What is $\mathbb{R}^n$ called in words?

Vector space over the reals? This doesn't make sense because obviously subspaces of $\mathbb{R}^n$ are also vector spaces that are also made up of real numbers.

Solution 1:

It may be helpful to expand $\mathbb{R}^n$ using the definition of the cartesian product as follows:

$\mathbb{R}^n$ $:=$ $\underbrace{\mathbb{R} \times \mathbb{R}... \times \mathbb{R}}_\text{$n$ many times}$ = {$(p_{1},...,p_{n})| p_{i} \in \mathbb{R}$}.

So $\mathbb{R}^n$ is, as the comments suggest, $n$-dimensional Euclidean space. Or more precisely, the set of all $n$-tuples of real numbers. We read the set $\mathbb{R}^n$ aloud as "R-N".