Linear Transformation of $\mathbb{R}^3 \to \mathbb{R}^R$

Question

In the highlighted text, this is an example of a linear mapping from $\mathbb{R}^3 \to \mathbb{R}$. I am not entirely sure what $\mathbb{R}^R$ is supposed to represent. For example, obviously $\mathbb{R}^n \neq \mathbb{R}^R$, but why is this differentiated?

Answer 1

$\Bbb R^{\Bbb R}$ means the set of all functions from $\Bbb R$ to $\Bbb R$. They even say so in your text:

the space $\Bbb R^{\Bbb R}$ of all real-valued functions on $\Bbb R$

This comes with a standard vector space structure, with addition of two functions and scalar multiplication of one function by a real number defined the obvious way.

In general, $X^Y$ for sets $X$ and $Y$ denotes the set of functions from $Y$ to $X$. Some times, like here, this also has implied structure (such as a vector space structure, or an ordering) inherited in some relatively obvious way from corresponding structure on $X$ or $Y$.

Note that if you interpret $n$ to mean a set of $n$ elements (either an arbitrary one, it a specific one), then the notation $\Bbb R^n$ also falls under this convention: an $n$-tuple of real numbers may be seen as a real-valued function on a set of $n$ elements.

Answer 2

Well, ${\Bbb R}^{\Bbb R} = \{f\mid f:{\Bbb R}\rightarrow{\Bbb R}\}$ is the set of all real-valued functions from $\Bbb R$ to $\Bbb R$.

Given some real-valued functions $f_1,f_2,f_3$, each triple of real numbers $(x_1,x_2,x_3)$ is mapped to the linear combination of functions $x_1f_1+x_2f_2+x_3f_3$ with $(x_1f_1+x_2f_2+x_3f_3)(x) = x_1f_1(x)+x_2f_2(x)+x_3f_3(x)$ for each real number $x$.