A Remark on Condensation of Singularities

Recently Alan D. Sokal in Amer. Math. Monthly 118 (2011), No. 5, 450–452, gave a very short and completely elementary proof of the uniform boundedness principle. The aim of this note is to point out that by using a similar technique one can give a short and simple proof of a stronger statement, namely a principle of condensation of singularities for certain double-sequences of non-linear operators on quasi-Banach spaces, which is a bit more general than a result of I. S. Gál from Duke Math. J. 20 (1953), No. 1, 27–35.