Exercise 4, Appendix 1

Assume that $k$ is a non-discrete complete valued field. Denote by $\mathscr{T}$ (resp. $\mathscr{Z}$) the Banach space (resp. Zariski) topology on V.

a) Show that $\mathscr{T}$ is finer than $\mathscr{Z}$ (and strictly finer if $V\not= 0$).

b) Show that every non-empty $\mathscr{Z}$ -open subset of V is $\mathscr{T}$-dense.

$^9$ For more details, cf. G. B. SELIGMAN, Modular Lie Algebras, Chap. V, §7, Springer-Verlag, 1967.