I have trouble following a proof of the induced Norm $||\cdot||_1$
The proof can be found here: http://nptel.ac.in/courses/Webcourse-contents/IIT-KANPUR/Numerical%20Analysis/numerical-analysis/kadalbajoo/lec1/fnode3.html
I'm at the step where we defined $$C=\max_{1\le j\le n}{ \sum^n_{i=1}|a_{ij}|}$$ which ist $||A||_1$.
They conclude $||A||_1 \le C$
And "to show this is an equality, we demonstrate an x for wich ${||Ax||_1 \over||x||_1} = C$ "
My Question is, how does finding an x shows this is an equality?
I don't know if I get your question correctly, but it seems like the induced norm is defined in the usual way, i.e. $$ \|A\|_1:=\sup_{x\ne 0}\frac{\|Ax\|_1}{\|x\|_1} $$ and the goal is to show that $$ C:=\max_{1\leqslant j\leqslant n}\sum_{i=1}^n \left|a_{ij}\right|=\sup_{x\ne 0}\frac{\|Ax\|_1}{\|x\|_1}$$ The inequality $\|A\|_1\leqslant C$ says essentially that for every $x$ you have $$ \frac{\|Ax\|_1}{\|x\|_1}\leqslant C $$ But you don't know yet if the $\sup$ is a $\max$. So if there exists an $x\ne 0$ such that $C=\frac{\|Ax\|_1}{\|x\|_1}$ , you're done.