EQUICONTINUITY AND SINGULARITIES OF FAMILIES OF MONOMIAL MAPPINGS

The starting-point for the present paper is the principle of condensation of the singularities of families consisting of continuous linear mappings that act between normed linear spaces. It is proved that this basic functional analytical principle can be generalized for families of continuous monomial mappings of degree n between topological linear spaces. The obtained principle yields a generalization of the principle of uniform boundedness published by I. W. Sandberg [IEEE Trans. Circuits and Systems CAS-32 (1985), 332-336] and recently rediscovered by R. Miculescu [Math. Reports (Bucharest) 5 (55) (2003), 57-59]. Furthermore, by applying the new nonlinear principle there are revealed Baire category properties of certain subsets of the normed linear space C[a, b] involved with Riemann-Stieltjes integrability.