We investigate possible quantifications of the Dunford-Pettis property. We show, in particular, that the Dunford-Pettis property is automatically quantitative in a sense. Further, there are two incomparable mutually dual stronger versions of a quantitative Dunford-Pettis property. We prove that L 1 spaces and C(K) spaces posses both of them. We also show that several natural measures of weak non-compactness are equal in L 1 spaces.2010 Mathematics Subject Classification. 46B03; 46B20; 47B07; 47B10.

basic properties. The most important fact is that many descriptive properties are stable with respect to perfect mappings, which allows us to transfer abstract Borel affine functions to the setting of compact convex sets. Simplicial function spaces are studied in Chapter 6. We discuss several classes of simplicial function spaces, namely the Bauer and Markov simplicial function spaces and spaces with boundary of type F σ. Among other results, the abstract Dirichlet problem for continuous and non-continuous functions is considered. Choquet simplices are presented at the end of the chapter. Next we generalize the basic concepts for function cones since they are indispensable in potential theory. We focus in particular on ordered compact convex sets. Analogues of faces in a non-convex setting, so-called Choquet sets, are investigated in Chapter 8. The main result is a characterization of simplicial spaces by means of Choquet sets. Suitably chosen families of closed extremal sets generate interesting boundary topologies on the set of extreme points. Chapter 9 studies these topologies and functions continuous with respect to them. It turns out that maximal measures induce measures on sets of extreme points that are regular with respect to boundary topologies. The last section is devoted to a study of a facial topology and facially continuous functions. Chapter 10 collects several deeper results on function spaces and compact convex sets. Among others, study of Shilov and James boundaries, Lazar's improvement of the Banach-Stone theorem, results on automatic boundedness of affine and convex functions, embedding of ℓ 1 in Banach spaces, metrizability of compact convex sets and their open images and some topological properties of the set of extreme points. The Lazar selection theorem and its consequences occupy the first part of Chapter 11. The second part is devoted to a presentation of Debs' proof of Talagrand's theorem on measurable selectors. Chapter 12 is concerned with two methods of constructing new function spaces: products and inverse limits. We show that both operations preserve simpliciality and describe resulting boundaries. The inverse limits lead to an interesting description of metrizable simplices as inverse limits of finite-dimensional simplices. The general results are illustrated by a construction of the Poulsen simplex and a couple of compact convex sets due to Talagrand. In Chapter 13, general results from Choquet's theory are applied to potential theory and several of its basic notions are investigated from this perspective. Important function cones and spaces appearing in potential theory are studied in detail, in particular, in connection to various solution methods for the Dirichlet problem. The functional analysis approach makes it possible to provide an interesting interpretation, for instance, of balayage and regular points in terms of representing measures and the Choquet boundary of suitable spaces and cones. The exposition covers potential theory for the Laplace equation and the heat equati...

We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in $L$-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space $X$ with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.Comment: 9 page

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We introduce two measures of weak non-compactness Ja E and Ja that quantify, via distances, the idea of boundary that lies behind James's Compactness Theorem. These measures tell us, for a bounded subset C of a Banach space E and for given x * ∈ E * , how far from E or C one needs to go to find. A quantitative version of James's Compactness Theorem is proved using Ja E and Ja, and in particular it yields the following result. Let C be a closed convex bounded subset of a Banach space E and r > 0. If there is an element x * * 0 in C w * whose distance to C is greater than r, then there is x * ∈ E * such that each x * * ∈ C w * at which sup x * (C) is attained has distance to E greater than 1 2 r. We indeed establish that Ja E and Ja are equivalent to other measures of weak non-compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-compactness can be equalities and when the inequalities are sharp.