Exercise 7, § 2

Let K be a valued division ring which is non-discrete and non-complete, let E be the topological vector subspace $K + Ka$ of $\hat{K}$ where $a \notin K$, and let F be the product space $K \times E$. In F, the subspace $M = K \times \{0\}$ is closed and of codimension 2. Let N be the complementary subspace to M in F generated by the vectors $(0, 1)$ and $(1, a)$; show that F is not the direct topological sum of M and N.