I do a lot of DSP (digital signal processing), and in compressing a signal and any number of other operations, I frequently need to constrain it to the range $[-1, 1]$. Now, I know that this isn't quite normalization, because normalization (at least in my typical use) implies $[0, 1]$.
In spite of this, I see the $[-1, 1]$ range so often I can't help but suspect that there is a formal name for it. What is it called?
In a normed space $X = (X,\|\cdot\|)$, the set $$B(0,1) = \{ x \in x : \|x\| < 1 \} $$ is called the [open] unit ball. In $\mathbb{R}^2$ this set is a disk (like a coin or pancake), and in $\mathbb{R}^3$ this set is a solid three-dimensional ball. The same definition still applies in $\mathbb{R}$, where $$ (-1,1) = B(0,1) = \{ x \in \mathbb{R} : |x|<1 \}.$$ Thus it is quite reasonable to call this set the one-dimensional unit ball. Note that, without modification, it is typically assumed that the unit ball is open. The question is about the closed interval $[-1,1]$, which is then the closed unit ball in $\mathbb{R}$, i.e. the set $$ [-1,1] = \overline{B(0,1)} = \{ x\in\mathbb{R} : |x| \le 1 \} $$ (note the weakening of the inequality).