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If

$\|f(x)-f(y)\|\geqslant \frac 1{2} \|x-y\|$ for any $x, y \in W$

then

$f$ is injective in $W$

How to prove this? If that inequality is right is it mean that the images are equal or not?

  • 0
    If the distance between two points is more than zero, then they're not the same point.2012-06-06
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    Just to make a comment, I think I saw this in a proof of the Inverse Function Theorem.2012-06-06
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    @AdenDong, you are [right](http://math.ucsd.edu/~nwallach/inverse[1].pdf). Thanks2012-06-06

1 Answers 1

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Assume $f(x) = f(y)$. What is $\|f(x)-f(y)\|$? What can you conclude about $\|x-y\|$?

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    Thanks. It's mean that the proof of this expression is in definition of injective function. $f(x) = f(y) => x=y $2012-06-06