A Simple Proof of the Borel Extension Theorem and Weak Compactness of Operators

Authors: Dobrakov I.¹; Panchapagesan T.V.²

Source: Czechoslovak Mathematical Journal, Volume 52, Number 4, December 2002 , pp. 691-703(13)

Publisher: Springer

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Abstract:

Let T be a locally compact Hausdorff space and let C₀(T) be the Banach space of all complex valued continuous functions vanishing at infinity in T, provided with the supremum norm. Let X be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of X-valued sigma

-additive Baire measures on T is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map u: C₀(T) rarr

X when C₀

X are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].

Keywords: weakly compact operator on C0(T); representing measure; lcHs-valued sigma -additive Baire (or regular Borel, or regular sigma -Borel) measures

Document Type: Research article

DOI: http://dx.doi.org/10.1023/B:CMAJ.0000027224.01146.63

Affiliations: 1: Mathematical Institute, Slovak Academy of Sciences, Scaron tefánikova 49, Bratislava, Slovakia tefánikova 49, Bratislava, Slovakia "> 2: Departamento de matemáticas, Facultad de Ciencias, Universidad de los Andes, Merida, Venezuela, Email: panchapa@ciens.ula.ve