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

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

What is the norm of $\frac{x}{\| x \|}$? Then use the scalar-linearity of the mapping $x\mapsto Ax$.
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 $A(kx)>kA(x)$.)