What does ${\mathbb R^n}$ mean? I know that $\mathbb R$ refers to all real numbers. But what does $\mathbb R^2$ mean? 
For that matter, what does $\mathbb R^n$ mean when $n$ equals any natural number?
 A: $\mathbb R^n$ refers to the space of all $n$-dimensional vectors $(x_1, x_2, \dots, x_n)$ where each coordinate $x_i$ is a real number i.e. $x_i \in \mathbb R$.
A: In general a set $X$ will have a Cartesian product with itself sometimes called $X^2$. We can again take the Cartesian product of $X^2$ and $X$ to get $X^3$
 and so on.
Another way to interpret this notation is by considering two sets $X$ and $Y$ then define $Y^X$ as the set of all function from $X$ to $Y$. Then we can interpret $X^2$ with $2$ meaning the set of $2$ elements rather than as the number $2$ itself. This can be done with $X^n$ being the set of function from a set of $n$ elements to $X$ since they are all unique up to isomorphism.
A: Without giving you a word definition, let's see if you understand this way:
$x\in\mathbb{R}$
$(x_1, x_2) \in \mathbb{R^2}$
$(x_1, x_2, x_3) \in \mathbb{R^3}$
Therefore,
$(x_1, x_2, x_3, \dots , x_n) \in \mathbb{R^n}$
where each $x_i$ is a member of the real numbers $\mathbb{R}$. 
Thus, When we say something is a member of $\mathbb{R}^n$, its saying that its a $n$ dimensional vector with elements of type $\mathbb{R}$
A: $\mathbb{R}^n$ is the set of all n-tuples with real elements. They are NOT a vector space by themselves, just a set. For a vector space, we would need an extra scalar field and 2 operations: addition between the vectors (elements of $\mathbb{R}^n$) and multiplication between the scalars and vectors. But usually we just denote the vector space of $\mathbb{R}^n$ over the $\mathbb{R}$, with the usual product and sum as $\mathbb{R}^n$ for simplicity reasons, but they are not the same.
