Tricks to interpret $\mathbb{R}^{\mathbb{R}}$ So I have the following question of how to interpret $\mathbb{R}^{\mathbb{R}}$. Is this just an "infinite" tuple of real numbers?
Since for example $ (x, y) \in \mathbb{R}^2$, $(x, y, z) \in \mathbb{R}^3$, etc. Is $(x_1, x_2, ... , x_n) \in \mathbb{R}^n$? Or is there another way that I should interpret this?
 A: Yes: it is the set of all functions from $\Bbb R$ into $\Bbb R$. In fact, a function $f\colon\Bbb R\longrightarrow\Bbb R$ can be seen as the element $\bigl(f(x)\bigr)_{x\in\Bbb R}$ of $\Bbb R^{\Bbb R}$. And an element $(a_\lambda)_{\lambda\in\Bbb R}$ of $\Bbb R^{\Bbb R}$ can be seen as the function$$\begin{array}{ccc}\Bbb R&\longrightarrow&\Bbb R\\\lambda&\mapsto&a_\lambda.\end{array}$$
A: @s.harp is right. For sets $A,\,B$, the set of functions from $A$ to $B$ is denoted $B^A$ because its cardinality is $|B|^{|A|}$ (this is trivial when both cardinalities are finite, but is more general the definition of exponentiation of cardinalities). So $\Bbb R^{\Bbb R}$ is the set of functions from $\Bbb R$ to $\Bbb R$.
A: In general, the notation $X^Y$ means the set of all functions from $Y$ to $X$. In particular, $\mathbb{R}^{\mathbb{R}}$ means all the real-valued functions on $\mathbb{R}$.
The set $\mathbb{R}^n$ can be thought as $\mathbb{R}^{\{1,2,\cdots,n\}}$, i.e., the set of all functions $f:\{1,2,\cdots,n\}\to\mathbb{R}$.
The set $\mathbb{R}^{\mathbb{R}}$ can be interpreted as an infinite Cartesian product.
