How should I understand $\mathbf F^\mathbf n$? I am trying to go through Axler's Linear Algebra Done Right. I am stuck in the first few pages because I can't wrap my head around fields. 
The definition of $\mathbf F^\mathbf n$ was given as:
the set of all lists of length $\mathbf n $ of elements of $\mathbf F$.
okay, for simple cases such as $\mathbf n$ = 2 or 3, I think I understand it 
$  \mathbf R^\mathbf 2 = \{(x_1, x_2): x_1, x_2 \in \mathbf R \} $ is basically any point on a plane, if n = 3, it's a point in a space and so on, but I can't understand expressions such as 
1) $\mathbf R^{[0,1]}$
2) $\mathbf R^{\mathbf R}$
how should I understand these expressions? 
1) seems to be any R between zero and one is that correct?
2) how is this different from plain R? 
 A: $A^B$ where $B$ is a set means all functions from $B$ to $A$.  Your ${\bf R}^2$ corresponds to ${\bf R}^{\{1,2\}}$, i.e. a member $(x_1, x_2)$ of ${\bf R}^2$ corresponds to the function $f$ from $\{1,2\}$ to $\bf R$ with $f(1) = x_1$ and $f(2) = x_2$.  In the case of ${\bf R}^{[0,1]}$, your function $f$ goes from $[0,1]$ to $\bf R$, i.e. it has to specify a value $f(x)$ for every 
real number $x$ from $0$ to $1$.
A: if $A$ and $B$ are sets then the symbol $A^B$ is defined to mean the set of all functions with domain $B$ and image contained in $A$.
thus $R^{[0,1])}$ signifies the set of functions $f:[0,1] \to \mathbb{R}$, i.e the set of all real-valued functions on the closed unit interval.
A: I'm sure he defines what $S^T$ means when $S$ and $T$ are sets. This is the collection of all functions from $T$ to $S$.
A: You can think of this $A^B$ as the set of functions $f:B\longrightarrow A$. Each such function is a 'list of length $B$ of elements of $A$' in the sense that it associates to each element of $B$ an element of $A$. The function $f$ 'encodes' the list.
