= Polar set
{title2=$B^\circ$}
For a real <dual pair> $(E,F)$ and $B\subseteq E$, the polar set is
$$
B^\circ=\{f\in F:|\langle x,f\rangle|\leq1\text{ for all }x\in B\}.
$$
The same absolute-value definition applies to a complex pairing. It collects the linear measurements uniformly bounded by one on $B$. Reversing the roles of $E,F$ defines the bipolar.
A polar is convex, balanced and closed in the topology of pointwise convergence on $E$, since it is an intersection of inverse images of closed scalar unit disks under linear evaluations. Inclusion reverses: $B\subseteq C$ implies $C^\circ\subseteq B^\circ$.
In $E=\mathbb R^2$ with the usual dot pairing, the polar of $[-1,1]^2$ is $\{(a,b):|a|+|b|\leq1\}$, because the maximum of $|ax+by|$ on that square is $|a|+|b|$. In a normed space paired with its continuous dual, the polar of the primal closed unit ball is exactly the dual closed unit ball by the definition of the operator norm.
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