Raising interval notation to a power when the intervals are not the same for each element We can raise a set written in interval notation to a power to indicate the set of vectors of values within that interval. For example:
$$[0,1]^3 = \{(x_1,x_2,x_3) \in \mathbb{R}^3 : 0 \leq x_1 \leq 1,0 \leq x_2 \leq 1,0 \leq x_3 \leq 1\}.$$
Can we do a similar thing, but with an interval that is not the same for every element? For example, can we say
$$[l,u]^3 = \{(x_1,x_2,x_3) \in \mathbb{R}^3 : l_1 \leq x_1 \leq u_1,l_2 \leq x_2 \leq u_2,l_3 \leq x_3 \leq u_3\},$$
where $l=(0,1,2)$ and $u=(3,4,5)$?
Or would that not be correct notation? Is there a standard notation way to write this? Or do I explicitly need to define it?
 A: The Cartesion product of two sets $A,B$ is defined as
\begin{align*}
A\times B=\{(a,b)|a\in A,b\in B\}
\end{align*}
If $A$ and $B$ are the same sets, $A=B$ we have for convenience the short notation
\begin{align*}
A^2=A\times A\tag{1}
\end{align*}
More general, if we consider sets $A_j, 1\leq j\leq n$ we can write
\begin{align*}
\color{blue}{\prod_{j=1}^n A_j}&=A_1\times\cdots\times A_n\tag{2}\\
&\color{blue}{=\{(a_1,\ldots,a_n):a_j\in A_j, 1\leq j\leq n\}}
\end{align*}
An interval $[0,1]=\{x\in\mathbb{R}|0\leq x\leq 1\}$ is a set and we can following (1) (and some other algebraic rules) conveniently write
\begin{align*}
[0,1]^3&=[0,1]\times[0,1]\times[0,1]\\
&=\{(x_1,x_2,x_3)\in\mathbb{R}^3: 0\leq x_j\leq 1, 1\leq j\leq 3\}
\end{align*}

If we consider the Cartesian product of different intervals we can conveniently use notation (2) and obtain
\begin{align*}
\color{blue}{\prod_{j=1}^3[l_j,u_j]}&\color{blue}{=[l_1,u_1]\times[l_2,u_2]\times [l_3,u_3]}\\
&=\{(x_1,x_2,x_3)\in\mathbb{R}:l_j\leq x_j\leq u_j, 1\leq j\leq 3\}
\end{align*}

Since here the endpoints of intervals are real numbers, a notation with $3$-tupels $l=(l_1,l_2,l_3), u=(u_1,u_2,u_3)$ as endpoints of intervals in $[l,u]^3$ is not admissible.
A: Usually this is denoted with $l,u\in \mathbb{R}^n$ as $[l,u]$, sometimes with the vectors in bold to distinguish them from scalar intervals. It's worth mentioning that you're using this notation whenever you use it first in a piece of writing, but usually it's clear from context.
