Nuclearity of a Class of Vector-valued Sequence Spaces

Abstract

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