Problem: Prove there is some such that a) , b) , and c) is not continuous on
let where for some be a basis for as a vector space over that means every element of can be written uniquely as where and all but at most finitely
many of are zero. define by the map is clearly well-defined and satisfies the conditions a) and b). to show that it also satisfies the condition c), choose
such that and also choose a sequence such that now define and see that but