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2010/2011 Integral Representations for a Class of Operators on \(L_E^1\)
Surjit S. Khurana
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Real Anal. Exchange 36(2): 417-420 (2010/2011).


Let \((X, \mathcal{A}, \mu)\) be a finite measure space, \(E\) a locally convex Hausdorff space, \(L_{E}^{1}\) the space of functions \(f: X \to E\) which are \(\mu\)-integrable by semi-norms, \(P(\mu, E)\) the space of Pettis integrable functions and \(P_{1}(\mu, E)\) those elements of \(P(\mu, E)\) which are measurable by semi-norms. We prove that a linear continuous mapping \( T: L_{E}^{1} \to E\) is of the form \(T(f)= \int g f d \mu\) (\(g \in L^{\infty}\)) if and only if \( h( T(f))=0\) whenever \( h\circ f=0 \) for any \( f \in L_{E}^{1}, h \in E'\). Similar results are proved for \(P(\mu, E)\) and \(P_{1}(\mu, E)\).


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Surjit S. Khurana. "Integral Representations for a Class of Operators on \(L_E^1\)." Real Anal. Exchange 36 (2) 417 - 420, 2010/2011.


Published: 2010/2011
First available in Project Euclid: 11 November 2011

MathSciNet: MR2476912

Primary: 28A32 , 28C05 , 46G10
Secondary: 28A40

Keywords: integrable by semi-norms , perfect measure , Pettis integration

Rights: Copyright © 2010 Michigan State University Press

Vol.36 • No. 2 • 2010/2011
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