Nuclear and integral polynomials on testing $\mathsf C^{(I)}, \;\; I$ uncountable

Abstract

We show that for $I$ an uncountable index set and $n\ge 3$ the spaces of all $n$-homogeneous polynomials, all $n$-homogeneous integral polynomials and all $n$-homogeneous nuclear polynomials are all different. Using this result we then show that the class of locally Asplund spaces is not preserved under uncountable locally convex direct sums nor is separably determined.