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where [tex]\frac{1}{p} + \frac{1}{q} = 1[/tex]

By using this definition of [tex]\left\| x^{*} \right\| _{p} = max_{ \left\| y \right\| _{p} =1} \left\| x^{*} y \right\| _{p} [/tex]

and Holder's inequality, I am able to prove that

[tex]\left\| x^{*} \right\| _{p} \leq \left\| x \right\| _{q}[/tex]

But how do you show the other side of the inequality?