Proposition 11.1.3.label Let $E$ be a TVS over $K \in \bracs{\real, \complex}$, then:

1. (1)

   There exists a unique translation-invariant uniformity $\fU$ on $E$ that induces the topology on $E$.

2. (2)

   For each neighbourhood $V \in \cn(0)$, let $U_{V} = \bracs{(x, y) \in E^2| x - y \in V}$, then for any fundamental system of neighbourhoods $\fB_{0}$ at $0$, $\fB = \bracs{U_V| V \in \fB_0}$ is a fundamental system of entourages for $\fU$.

The space $E$ will always be assumed to be equipped with its translation-invariant uniformity.