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”.