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I would be glad if anyone could provide me example of a Polar Set . Polar set is defined as follows: Given a dual pairing $(X,Y)$ the polar set or polar of a subset $A$ of $X$ is the set $A^0$ of $Y$ such that :

$A^0 = \{y \in X : \sup |\langle x,y\rangle|\le1 \}$

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Let $E$ be a subspace of a normed linear space $X$ with topological dual $X^\ast$. Then: $E^0=\{f\in X^\ast:|f(x)|\leq 1,\,\forall x\in E\}=\{f\in X^\ast:f=0\text{ on } E\}$ This is known as the annihilator of $E$ in $X^\ast$.

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Take $X=Y=R$. Let $A=[a,b]$. Then

$A^0= \{ y \in \mathbb{R} : |xy|\leq 1\quad \forall x \in A \}= [-c,c] $

where $c =\max \{ |a|, |b| \}$

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Let $X$ be a normed space with norm-unit ball $B$ and continuous dual $X^*$. Then $ B^\circ = \{ f \in X^*: |f(x)| \leq 1, \forall x \in B\} $ is precisely the norm-unit ball $B^*$ in $X^*$. So the concept of a polar is the natural extension of norm-neighbourhoods of 0 to topological vector spaces.