so I have learned to proof the 1-Norm,2-Norm and p-Norm.
I know that max is a function, which returns the maximum in a list/array.
So in this example I have: $\|A\|_\infty$
$$ \|A\|_\infty := \max_{1\le i\le n}\Biggl(\sum_{j=1}^n |a_{ij}|\Biggr) $$
I know the axioms and I can use them on the previously mentioned norms except the Inf-Norm :/
How do I handle max?
My attempt: Let $$ A=\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ with $\|A\|_\infty= \max(|a|+|b|+|c|+|d|)$
Well thats it, I usually show that it has to be equal or greater then $0$, since the squareroot only allows in $\mathbb{R}^{n\times n}$ non complex solutions,
hence $\|A\|_\infty = 0$ implies
$$ A=\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix} $$ but in my example there is no $\sqrt{(a_{ij})^2}$
Also, would I simply take the maximum of i? like the biggest i? Since in the norm my matrix cant be infinite itself, it would make sense to me, but I am most of the time wrong anyways :1
I also found Infinity matrix norm example
But this seems like analysis + linear algebra, since supremum is a analysis function, hence I am even more confused.
Maybe max returns the biggest (total) number in my matrix A?