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On coincidence of Pettis and McShane integrability

Marián J. Fabián (2015)

Czechoslovak Mathematical Journal

R. Deville and J. Rodríguez proved that, for every Hilbert generated space X , every Pettis integrable function f : [ 0 , 1 ] → X is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space X and a scalarly null (hence Pettis integrable) function from [ 0 , 1 ] into X , which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from [ 0 , 1 ] (mostly) into C ( K ) spaces. We focus in more detail on the behavior...

On convex functions in c0(w1).

Petr Hájek (1996)

Collectanea Mathematica

It is proved that no convex and Fréchet differentiable function on c0(w1), whose derivative is locally uniformly continuous, attains its minimum at a unique point.

On Denjoy type extensions of the Pettis integral

Kirill Naralenkov (2010)

Czechoslovak Mathematical Journal

In this paper two Denjoy type extensions of the Pettis integral are defined and studied. These integrals are shown to extend the Pettis integral in a natural way analogous to that in which the Denjoy integrals extend the Lebesgue integral for real-valued functions. The connection between some Denjoy type extensions of the Pettis integral is examined.

On Denjoy-Dunford and Denjoy-Pettis integrals

José Gámez, José Mendoza (1998)

Studia Mathematica

The two main results of this paper are the following: (a) If X is a Banach space and f : [a,b] → X is a function such that x*f is Denjoy integrable for all x* ∈ X*, then f is Denjoy-Dunford integrable, and (b) There exists a Dunford integrable function f : [ a , b ] → c 0 which is not Pettis integrable on any subinterval in [a,b], while ʃ J f belongs to c 0 for every subinterval J in [a,b]. These results provide answers to two open problems left by R. A. Gordon in [4]. Some other questions in connection with Denjoy-Dundord...

On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function

Luděk Zajíček (1997)

Commentationes Mathematicae Universitatis Carolinae

We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gâteaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gâteaux differentiable bump function. In particular, our result implies the following theorem: If d is a distance function determined by a closed subset A of a Banach space X with a uniformly Gâteaux differentiable norm, then the set of points of X ∖ A at which d is not Gâteaux differentiable is not only a first category set, but it is even σ -porous...

On Fréchet differentiability of convex functions on Banach spaces

Wee-Kee Tang (1995)

Commentationes Mathematicae Universitatis Carolinae

Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function f defined on a separable Banach space are studied. The conditions are in terms of a majorization of f by a C 1 -smooth function, separability of the boundary for f or an approximation of f by Fréchet smooth convex functions.

On Gateaux differentiable bump functions

Francisco Hernández, Stanimir Troyanski (1996)

Studia Mathematica

It is shown that the order of Gateaux smoothness of bump functions on a wide class of Banach spaces with unconditional basis is not better than that of Fréchet differentiability. It is proved as well that in the separable case this order for Banach lattices satisfying a lower p-estimate for 1≤ p < 2 can be only slightly better.

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