Is it correct to say that for matrices: $\|AB\|_{max} \leq \|A\|_{\infty} \|B\|_{max}$?

Question

Is it correct to say that for matrices: $\|AB\|_{max} \leq \|A\|_{\infty} \|B\|_{max}$?

Where, $\|B\|_{max}$ is the max-norm of matrix $B$, that is, maximum absolute value present in matrix $B$.

EDIT 1: Infinity norm of matrix $A$ is defined as: $\|A\|_{\infty}=\max_{1 \leq i \leq m}\sum_{j=1}^{n}|a_{ij}|$

Answer 1

I will assume the matrices are finite dimensional. $|(AB)_{ij}|=|\sum_{k=1}^nA_{ik}B_{kj}|\leq \sum_{k=1}^n |A_{ik}B_{kj}|\leq ||B||_{max}\sum_{k=1}^n |A_{ik}|\leq ||B||_\max||A||_\infty$

Therefore $||AB||_{\max}\leq ||A||_\infty||B||_\max$