Let \(X\) be a completely regular Hausdorff space, \( C_{b}(X) \) be the space of all scalar-valued, bounded continuous functions on \(X\) with the strict topology \(\beta\) (some authors call the topology \(\beta_{t}\)), \((E, \xi)\) a Hausdorff locally convex space with \(E', E''\) its dual and bidual, and \(\xi_{\varepsilon}\) the natural topology on \(E''\). \(\mathcal{B}o\) denotes all Borel subsets of \(X\) and \(M(X)\) all scalar-valued, regular Borel measures on \(X\) with a norm topology. Let \(\mathcal{M} \subset M(X)\) be bounded. When \(X\) is compact, A. Grothendieck in his celebrated paper [Can. J. Math. 5, 129–173 (1953; Zbl 0050.10902)] gave many equivalent characterizations for \(\mathcal{M}\) to be relatively weakly compact. When \(X\) is a \(k\)-space, the author extends it to the present setting.
When \(T: (C_{b}(X), \beta) \to (E, \xi)\) is a weakly compact linear mapping, it is well known that it can be represented by an \(E\)-valued regular Borel measure on \(X\). The author extends this to the case when \(T\) is just assumed to be linear and continuous and proves that the representing measure is \(E''\)-valued and \(\xi_{\varepsilon}\)-tight. Also, for a \(k\)-space, some equivalent characterizations for \(T\) to be weakly compact are proved.
Several corollaries and related additional results are established.