In matrix analysis, a matrix norm can be induced by a vector norm by the following.
$$|||A|||_\alpha = \max _{||x||_\alpha =1} ||Ax||_{\alpha}$$ where $||\cdot||_\alpha$ is a vector norm and $A$ is a square matrix.
One can verify that the $|||\cdot|||_\alpha$ is a matrix norm.
However suppose there is another vector norm $|||\cdot||_\beta$ and define
$$|||A|||_{\alpha, \beta} = \max _{||x||_\alpha =1} ||Ax||_\beta.$$
Is the above still a matrix norm? What I cannot establish is the submultiplicative property, i.e.
$$|||AB|||_{\alpha, \beta} \leq |||A|||_{\alpha, \beta}|||B|||_{\alpha, \beta}$$
where $A$ and $B$ are square matrices of the same size.
If not, can someone give me a counterexample that makes the above false?
Matrix norms defined like this are known as consistent norms. Consistent norms are not necessarily submultiplicative. Here is an easy way to construct a non-submultiplicative consistent norm: just define $\|\cdot\|_\beta=\epsilon\|\cdot\|_\alpha$ for some sufficiently small $\epsilon>0$. Then $|||\cdot|||_{\alpha,\beta}=\epsilon|||\cdot|||_\alpha$ and hence $$ |||I^2|||_{\alpha,\beta}=\epsilon|||I|||_\alpha >\epsilon^2|||I|||_\alpha=|||I|||_{\alpha,\beta}^2 $$ when $\epsilon$ is small enough.