What is the formal definition of the set $E^n$? I'm currently in a non-linear optimization course, and I can't seem to find, even in the book, the definition of $E^n$, which is clearly some set. Please see the picture below context.
$\textbf{E}^n$ is used to denote Euclidean $n$-space, that is $\mathbb{R}^n$ equipped with the Euclidean norm $\Vert\cdot\Vert:\mathbb{R}^n\to\mathbb{R}^+$, $ \Vert\textbf{x}\Vert=\sqrt{\textbf{x}\cdot\textbf{x}}$, and distance function $d(\textbf{x},\textbf{y})=\Vert\textbf{x}-\textbf{y}\Vert$

The notation $\textbf{E}^n$ is used to distinguish a Euclidean space from a non-Euclidean real spaces. For example, $\textbf{H}^n$ is used to denote a real hyperbolic space, which may be constructed from $\mathbb{R}^{n+1}$ with the distance function $d(\textbf{x},\textbf{y})=\text{arcosh}{B(\textbf{x},\textbf{y})}$.
A set is a well-defined collection of distinct objects. ... The most basic properties are that a set can have elements and that two sets are equal (one and the same) if and only if every element of each set is an element of the other; this property is called the extensionality of sets.