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**math8** I was thinking about this problem again, and since I am able to prove $\displaystyle \left\| x^{*} \right\|_{p} \leq \left\| x \right\|_{q} $ where $\displaystyle \frac {1}{p} + \frac {1}{q} =1$, by switching the roles of p and q, can we say that it follows that $\displaystyle \left\| x^{*} \right\|_{q} \leq \left\| x \right\|_{p} $ and therefore that $\displaystyle \left\| x^{*} \right\|_{q} = \left\| x \right\|_{p} $ follows from the fact that $\displaystyle \left\| x \right\|_{p} = \left\| x^{*} \right\|_{p}$ and that $\displaystyle \left\| x \right\|_{q} = \left\| x^{*} \right\|_{q} $ ?