p norm of a matrix

Question

i have the definition $||A||_p =max_{x{_\neq 0}}\frac{||Ax||_p}{||x||_p}$ and have to show that $max_{x{_\neq 0}}\frac{||Ax||_p}{||x||_p}=max_{||x||_p=1}||Ax||_p$. I would be very grateful if someone could give me some hints how to solve this problem.

Answer 1

RHS $\leq $ LHS is obvious. Given $x \neq 0$ define $y =\frac 1 {\|x\|_p} x$. Verify that $\|y\|_p=1$. This gives RHS $\geq {\|Ay\|}$. If you write this in terms of $x$ you will get RHS $\geq \|Ax\|_p$. Take max over $x$ to get RHS $\geq $ LHS.