Remark

— 1) If $E$ is a semi-normed space, the balls form a fundamental system of bounded neighbourhoods of $0$ in $E$. Conversely, if $E$ is a locally convex topological vector space, and if there exists a bounded neighbourhood of $0$ in $E$, this neighbourhood contains a convex balanced neighbourhood $W$, and the gauge of $W$ is then a semi-norm defining the topology of $E$.

Thus, if $E$ is locally convex and metrizable, and if its topology cannot be defined by a single norm, then there exists no distance on $E$ defining its topology and such that the bounded subsets for $d$ (GT, IX, § 2, No. 2) are the bounded subsets of $E$. More precisely, for every distance $d$ on $E$, which is translation invariant and which defines the topology of $E$, the bounded subsets of $E$ are bounded for $d$ (III, p. 38, exerc. 3), but the converse is false.

2) Let $M$ be a vector subspace of $E$ endowed with the induced topology. In order that a subset of $M$ be bounded in $M$, it is necessary and sufficient that it be bounded in $E$.

3) Let $N$ be the intersection of all neighbourhoods of $0$ in $E$, so that $\tilde{E} = E/N$ is the Hausdorff vector space associated with $E$. Then $N$ is bounded; if $\pi : E \to \tilde{E}$ is the canonical homomorphism then a subset $B$ of $E$ is bounded if and only if $\pi(B)$ is bounded.

4) Let $E$ be a Hausdorff locally convex space; then for every $x \neq 0$ in $E$, there exists a continuous semi-norm $p$ such that $p(x) \neq 0$; this semi-norm is not bounded on the real half-line $\mathbf{R}_+ . x$ generated by $x$. Hence no non-null subspace of $E$ is bounded. In particular, a bounded subset does not contain any line.