Weak Integrals and Bounded Operators in Topological Vector Spaces


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.

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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.

Conflicts of Interest

The authors declare no conflicts of interest.


[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.

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