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.
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 which is not Pettis integrable on any subinterval in [a,b], while belongs to 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...
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 is a distance function determined by a closed subset of a Banach space with a uniformly Gâteaux differentiable norm, then the set of points of at which is not Gâteaux differentiable is not only a first category set, but it is even -porous...
Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function defined on a separable Banach space are studied. The conditions are in terms of a majorization of by a -smooth function, separability of the boundary for or an approximation of by Fréchet smooth convex functions.
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.
Let be a bounded domain of Lyapunov and a holomorphic function in the cylinder and continuous on . If for each fixed in some set , with positive Lebesgue measure , the function of can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then can be holomorphically continued to , where is some analytic (closed pluripolar) subset of .