Nuclearity of a class of vector-valued sequence spaces

In this note, we deal with a perfect sequence space lambda and a convex bornological space E to introduce and study the space lambda(E) of all totally lambda-summable sequences from E. We prove that lambda(E) is complete if and only if lambda and E are complete, nuclear if and only if lambda and E are nuclear, and we make use of a result of Ronald C. Rosier [10] to give a similar characterization of the nuclearity of the space.{E} of all absolutely lambda-summable sequences in a locally convex E.