I am studying robust control system, and get confused with the following two definitions of infinity norm. ($G(j\omega)$ is the transfer function of a MIMO system)
[1] $$\left \| G \right \| _\infty = \max \limits _{1 \leq i \leq m} \sum _{j=1} ^n | g_{ij} |$$ [2] $$\left \| G \right \| _\infty = \sup \limits _{\omega} \bar \sigma (G(j\omega))$$
What is the difference between these two different definiation? I am wondering whether the definition [1] is just the infinity norm of the TF $G(j\omega)$ at a specified frequency $\omega_0 $.