Example

If K is a non-discrete valued division ring then the space $\mathscr{B}(I ; K)$ (I, p. 4, Example) is complete (GT, X, § 3.1, cor. 1). This is also true for the space $\ell_K^1(I)$ (I, p. 4, Example) with the norm $\|x\|_1 = \sum_{i \in I} |\xi_i|$: for, if $x_n$ is a Cauchy sequence in this space and $x_n = (\xi_{m_i})_{i \in I}$, then for all $i \in I$

thus, for each $i \in I$, the sequence $(\xi_{m_i})_{n \geq 1}$ converges to a limit $\xi_i$ in K. Further, for each finite subset J of I

and it follows immediately that there exists a constant $a > 0$, independent of J, m, n such that $\sum_{i \in J} |\xi_{m_i} - \xi_{n_i}| \leq a$. Letting m tend to $+ \infty$, we deduce $\sum_{i \in J} |\xi_i - \xi_{m_i}| \leq \varepsilon$ from which $\sum_{i \in I} |\xi_i| \leq a + \|x_n\|_1$, which shows that $z = (\xi_i)_{i \in I}$ belongs to $\ell_K^1(I)$; further, for all $\varepsilon > 0$, there exists $n_0$ such that for $n \geq n_0$ and for every finite set J of I, we have $\sum_{i \in J} |\xi_i - \xi_{m_i}| \leq \varepsilon$; passing to the limit with respect to the directed set of finite subsets of I, we see that $\|z - x_n\|_1 \leq \varepsilon$ for $n \geq n_0$, which shows that $z$ is the limit of the sequence $(x_n)$ in the normed space $\ell_K^1(I)$.

Let $K$ be a Hausdorff topological division ring, $E$ a topological vector space on $K$ and suppose that the completed ring $\hat{K}$ is a division ring (this is so when $K$ is a valued division ring, GT, IX, § 3.3) then the Hausdorff completion $\hat{E}$ of $E$ carries the structure of a complete topological vector space on $\hat{K}$ (GT, III, § 6.5); we say that $\hat{E}$, with this structure, is the Hausdorff completion of the topological vector space $E$, or simply the completion of $E$ when $E$ is Hausdorff.