A Note on Weak Differentiability of Pettis Integrals
نویسنده
چکیده
Pettis raised the question whether or not separability of the range space implies almost everywhere weak differentiability of Pettis integrals. Phillips has given an example which answers this question in the negative. His construction is based on a sequence of orthogonal vectors in Hilbert space. We present here a different example of the same type of function. Our basic construction is that of a function defined to the space C. Using that function as a basis, we are able to give a specific construction of such a function defined to each member of a large class of Banach spaces.
منابع مشابه
A Second Note on Weak Differentiability of Pettis Integrals
In a recent paper the author proved that if Q is any compact metric space containing non-denumerably many points and C(Q) is the Banach space of all continuous functional over ft, then there is a Pettis integrable function from the unit interval to C(0) whose integral fails to be weakly differentiable on a set of positive measure. The purpose of this note is to obtain the same result, assuming ...
متن کاملNowhere Weak Differentiability of the Pettis Integral
For an arbitrary in nite-dimensional Banach space X, we construct examples of strongly-measurable X-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly di erentiable; thus, for these functions the Lebesgue Di erentiation Theorem fails rather spectacularly. We also relate the degree of nondi erentiability of the inde nite Pettis integral to the cotype of X, fr...
متن کاملOrder Almost Dunford-Pettis Operators on Banach Lattices
By introducing the concepts of order almost Dunford-Pettis and almost weakly limited operators in Banach lattices, we give some properties of them related to some well known classes of operators, such as, order weakly compact, order Dunford-Pettis, weak and almost Dunford- Pettis and weakly limited operators. Then, we characterize Banach lat- tices E and F on which each operator from E into F t...
متن کاملSET - VALUED CHOQUET - PETTIS INTEGRALS Chun - Kee
In this paper, we introduce the Choquet-Pettis integral of set-valued mappings and investigate some properties and convergence theorems for the set-valued Choquet-Pettis integrals.
متن کاملBanach lattices with weak Dunford-Pettis property
We introduce and study the class of weak almost Dunford-Pettis operators. As an application, we characterize Banach lattices with the weak Dunford-Pettis property. Also, we establish some sufficient conditions for which each weak almost Dunford-Pettis operator is weak Dunford-Pettis. Finally, we derive some interesting results. Keywords—eak almost Dunford-Pettis operator, almost DunfordPettis o...
متن کامل