Looking for notation of set of all entries of some matrix?

Question

I'm busy writing my thesis, and I'm looking for some concise notation to denote the supremum of the matrix entries of, say $A \in M_n(\mathbb{R})$. How should I do this?

Looking for something like $$\sup_{a_{i,j} \in A}|a_{i,j}|$$ but the notation $a_{i,j} \in A$ in reality doesn't make much sense in my opinion. What else can I do?

EDIT: Even more ideally I want to denote $\sup_{a_{i,j}\in (A-B)}|A - B|$, but I might just introduce general notation for the "norm" to simplify this.

Answer 1

The defined quantity is not a "norm", it is a norm (not an operator norm though and not sub-multiplicative). I'm not aware of a standard notation for this quantity, but $\|\cdot\|_M$ or $\|\cdot\|_{\max}$ look suitable.

Answer 2

Although not really a notation but a combination of notations, the quantity in question is $\|\operatorname{vec}(A)\|_\infty$.