I'm having struggle with proving that function given by $max(|a_{ij}|)$ for a 3-by-3 matrix A is norm.
Showing positivity and homogeneity is trivial, but I'm struggling with triangle inequality property.
According to the definition $||A+B|| = max(|a_{ij}+b_{ij}|)$, but $||A||$ might just as well be $max(|a_{kl}|)$, with $k$ and $l$ different from $i$ and $j$. The same goes with $||B||$. How to show that regardless of that the property still holds?
Using the common notation for the matrix maximum norm $$ \Vert A \Vert_\infty = \max_{i, j} |a_{ij}| $$ we have that for all possible indices $i, j$ $$ |a_{ij} + b_{ij}| \le |a_{ij}| + | b_{ij}| \le \Vert A \Vert_\infty + \Vert B \Vert_\infty \, . $$ The last expression does not depend on $i,j$, therefore $$ \Vert A+B \Vert_\infty \le \Vert A \Vert_\infty + \Vert B \Vert_\infty \, . $$