Remark

— Every neighbourhood of 0 in $G$ in the topology $\sigma(G, F)$ contains a neighbourhood $V$ defined by a finite number of inequalities of the form $|\langle x_i, y \rangle| \leq 1$ ($1 \leq i \leq n$), where the $x_i$ are arbitrary points of $F$. If $A$ is the symmetric convex envelope of the set of the $x_i$, then $V$ is the polar set $A^\circ$ of $A$ in $G$. We can say that the polars in $G$ of finite symmetric sets in $F$ (or of their convex envelopes) form a fundamental system of neighbourhoods of 0 in $G$ for $\sigma(G, F)$. If the duality is separating in $F$, these convex envelopes are compact for $\sigma(F, G)$ (II, p. 14, cor. 1 of prop. 15), and of finite dimensions. Conversely every compact, convex set of finite dimension $C$ in $F$ (with the $\sigma(F, G)$ topology) is contained in the convex envelope of a finite subset of $F$. For, let $M$ be a vector subspace of finite dimension containing $C$. If $(e_i)_{1 \leq i \leq n}$ is a basis of $M$, we can suppose that $C$ is contained in the closed parallelope centre 0 and constructed on the vectors of the basis $e_i$ (GT, VI, § 1.3); now it is immediate that this parallelope is the convex envelope of the points $\sum \limits_{i=1}^n \varepsilon_i e_i$ with $\varepsilon_i = \pm 1$.

Thus we can say that (if $\sigma(F, G)$ is Hausdorff) the polars of finite dimensional, convex, compact sets in $F$ (for $\sigma(F, G)$ or for any Hausdorff locally convex topology finer than $\sigma(F, G)$ on $F$) form a fundamental system of neighbourhoods of 0 for $\sigma(G, F)$.