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As we know $\|x\|_{\infty}$ returns the maximum "absolute" value through the vector . I'm looking for a symbol to show the actual value that can be also negative: $ ?x?=\max\{x_1,...x_N\}$

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$\max_{1\leq i \leq N} x_i$ is the standard notation, and note that the symbol $||\cdot||:\mathbb{R}^n \rightarrow \mathbb{R}$ usually denotes a norm, which max is not.

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You should use $\max x_i$ or $\max_i x_i$.

It is not a norm, so you shouldn't use $\|x\|_\square$ or anything like that. Maybe you could define a notation like $(x)_{max}$, but I don't think there already is one.

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