Remark

— 1) The topology defined by a finite set of semi-norms $p_i$ ($1 \leq i \leq n$) on $E$, can be defined by the single semi-norm $p = \sup_{1 \leq i \leq n} p_i$. But a topology defined by an infinite set of semi-norms cannot, in general, be defined by a single semi-norm (III, p. 37, exerc. 2).

2) Let $(\mathscr{T}_i)_{i \in I}$ be a family of topologies on a vector space $E$ over $K$, each of which is defined by a family of semi-norms $\Gamma_i$. Then the topology defined by the set of semi-norms $\Gamma = \bigcup_{i \in I} \Gamma_i$ is the upper bound of the topologies $\mathscr{T}_i$.

3) If $\Gamma_0$ is a set of semi-norms directed by the increasing order relation defined between two semi-norms $p, q$ on $E$ by « there exists $\lambda > 0$ such that $p \leq \lambda q$ », then a fundamental system of neighbourhoods of 0, for the topology defined by $\Gamma_0$, is obtained by taking the sets $V(p, \alpha)$ where $p \in \Gamma_0$ and $\alpha > 0$. If $\Gamma$ is any set of semi-norms on $E$, then a filtered set of semi-norms, defining the same topology as $\Gamma$, is the set $\Gamma_0$ of upper envelopes of all finite families of semi-norms belonging to $\Gamma$.

4) Even if $K = \mathbf{R}$, the topology of a topological vector space over $K$ cannot always be defined by a set of semi-norms (cf. II, p. 24, corollary).