Unsure of how to interpret the set $\mathbb{R}^X$ where $X$ is a real vector space? I recently came across the notation $\mathbb{R}^X$ and I'm not exactly sure what it means or how to 'visualize it'. The text it comes from is the following:

Let $X$ be a real vector space and let $\mathcal{F} \subset
 \mathbb{R}^X$ be a set of real-valued functionals on $X$.

How come the set of real functionals is a subset of $\mathbb{R}^X$? Is it possible to demonstrate why this is so with some simple example?
 A: By definition $\mathbb R^X$ is the set of all functions $f:X\to\mathbb R$. Again by definition a functional of $X$ is function from $X$ to $\mathbb R$. Therefore $\mathbb R^X$ is the set of functionals from $X$ to $\mathbb R$, and $\mathcal F$ is some set of functionals.
A: The notation $B^A$ for sets $A$ and $B$ represent the set of all functions from $A$ to $B$.
It is a generalization of the meaning of the notation $X^n:=\overbrace{X\times X\times\cdots\times X}^{n\text{ times}}$ where we can visualize it as the set of functions from $\{1,2,\ldots,n\}$ to $X$.
In your case the set of functionals in a vector space $V$ are defined as the  maps such that $V\to \Bbb F$, where $\Bbb F$ is the field of the vector space $V$.
Then clearly any set of functionals on $V$ is contained in $\Bbb F^V$.
A: If you can visualize $X$ (say, as some set of points) then you can visualize each individual element of $\mathbb{R}^X$ as a real number attached to each element of $X$. 
For example, $\mathbb{R}^\mathbb{N}$ is the set of functions $f : \mathbb{N} \to \mathbb{R}$, which we often think of as real valued sequences $(x_1,x_2,x_3,...)$, which we can visualize by attaching the real number $x_1$ to the natural number $1$, then attaching the real number $x_2$ to the natural number $2$, and so on.
For another example, suppose $X$ is the real vector space $\mathbb{R}^2$. Then you can visualize an element of $\mathbb{R}^X$ by attaching a real number to each element of the plane.
