## Real Analysis Exchange

### Integral Representations for a Class of Operators on $L_E^1$

Surjit S. Khurana

#### Abstract

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 $P_{1}(\mu, E)$.

#### Article information

Source
Real Anal. Exchange, Volume 36, Number 2 (2010), 417-420.

Dates
First available in Project Euclid: 11 November 2011

https://projecteuclid.org/euclid.rae/1321020509

Mathematical Reviews number (MathSciNet)
MR2476912

#### Citation

Khurana, Surjit S. Integral Representations for a Class of Operators on $L_E^1$. Real Anal. Exchange 36 (2010), no. 2, 417--420. https://projecteuclid.org/euclid.rae/1321020509

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