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In this paper, a vector topology is introduced in the vector-valued sequence space $\mathit{\text{BMC}}\left(X\right)$ and convergence of sequences and sequentially compact sets in $\mathit{\text{BMC}}\left(X\right)$ are characterized.

In this paper, we show the representation of Köthe dual of Banach sequence spaces ${\ell }_p\left[X\right]$ $(1\le p<\infty )$ and give a characterization of that the spaces ${\ell }_p\left[X\right]$ $(1<p<\infty )$ are Grothendieck spaces.