Barrelled and bornological function spaces
A subset \(K\) of a completely regular topological space \(Y\) is \(t\)-bounded if \(f(K)\) is a bounded subset of \(\mathbb{R}\) for every real continuous function \(f\) on \(Y\). Let \(X\) be a completely regular topological space. \(\nu X\) is the real-compactification of \(X\). Let \(\lambda\) be a cover of \(X\) such that every element of \(\lambda\) is closed and \(t\)-bounded in \(X\) and \(\lambda\) is invariant with respect to finite unions. \(\nu_\lambda(X):= \bigcup\{\text{cl}_{\nu X}(K): K\in\lambda\}\) with the topology inherited from \(\nu X\). If \(Z\) is a subspace of \(\nu X\) containing \(\nu_\lambda(X)\), then a set \(B\subset Z\) is \(\lambda\) bounded in \(Z\) if every lower semi-continuous function \(\varphi: Z\to [0,\infty)\) is bounded on \(B\) provided it is bounded on each \(K\in\lambda\). If \(E\) is a normed space, \(C_\lambda(X, E)\) denotes the set of all \(E\)-valued continuous maps on \(X\) with the topology of uniform convergence on members of \(\lambda\). The authors give the following results: 1. Let \(E\) be a Banach space. Then \(C_\lambda(X, E)\) is barrelled if and only if every \(t\)-bounded subset of \(\nu_\lambda(X)\) is contained in \(\text{cl}_{\nu X}(K)\) for some \(K\in\lambda\). 2. \(C_\lambda(X, E)\) is quasi-barrelled if and only if every \(\lambda\)-bounded subset of \(\nu_\lambda(X)\) is contained in \(\text{cl}_{\nu X}(K)\) for some \(K\in\lambda\). 3. \(C_\lambda(X, E)\) is bornological if and only if every \(\lambda\)-bounded subset of \(\nu X\) is contained in \(\text{cl}_{\nu X}(K)\) for some \(K\in\lambda\). 4. Let \(E\) be a Banach space. Then \(C_\lambda(X, E)\) is ultrabornological if and only if every \(t\)-bounded subset of \(\nu X\) is contained in \(\text{cl}_{\nu K}(K)\) for some \(K\in\lambda\).
scientific article; zbMATH DE number 1423635
scientific article