Weak continuity of a superposition operator in sequence spaces

Under study are the conditions of weak continuity of a superposition operator in a sequence space. We give the conditions for the weak continuity of the superposition operator be equivalent to its affinity. At the same time, in the space of vanishing sequences each bounded continuous function generates a weakly continuous superposition operator. We demonstrate by example that the hypothesis of boundedness is essential and show that in an arbitrary finite-dimensional space of sequences there always is a superposition operator that is weakly continuous but fails to be representable as a sum of an affine operator and a finite-rank operator.