Abstract: In this paper, we investigate the $\lambda$-nuclearity in the system of completely 1-summing mapping spaces $(\Pi_{1}(\cdot, \cdot), \pi_{1})$. In Section 2, we obtain that $\mathbb{C}$ is the unique operator space that is nuclear in the system $(\Pi_{1}(\cdot, \cdot), \pi_{1})$. We generalize some results in Section 2 to $\lambda$-nuclearity in Section 3.

Key words: λ-nuclearity, completely 1-summing mapping space