receiving 445 citations.

Fréchet differentiability of convex functions

Abstract: A continuous convex function of one real variable is differentiable, except perhaps at a countable subset of its interval of continuity. The present paper deals with generalizat ions of this e lementary s ta tement to convex functions which are defined on some Banach space E, and continuous in the norm topology, with \"differentiable\" replaced either by \"Frdchet differentiable\" or \"Gateaux differentiable\". Since for E = L ~ ( 0 , 1 ) the very norm funct ion/ (x) = Ilxll for x in E, which is convex and continuous on all of E, is nowhere even G~teaux differentiable (Mazur [13]), this amounts to a classification of the category of all Banach spaces depending upon whether certain differentiability s ta tements hold. Therefore we say tha t a Banach space is a strong di//erentiability space (SDS) if the following theorem holds for it.

Averaged norms

Abstract: A method to construct an equivalent norm with both a rotundity and a smoothness property in a Banach space having two different equivalent norms, one with the rotundity and one with the smoothness property.

Farthest points in reflexive locally uniformly rotund Banach spaces

Abstract: IfS is a bounded and closed subset of a Banach spaceB, which is both reflexive and locally uniformly rotund, then, except on a set of first Baire category, the points inB have farthest points inS.

Sets with unique farthest points

Abstract: If a certain set in ℝ n has the property, that in some unsymmetric norm each point in ℝ n has a unique farthest point in this set, then it consists of exactly one point

The Convenient Setting of Global Analysis

Abstract: Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifolds Infinite dimensional differential geometry Manifolds of mappings Further applications References Index.

On the maximality of sums of nonlinear monotone operators

Abstract: is called the effective domain of F, and F is said to be locally bounded at a point x e D(T) if there exists a neighborhood U of x such that the set (1.4) T(U) = (J{T(u)\ueU} is a bounded subset of X. It is apparent that, given any two monotone operators Tx and T2 from X to X*, the operator F», + T2 is again monotone, where (1 5) (Ti + T2)(x) = Tx(x) + T2(x) = {*? +x% I xf e Tx(x), xt e T2(x)}. If Tx and F2 are maximal, it does not necessarily follow, however, that F», + T2 is maximal—some sort of condition is needed, since for example the graph of Tx + T2 can even be empty (as happens when D(Tx) n D(T2)= 0). The problem of determining conditions under which Tx + T2 is maximal turns out to be of fundamental importance in the theory of monotone operators. Results in this direction have been proved by Lescarret [9] and Browder [5], [6], [7]. The strongest result which is known at present is :

Nonconvex minimization problems

Abstract: I. The central result. The grandfather of it all is the celebrated 1961 theorem of Bishop and Phelps (see [7], [8]) that the set of continuous linear functionals on a Banach space E which attain their maximum on a prescribed closed convex bounded subset X c E is norm-dense in £*. The crux of the proof lies in introducing a certain convex cone in E, associating with it a partial ordering, and applying to the latter a transfinite induction argument (Zorn's lemma). This argument was later used in different settings by Brondsted and Rockafellar (see [9]) and by F. Browder (see [11]). The various situations can be adequately summarized in a diagram:

Functional Analysis and Infinite-Dimensional Geometry

Abstract: Preface * 1 Basic Concepts in Banach Spaces * 2 Hahn-Banach and Banach Open Mapping Theorems * 3 Weak Topologies * 4 Locally Convex Spaces * 5 Structure of Banach Spaces * 6 Schauder Bases * 7 Compact Operators on Banach Spaces * 8 Differentiability of Norms * 9 Uniform Convexity * 10 Smoothness and Structure * 11 Weakly Compactly Generated Spaces * 12 Topics in Weak Toplogy * References * Index