Exercise 6, § 3

Soit E a left vector space over a field K, admitting a countable basis ($a_n$). Let $(r_n)$ be a decreasing sequence of numbers $> 0$, tending to 0; put $\|0\| = 0$, and for every element $x = \sum_k t_k a_k \neq 0$ in E, put $\|x\| = r_h$, if $h$ is the smallest of the indices $k$ such that $t_k \neq 0$. Now prove that $\|x - y\|$ is an invariant distance on the additive group E, and that the topology which it defines on E does not depend on the sequence $(r_n)$ (decreasing and tending to 0) chosen. Deduce that, if one extends the definitions of a norm and of a normed space (IX, p. 31, def. 5 and p. 32, def. 6) to the case where the absolute value considered on the field of scalars is improper, Prop. 8 (IX, p. 32) and Th. 1 (IX, p. 35) are no longer valid.