Exercise 6, § 3

Let $E$ be a left vector space over a division ring $K$, with a countable base $(a_n)$. Let $(r_n)$ be a decreasing sequence of numbers $> 0$, tending to $0$. For each $x = \sum_k t_k a_k \neq 0$ of $E$, put $||x|| = r_h$ where $h$ is the smallest of the indices $k$ such that $t_k \neq 0$, and put $||0|| = 0$. Show that $||x - y||$ is an invariant metric on the additive group $E$, and that the topology it defines on $E$ is independent of the sequence $(r_n)$ (decreasing and tending to $0$) chosen. Hence show that if the definitions of a norm and a normed space (Definitions 5 and 6) are extended to the case where the absolute value on the scalars is improper, Proposition 7 and Theorem 1 are no longer valid.