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Can someone explain why the term in the first {} equals the second term in the {} :

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What is the norm of $\frac{x}{\| x \|}$? Then use the scalar-linearity of the mapping $x\mapsto Ax$.

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For every $x\in K^n$ there is a normalized vector $\frac{x}{\|x\|}$; using the axioms for matrixnorms we have $\|A\frac{x}{\|x\|}\|=\frac{\|Ax\|}{\|x\|}$

Therefore, it is sufficient to only consider the maximum over vectors of norm 1. (As a matrix represents a linear function, there cannot be a $k\in K$, for which $A(kx)>kA(x)$.)

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@0x90 en.wikipedia.org/wiki/Axiom thank me later ~ –  CBenni Dec 22 '12 at 13:04
    
    
@0x90 what about that? Just some painfully inprecise reading script? –  CBenni Dec 22 '12 at 13:27
    
That is a little bit more relevant in compare to your link into wiki Axiom ... that explains my question :) thanks –  0x90 Dec 22 '12 at 13:35
    
Dont forget to mark your question as answered ;) –  CBenni Dec 22 '12 at 13:46
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