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I know the definition of the polar set is: $$C^0 = \left\{ y \in \mathbb{R}^n \mid y^Tx \leq 1 , \forall x \in C\right\},$$ and the dual norm is defined as: $$||z||_* = \text{sup} \left\{ z^Tx \mid ||x|| \leq 1\right\}.$$

But I cant see any connection between the definition. How can I show that the polar of the unit ball for a general norm $||\cdot||$ is the unit ball of the dual space? Where do I start from? Thanks.

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    The depth of your insight is not so clear from the question. Does it help that '$\| x \| \leq 1$' is the same as '$\forall x \in C$' in this context? – 2017-02-21
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    still unable to see the connection. How would you define the 'unit ball of the dual norm'? – 2017-02-21
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    In this context, the dual space of $\mathbb{R}^n$ is: $\mathbb{R}^n$ equipped with the norm $\| \cdot \|_*$. Hence, the unit ball of the dual space is $\{ z \in \mathbb{R}^n \, | \, \|z\|_* \leq 1 \}$. – 2017-02-21

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