Giles skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. In the first part of this article the following result was proved. Radonnikodym property and area formula 3 as in the. Some characterizations of the analytic radonnikodqm property in.
Examples of radonnikodym spaces are re exive banach spaces and separable dual spaces. If x is a banach space, then x has the radonnikodym property. This is by no means a comprehensive discussion of the radonnikodym property. The radon nikodym property in conjugate banach spaces by charles stegalll abstract. A good resource on the radonnikodym property in reflexive.
The radonnikodym property for banach spaces a survey of geometric aspects, in functional analysis. Nikodym property then need x imbed in a separable conjugate. Then g is a proper lower semicontinuous convex function which is also epipointed, since g. Introduction the radonnikodym property rnp for banach spaces, which led to a remarkable theory developed largely between 1965 and 1980. Maynard, a geometric characterization of banach spaces having the radonnikodym property, trans. The radonnikodym property and conjugate banach spaces ii, 1981 by stegall venue. The radonnikodym property and conjugate banach spaces ii. Differentiability of lipschitz maps from metric measure spaces to banach spaces with the radon nikodym property jeff cheeger and bruce kleiner abstract.
On metric characterizations of the radonnikodym and. Metric spaces nonembeddable into banach spaces with the radon. The analytic radonnikodym property for a complex banach space x is charac terized in. Banach space theory and its applications proceedings of. Metric spaces nonembeddable into banach spaces with the radon nikodym property and thick families of geodesics mikhail ostrovskiiy march 20, 2014 abstract.
Remark 18 formula 14 also fails in banach spaces without the radonnikodym property. Regularized minimization of convex functions in radon nikodym. Complex measures, radonnikodym theorem and the dual of l. Zizler, differentiability of convex functions and the convex point of continuity property in banach spaces, to appear in israel j. Surveys and recent results, proceedings of the paderborn conference on functional analysis ed. Banach spaces having the radonnikodym property and numerical index 1 miguel mart in communicated by jonathan m. Im looking for a good resource that builds the theory of the radonnikodym property. Pdf approximative compactness and radonnikodym property in. On conjugate banach spaces with the radonnikodym property. Some characterizations of the analytic radonnikodym property.
Metric spaces nonembeddable into banach spaces with the radonnikodym property and thick families of geodesics mikhail ostrovskiiy march 20, 2014 abstract. We provide a metric area formula that applies to these mappings and more generally to all almost everywhere metrically differentiable lipschitz mappings defined on a carnot group. Odell and rosenthal or showed that the double dual of a separable banach space e with the weak. Im not particularly interested in the measuretheoretic characterisation. Differentiability of lipschitz maps from metric measure. Comparable differentiability characterisations of two classes of banach spaces volume 56 issue 2 j. A note on trees in conjugate banach spaces sciencedirect.
A banach space, x is said to have the radonnikodym property. Abstract a banach space x over the field of real numbers ffi. The dual of a banach space x has the radonnikodym property if and only if for every closed, linear separable subspace y of x, y is separable. A geometric characterization of the radonnikodym property in. It is known that the space has the radonnikodym property, but and the spaces.
The radonnikodym property and weighted trees in banach spaces. Spaces with radonnikodym property include separable dual spaces this is the dunfordpettis theorem and reflexive spaces, which include, in particular, hilbert spaces. Im looking for a good resource that builds the theory of the radon nikodym property. Asplund spaces, kadecklee property, rotund norm, locally uniform convex norm, slices 1. Notes on sobolev spaces peter lindqvist norwegian university of science and technology.
Stegall, the radon nikodym property of conjugate banach spaces, trans. We characterize conjugate banach spaces x having the radon nikodym property as those spaces such that any separable subspace of x has a separable conjugate. Let us mention that, in rnp spaces, the set expcof strongly exposing functionals of a nonempty closed convex bounded set cis dense in x. They called the analytic radonnikodym property the. The radonnikodym and the kreinmilman properties in. A unifying radonnikodym theorem through nonstandard hulls zimmer, g. Comparable differentiability characterisations of two. We characterize conjugate banach spaces x having the. Rosenthal, the heredity problem for weakly compactly generated banach spaces, to appear. Representation of operators with martingales and the radon.
The dual of a banach space x has the radon nikodym property if and only if for every closed, linear separable subspace y of x, y is separable. If x is a separable banach space possessing the radon. The problem of metric characterization of the radon nikodym property and reexivity can be understood and approached in several di erent ways. The choice of notation and the name of the function reflects the fact that the function is analogous to a. A simple proof is given of the following result due to stegall. Phelps, the radonnikodym property and dentable sets in banach spaces, p roc.
Metric spaces nonembeddable into banach spaces with the. In fact, if a l 1space contains an isomorphic copy of all l 1spaces with the radonnikodym. We prove a rademachertype theorem for lipschitz mappingsfrom a subset of a carnot group to a banach homogeneous group, equipped with a suitably weakened radon nikodym property. It is shown that a banach space has the radonnikodym property if and only if it. If y is a banach space and the generalization of the radonnikodym theorem also holds, mutatis mutandis, for functions with values in y, then y is said to have the radonnikodym property. Edgar department of mathematics, the ohio state university, columbus, ohio 43210 communicated by c. Boundary values of functions in vectorvalued hardy spaces. For lp spaces, we will use the radonnikodym theorem to show that lpx may be identi ed with lp0x for 1 pdf b. Regularized minimization of convex functions in radon. Nearly dentable banach spaces and applications article pdf available in journal of function spaces 20153. Indeed let x c 0n and let g be the indicator function of the closed unit ball of x. The radonnikodym and the kreinmilman properties in banach. Phelps, the radon nikodym property and dentable sets in banach spaces, p roc. P every nonempty relatively weakly open subset of the closed unit ball of x has.
Boundary values of vectorvalued hardy spaces on nonsmooth domains and the radonnikodym property ocamposalgado, hugo, riveranoriega, jorge, and san martin, luis, banach journal of mathematical analysis, 2016. This lemma shows that weakly compactly genera ted conjugate spaces have the. Danilevich to show that for any real banach lattice e, the clifford module x a n. Pdf approximative compactness and radonnikodym property. A duality theory for banach spaces with the convex point. Radon nikodym property and area formula for banach homogeneous group targets valentino magnani and tapio rajala abstract. Stegall, the radonnikodym property of conjugate banach spaces, trans. A variational henstock integral characterization of the radonnikodym property, illinois j. The radonnikodym theorem is used in most proofs of the representation theorem.
The weak radonnikodym property in conjugate banach spaces. In case of unbounded sets, one has the following result. The problem of metric characterization of the radonnikodym property and reexivity can be understood and approached in several di erent ways. Pdf an analytic radonnikodym property related to systems. Representation of operators with martingales and the radonnikodym property. We prove a rademachertype theorem for lipschitz mappings from a subset of a carnot group to a banach homogeneous group, equipped with a suitably weakened radonnikodym property. More facts about conjugate banach spaces with the radon nikodym property by c. Comparable differentiability characterisations of two classes. Pdf representation of operators with martingales and the. More facts about conjugate banach spaces with the radon. The theory concerned with rnp is an impressive example. Approximative compactness and radonnikodym property in w. The radonnikodym property in conjugate banach spaces.
Banach space theory and its applications proceedings of the first romanian gdr seminar held at bucharest, romania, august 31 september 6, 1981. More facts about conjugate banach spaces with the radonnikodym property by c. We show that a geodesic metric space which does not admit bilipschitz embeddings into banach spaces with the radonnikodym property does not nec. Connections between metric characterizations of superreflexivity and the radonnikodym property for dual banach spaces. Rnp if and only if every separable, linear subspace of x has a separable dual. A duality theory for banach spaces with the convex pointof. Satellite conference on banach spaces icm2006 on operators that preserve the radonnikodym. Journal of functional analysis 80, 349357 1988 some characterizations of the analytic radon nikodym property in banach spaces p. A more direct proof in the one dimensional case is given in r. Let x be a banach space with the radon nikod ym property. Relatively weakly open sets in closed balls of banach.
Stegall, the radonnikodym property in conjugate banach spaces, to appear. Id like the geometry of banach spaces version, involving strongly exposed points of convex sets. On metric characterizations of the radonnikodym and related. Radon nikodym property as those spaces such that any separable subspace of x has a. A banach space x has rnp if and only if every bounded subset of x is dentable. Newest radonnikodym questions mathematics stack exchange. We prove the differentiability of lipschitz maps x v, where x denotes a pi space, i. The existence of in nitedimensional extremely noncomplex banach spaces had been asked in 15, question 4. Lp spaces a linear functional is bounded if and only if it is continuous. Stegall,the radonnikodym property in conjugate banach spaces, trans. The radonnikodym property 215 bounded, nondentable subset then it has a bounded, nonsdentable subset. The radonnikodym property in conjugate banach spaces by charles stegalll abstract. Radonnikodym property if and only if any operator from l to x that factors through l. Thus if the set of extreme points in the unit ball a1 is closed open, then the set of extreme points in b1 is also closed open if there is a linear isometry.
A geometric characterization of the radonnikodym property. We show that a geodesic metric space which does not admit bilipschitz embeddings into banach spaces with the radon nikodym property does not nec. Then t preserves extreme points, namely t e is an extreme point of b1 if and only if e is an extreme point of a1. These results may be combined to give the following result. We characterize conjugate banach spaces x having the radonnikodym property as those spaces such that any separable subspace of x has a separable conjugate. Mathematical institute, university of amsterdam, roetersstraat is. The weak radon nikodym property in conjugate banach spaces.
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