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Two norms $\| x \|_\alpha$ and $\| x \|_\beta$ are said to be equivalent if there exists positive real numbers $C$ and $D$ such that

$C\|x\|_\alpha\leq\|x\|_\beta\leq D\|x\|_\alpha$

does this mean that there also exists positive real numbers $E$ and $F$ such that

$E\|x\|_\beta\leq\|x\|_\alpha\leq F\|x\|_\beta \qquad ?$

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    Yes, because $C\|x\|_\alpha\leq\|x\|_\beta\leq D\|x\|_\alpha \leq C^{-1}D\|x\|_\beta$ and divide the last three terms by $D$.2012-03-30

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$\text{Try}\ E=1/D\ \text{and}\ F=1/C.$