A dual characterisation of the Radon-Nikodym property

Journal of Mathematical Analysis and Applications, 2013

It is well known that every bounded below and non increasing sequence in the real line converges. We give a version of this result valid in Banach spaces with the Radon-Nikodym property, thus extending a former result of A. Procházka.

Journal of Mathematical Analysis and Applications

Bulletin of the Australian Mathematical Society, 1995

For a continuous convex function on an open convex subset of any Banach space a separability condition on its image under the subdifferential mapping is sufficient to guarantee the generic Frechet differentiability of the function. This gives a direct insight into the characterisation of Asplund spaces.

2013

In this paper we present a characterization of Banach spaces possessing the Radon-Nikodym property in terms of the limit average range of additive interval functions defined on [0, 1] and taking values in a Frechet space.

Pure and Applied Mathematics, 2018

We have introduced two new notions of convexity and closedness in functional analysis. Let X be a real normed space, then C(⊆ X) is functionally convex (briefly, F-convex), if T (C) ⊆ R is convex for all bounded linear transformations T ∈ B(X, R); and K(⊆ X) is functionally closed (briefly, F-closed), if T (K) ⊆ R is closed for all bounded linear transformations T ∈ B(X, R). By using these new notions, the Alaoglu-Bourbaki-Eberlein-Šmuljan theorem has been generalized. Moreover, we show that X is reflexive if and only if the closed unit ball of X is F-closed. James showed that for every closed convex subset C of a Banach space X, C is weakly compact if and only if every f ∈ X * attains its supremum over C at some point of C. Now, we show that if A is an F-convex subset of a Banach space X, then A is bounded and F-closed if and only if every element of X * attains its supremum over A at some point of A.

Communications of the Korean Mathematical Society

In this note we present sufficient conditions for the existence of Radon-Nikodym derivatives (RND) of operator valued measures with respect to scalar measures. The RND is characterized by the Bochner integral in the strong operator topology of a strongly measurable operator valued function with respect to a nonnegative finite measure. Using this result we also obtain a characterization of compact sets in the space of operator valued measures. An extension of this result is also given using the theory of Pettis integral. These results have interesting applications in the study of evolution equations on Banach spaces driven by operator valued measures as structural controls.

Mathematical Programming, 2009

As well known, the Moreau-Rockafellar-Robinson internal point qualification condition is sufficient to ensure that the infimal convolution of the conjugates of two extended-real-valued convex lower semi-continuous functions defined on a locally convex space is exact, and that the subdifferential of the sum of these functions is the sum of their subdifferentials. This note is devoted to proving that this condition is, in a certain sense, also necessary, provided the underlying space is a Banach space. Our result is based upon the existence of a non-supporting weak -closed hyperplane to any weak -closed and convex unbounded linearly bounded subset of the topological dual of a Banach space.

Israel Journal of Mathematics, 2005

We prove that there exists a Lipschitz function from ℓ 1 into IR 2 which is Gâteaux-differentiable at every point and such that for every x, y ∈ ℓ 1 , the norm of f ′ (x) − f ′ (y) is bigger than 1. On the other hand, for every Lipschitz and Gâteaux-differentiable function from an arbitrary Banach space X into IR and for every ε > 0, there always exists two points x, y ∈ X such that f ′ (x)−f ′ (y) is less than ε. We also construct, in every infinite dimensional separable Banach space, a real valued function f on X, which is Gâteaux-differentiable at every point, has bounded non-empty support, and with the properties that f ′ is norm to weak * continuous and f ′ (X) has an isolated point a, and that necessarily a = 0.

Journal of Mathematical Analysis and Applications, 1973

Introduction. Let J be a <r-field of subsets of an abstract set M and let m(e) be a non-negative measure function defined on J. The classical Radon-Nikodym theorem [17, p. 36](1) states that, if M is the union of a countable number of sets of finite measure, then a necessary and sufficient condition for a completely additive real function R(e), defined over J, to be a Lebesgue integral (with respect to m(e)) is that R(e) be absolutely continuous relative to m(e). Our purpose is to extend this theorem to functions with values in an arbitrary Banach space and apply the resulting theorem to obtain an integral representation for the general bounded linear transformation on the space of summable functions to an arbitrary Banach space. A number of writers [4, 6, 7, 8, 11, 12, 13, 14] have obtained similar extensions; however they have all imposed restrictions either on the Banach space or on the completely additive functions considered. The theorem proved here is free of all such restrictions. It is evident that any such generalization of the Radon-Nikodym theorem will involve a corresponding generalization of the Lebesgue integral, of which there are many. A variation of an integral studied in detail by B. J. Pettis(2) will be used here. A point function x(p) defined on ¥ to a Banach space X is said to be Pettis integrable [12] provided there exists a function X(e) on J to Ï such that, for each element x of the space 3-adjoint to ï and each element e of J, the function x(x(p)) is Lebesgue integrable on the set e to the value x(X(e)). Whenever X(e) exists, it is completely additive and absolutely continuous relative to m(e). On the other hand, Pettis [12, p. 303] gave an example of a completely additive function which is absolutely continuous but is not an integral in his sense. This shows that the ordinary Pettis integral cannot appear in a general Radon-Nikodym theorem. However, without changing essentially the definition or general properties of the integral, we can enlarge the class of functions admissable for integration (so that it contains certain functions other than point functions) and thus obtain an integral which will serve our purposes. The class of functions which we will admit for integration consists of all multivalued set functions x(e) defined for elements of J having finite, nonzero Except for §5, the contents of this paper were presented to the Society, September 12, 1943. The results in §5 were presented February 27, 1944, under the title Representation of linear transformations on summable functions.

Operators and Matrices, 2012

It is a classical fact, due to Day, that every separable Banach space admits an equivalent Gâteaux smooth renorming. In fact, it admits an equivalent uniformly Gâteaux smooth norm, as was shown later byŠmulyan. It is therefore rather unexpected that the existence of Gâteaux smooth renormings satisfying various quantitative estimates on the directional derivative has rather strong structural and geometrical implications for the space. For example, by a result of Vanderwerff, if the directional derivatives satisfy a p-estimate, where p varies arbitrarily with respect to the point and the direction in question, then the Banach space must be an Asplund space. In the present survey paper, we discuss the interplay between various types of Gâteaux differentiability of norms and extreme points with the geometry of separable Banach spaces. In particular, we present various characterizations of Asplund, reflexive, superreflexive, and other classes of separable Banach spaces, via smooth as well as rotund renormings. We also include open problems of various levels of difficulty, with the hope of stimulating research in the area of smoothness and renormings of Banach spaces. In nonlinear analysis, the differentiability of norms plays an important role. The most important type of differentiability is that of Fréchet differentiability. However, in many instances it suffices to use weaker forms of differentiability, i.e., variants of the Gâteaux differentiability (that are more often accessible). This happens especially when some convexity arguments can be combined with Baire category techniques. The present paper surveys some of these results and discusses several ideas and constructions in their proofs. We focus on the interplay of these concepts with the geometry of separable spaces, for example with problems on containment of c 0 or 1 , with superreflexivity, the Radon-Nikodým property, etc. Several open problems in this area are discussed. We refer to, e.g., [Gode], [DGZb], [Fab], [AlKal06], [BoVa10], and [FHHMZ] for all unexplained notions and results used in this note.

Pacific Journal of Mathematics, 1973

Local convexity appears-by the Hahn-Banach theorem-as a sufficient condition for the (topological) dual of a topological vector space to separate points from closed subspaces. The aim in the present article is to obtain necessary conditions, in terms of local convexity, for the latter statement to hold for a metrizable topological vector space. In particular, certain classes of such spaces are found, for which local convexity is, really, a necessary condition for the dual to separate points from closed subspaces. The course of proof goes via consideration of the more general question how two metrizable vector space topologies on a linear space must be related to each other, given that the class of linear subspaces which are closed in one of them is larger than the class of those closed in the other.

Studia Mathematica, 2018

We use the smooth variational principle and a standard renorming to give a short direct proof to the classical Bishop-Phelps-Bollobás theorem on the density of norm-attaining functionals for weakly compactly generated Banach spaces. Then we show that a slight adjustment of a known Preiss-Zajíček differentiability argument provides for a simple, useful characterization of individual norms on separable Banach spaces admitting residual sets of norm-attaining functionals in terms of Fréchet differentiability of their dual norms.

Journal of the London Mathematical Society, 2000

It is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on % " (Γ) and % _ (Γ) are Fre! chet differentiable on a dense G δ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein-Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.

We study the equivalence property of the I- and II-Radon-Nikodym (resp. I- and II-Complete Continuity) property types of Banach spaces. As a bi-product, we obtain a proof of the following F. Lust-Picard&#39;s conjecture: a subset $$\Lambda$$ of a discrete abelian group is a Rosenthal set if and only if $$C_\Lambda \notcontain c_0$$.

Studia Mathematica, 2009

An exact Radon-Nikodym derivative is obtained for a pair (I, J) of positive linear functionals, with J absolutely continuous with respect to I, using a notion of exhaustion of I on elements of a function algebra lattice.