Exercise 6, § 1

Let K be a non-discrete valued division ring. Show that, for the space $\ell_K^1(\mathbf{N})$ of absolutely summable sequences $x = (\xi_n)$ of elements of K, the norms $\|x\|_1 = \sum_{n=0}^\infty |\xi_n|$ and $\|x\| = \sup_n |\xi_n|$ are not equivalent (*cf. GT, IX, § 3.3, prop. 7); show that $\ell_K^1(\mathbf{N})$ with the norm $\|x\|$ is never complete even if K is complete; what is its closure in $\mathscr{B}_K(\mathbf{N})$?