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You probably know that matrix norms can be defined by the vector norms in the following way: $$||A||:= \max_{x\neq 0} \frac{||Ax||}{||x||}$$ for a matrix $A$. So you just look up the definition for the infinity or 2-norm and plug it into the expression above. When doing this you will further realize that the following equalities ...

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I am not sure about your application -- and we say the $L^2$ norm of a function and not a system. But for simplicity I will explain the concepts for real valued functions. Consider an open domain $\Omega$ and a function $f:\Omega \to \mathbb{R}$. We say that $f \in L^2(\Omega)$ if $||f||_{L^2(\Omega)} < \infty$ where ||f||^2_{L^2(\Omega)} ...

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The advantage of that definition is that computing relres comes "for free" from the GMRES iteration. You could switch to the other definition without the preconditioner, but then you'd have to use more operations to compute it. I don't think you can get anything significantly faster than trivially computing $b-Ax$ from scratch at every step. As far ...

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