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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 $D$. –  Ravi Donepudi Mar 30 '12 at 11:57
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up vote 1 down vote accepted

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

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