Journal of the Australian Mathematical Society - Series A Vol. 65 Part 2 (1998) Operators and the space of integrable scalar functions with respect to a Fréchet-valued measure Antonio Fernández Departamento Matemática Aplicada II Escuela Superior de Ingenieros Camino de los Descubrimientos, s/n 41092-Sevilla Spain e-mail: anfercar@matinc.us.es and Francisco Naranjo Abstract: We consider the space $L^{1}(\nu ,X)$ of all real functions that are integrable with respect to a measure $\nu$ with values in a real Fréchet space X. We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of $% \ell ^{\infty }$ in the space of all linear continuous operators from a complete DF-space Y to a Fréchet lattice E with the Lebesgue property and the coincidence of this space with some ideal of compact operators. We give sufficient conditions on the measure $\nu$ and the space X that imply that $L^{1}(\nu ,X)$ has the Dunford-Pettis property. Applications of these results to Fréchet AL-spaces and Köthe sequence spaces are also given. PDF file size: 118K © Copyright 1998, Australian Mathematical Society