The Journal of the Indian Mathematical Society

N. D. Chakraborty ¹, M. Sahu ¹, B. Sen ¹

Affiliations
1 Department of Mathematics, The University of Burdwan, Burdwan-713104 (W.B.), IN

Abstract

Let (Ω,Σ,μ) be a probability space and let N⊂ca (Σ) be a bounded family of positive measures and X be a Banach space. Let P₁(N,X) be the Pettis-type spaces with respect to N. Assuming that X is weakly sequentially complete, we prove the completeness of P₁(N,X) with respect to the Pettis semi-norm. Also we prove the Vitali’s convergence theorem, Lebesgue dominated theorem and a necessary and sufficient condition for a function to belong to P₁(N,X).

Keywords

Lebesgue-Type Spaces, Pettis-Type Spaces, Vitali’s Convergence Theorem, Lebesgue Dominated Convergence Theorem, Young’s Function, Orlicz Spaces.

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Sk. Jaker Ali ¹, N. D. Chakraborty ¹

Affiliations
1 Department of Mathematics, University of Burdwan, Burdwan—713104, West Bengal, IN

Abstract

The notion of weak Radon-Nikodym property in a Banach space was first introduced by Musial in his famous paper [9]. The aim of this paper is to study the vector-valued harmonic functions defined on the open unit disc in the complex plane with values in a Banach space and to find how they are related with the weak Radon-Nikodym property.

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