Weak Integrals and Bounded Operators in Topological Vector Spaces

ABSTRACT

Let *X* be a
topological vector space and let *S* be a locally compact space. Let us consider the function space of all
continuous functions , vanishing outside a compact set of *S*, equipped with an appropriate topology. In
this work we will be concerned with the relationship between bounded operators* *, and *X*-valued integrals on . When *X* is a Banach space, such relation has been
completely achieved via Bochner integral in [1]. In
this paper we investigate the context of locally convex spaces and we will focus
attention on weak integrals, namely the Pettis integrals. Some
results in this direction have been obtained, under some special conditions on
the structure of *X* and
its topological dual *X*^{}*. In this work we consider the case of a semi reflexive locally convex space and prove that each Pettis integral with respect to a signed
measure *μ*, on *S *gives rise to a unique bounded operator , which has the given Pettis integral form.

Cite this paper

L. Meziani and S. Alsulami, "Weak Integrals and Bounded Operators in Topological Vector Spaces,"*Advances in Pure Mathematics*, Vol. 3 No. 5, 2013, pp. 475-478. doi: 10.4236/apm.2013.35068.

L. Meziani and S. Alsulami, "Weak Integrals and Bounded Operators in Topological Vector Spaces,"

References

[1] L. Meziani, “Integral Representation for a Class of Vector Valued Operators,” Proceedings of the American Mathematical Society, Vol. 130, No. 7, 2002, pp. 2067-2077. doi:10.1090/S0002-9939-02-06336-0

[2] S. Kakutani, “Concrete Representation of Abstract (M). Spaces,” Annals of Mathematics, Vol. 42, No. 4, 1941, pp. 994-1024. doi:10.2307/1968778

[3] J. Diestel and J. J. Uhl, “Vector Measures,” Mathematical Surveys, No. 15, 1977.

[4] L. Meziani, “A Theorem of Riesz Type with Pettis Integrals in TVS,” Journal of Mathematical Analysis and Applications, Vol. 340, No. 2, 2008, pp. 817-824. doi:10.1016/j.jmaa.2007.09.005

[5] A. Wilanski, “Modern Methodsin Topological Vector Spaces,” McGraw-Hill, New York, 1978.

[1] L. Meziani, “Integral Representation for a Class of Vector Valued Operators,” Proceedings of the American Mathematical Society, Vol. 130, No. 7, 2002, pp. 2067-2077. doi:10.1090/S0002-9939-02-06336-0

[2] S. Kakutani, “Concrete Representation of Abstract (M). Spaces,” Annals of Mathematics, Vol. 42, No. 4, 1941, pp. 994-1024. doi:10.2307/1968778

[3] J. Diestel and J. J. Uhl, “Vector Measures,” Mathematical Surveys, No. 15, 1977.

[4] L. Meziani, “A Theorem of Riesz Type with Pettis Integrals in TVS,” Journal of Mathematical Analysis and Applications, Vol. 340, No. 2, 2008, pp. 817-824. doi:10.1016/j.jmaa.2007.09.005

[5] A. Wilanski, “Modern Methodsin Topological Vector Spaces,” McGraw-Hill, New York, 1978.