1. A simple proof of the Borel extension theorem and weak compactness of operators
- Creator:
- Dobrakov, Ivan and Panchapagesan, Thiruvaiyaru V.
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- weakly compact operator on $C_0(T)$, representing measure, and lcHs-valued $\sigma $-additive Baire (or regular Borel; or regular $\sigma $-Borel) measures
- Language:
- English
- Description:
- Let $T$ be a locally compact Hausdorff space and let $C_0(T)$ 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 $u\: C_0(T) \rightarrow X$ when $c_0 \lnot \subset X$ 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].
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public