On a theorem of Daneš and the principle of equicontinuity. (English) Zbl 0603.46013
The purpose of this paper is to establish an appropriate principle of equicontinuity for topological groups without the use of Baire’s category argument. J. Daneš published a paper in ibid. V. Ser. A 13, 282-286 (1976; Zbl 0345.22003) containing theorems from which the Banach-Steinhaus theorem on the condensation of singularities and other interesting results may be derived. In this note we show that a theorem due to Daneš can be proved under weaker hypothesis. Our result enables us to prove another theorem of Daneš under weaker conditions; it also enables us to prove the following version of the principle of equicontinuity for topological groups.
”Let (G,\(\tau)\) be a commutative topological group of the second category and let (H,\(\zeta)\) be a commutative group whose topology is determined by a family of \(\alpha\)-quasinorms (a quasi-norm p on G is an \(\alpha\)-quasi-norm if \(p(2x)=2^{\alpha}p(x)\) for \(x\in G\) and a fixed positive number \(\alpha)\). Suppose that \(\{\phi_ n:=1,2,...\}\) is a sequence of continuous homomorphisms from G to H such that, for each \(x\in G\), the set \(\{\phi_ n(x):n=1,2,...\}\) is \(\zeta\)-bounded. Then the sequence \(\{\phi_ n\}\) is equicontinuous”.
”Let (G,\(\tau)\) be a commutative topological group of the second category and let (H,\(\zeta)\) be a commutative group whose topology is determined by a family of \(\alpha\)-quasinorms (a quasi-norm p on G is an \(\alpha\)-quasi-norm if \(p(2x)=2^{\alpha}p(x)\) for \(x\in G\) and a fixed positive number \(\alpha)\). Suppose that \(\{\phi_ n:=1,2,...\}\) is a sequence of continuous homomorphisms from G to H such that, for each \(x\in G\), the set \(\{\phi_ n(x):n=1,2,...\}\) is \(\zeta\)-bounded. Then the sequence \(\{\phi_ n\}\) is equicontinuous”.
MSC:
46A99 | Topological linear spaces and related structures |
22A10 | Analysis on general topological groups |
46A08 | Barrelled spaces, bornological spaces |