학술저널
A Note on the Pettis Integral and the Bourgain Property
- 충청수학회
- Journal of the Chungcheong Mathematical Society
- Volume 5, No. 1
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1992.06159 - 165 (7 pages)
- 0
In 1986, R. Huff [3] showed that a Dunford integrable function is Pettis integrable if and only if T : x* →L₁ (μ) is weakly compact operator and {T(K(F,e)) I F ⊂ X, F: finite and ε > 0} = {0}. In this paper, we introduce the notion of Bourgain property of real valued functions formulated by J. Bourgain [2]. We show that the class of pettis integrable functions is linear space and if fis bounded function with Bourgain property, then T: x** → L₁(μ) by T(x**) = x** f is weak* - to - weak linear operator. Also, if operator T : L₁ (μ) → x* with Bourgain property, then we show that f is Pettis representable.
ABSTRACT
I. Introduction
II. Pettis integral
III. Pettis representable
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