Definition 3

Let $E$ be a topological vector space. A subset $A$ of $E$ is said to be bounded if it is absorbed by every neighbourhood of $0$ in $E$ (I, p. 7, def. 4).

In order that $A$ be bounded, it is sufficient that $A$ be absorbed by every neighbourhood of a fundamental system of neighbourhoods of $0$. Since there exists a fundamental system of balanced neighbourhoods of $0$ (I, p. 7, prop. 4), this is the same as saying that, for every neighbourhood $V$ of $0$ in $E$, there exists $\lambda \in K$ such that $A \subset \lambda V$.

Suppose the topology of $E$ is defined by a fundamental system $\Gamma$ of semi-norms (II, p. 3); then a subset $A$ of $E$ is bounded if and only if every semi-norm $p \in \Gamma$ is bounded on $A$.

In particular, if $E$ is a semi-normed space, a subset $A$ of $E$ is bounded if and only if it is contained in a ball. In other words, if $E$ is normed this means that $A$ is bounded for the metric space structure of $E$ (GT, IX, § 2, No. 2).