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I'm asked to prove that if

$T:\mathbb{R}^m \rightarrow \mathbb{R}^n$

is a linear transformation, there exists a number $M$ such that

$\|T(h)\|\le M\|h\|$ for $h\in \mathbb{R}^m$.

I'm not sure whether I'm being asked to show that such a number (that is independent of $h$) exists or to show that such a $M$ exists for each $h$. Can someone please clarify?

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Note that if $M$ could depend on $h$, the statement would be really really obvious -- for any real numbers $x$ and $y$, there exists an $M$ such that $x \leq My$... –  Micah May 19 '12 at 1:14

2 Answers 2

You are being asked to show that such a number exists which is independent of $h$.

Hint: Note that, since $T$ is linear, you need only show that there is some $M$ such that $\|T(h)\|\leq M$ when $\|h\|=1$. Furthermore, the set $\{h:\|h\|=1\}$ is compact.

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It actually says "show that there is a number $M$ such that...". So yes, you are supposed to find a single $M$ such that the inequality is satisfied for every $h$.

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