Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

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

share|cite|improve this question
1  
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 Mar 30 '12 at 11:57
up vote 1 down vote accepted

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

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.