Consider the following norm: $$ ||A||_F = \left( \sum^m_{i=1} \sum^n_{j=1} |A_{ij}|^p \right)^{1/p}. $$
I would like to prove that it is indeed a norm by proving the triangle inequality:
$$||A+B|| = \left( \sum^m_{i=1} \sum^n_{j=1} |A_{ij}+B_{ij}|^p \right)^{1/p}. $$ Applying the triangle inequality for absolute values and using the binomial expansion we get:
$||A+B||^p \leq ||A||^p + ||B||^p + p\sum^m_{i=1} \sum^n_{j=1} |A_{ij}|^{p-1}|B_{ij}|$ + ... + $p\sum^m_{i=1} \sum^n_{j=1} |A_{ij}||B_{ij}|^{p-1}$
I would like to use Holder's inequality, however, I am completely stuck on how. If I can reduce it to $$||A+B||^p \leq (||A|| + ||B||)^p $$ then I am done
Any help would be appreciated.
As I understood, $A=(A_{ij})$ and $B=(B_{ij})$. Then
For $1\le p\le+\infty$ the claim follows from Minkowski inequality.
For $0<p<1$ if fails even for $1\times 2$ matrices over $\mathbb R$, because $\|(1,1)\|_F=2^{1/p}>2\cdot 1^{1/p}=\|(1,0)\|_F+\|(0,1)\|_F$.
For $p<0$, $\|A \|_F$ is undefined when $A_{ij}=0$.