What does the vector space R^[0,1] mean? While reading a book Linear Algebra Done Right, I came to knew that a vector space $\mathbf{R}^n$ represents a space with dimensions as $(x_1, x_2, ...,x_n)$, but there were other vector spaces that I could not understand.
There was a statement as Ref: 1.35

The set of continuous real-valued functions on the interval $[0,1]$ is a subspace of $\mathbf{R}^{[0,1]}$

What kind of space does $\mathbf{R}^{[0,1]}$ represent? Is this a space that can continuously be from $0$ dimension to $1$ dimension?
Another statement, Ref: 1.35

The set of differentiable real-valued functions on $\mathbf{R}$ is a subspace of $\mathbf{R}^\mathbf{R}$

What kind of space is $\mathbf{R}^\mathbf{R}$?
Similarly, there were other subspaces as, $\mathbf{R}^{(0,\ 3)}$ and $\mathbf{R}^{(-4,\ 4)}$ 
Explain me how can I visualize such spaces. If you can explain with the proof too, that will be great.

The set $\mathbb{R}^{[0,1]}$ is the set of all functions from $[0,1]$ into $\mathbb R$ and the set $\mathbb{R}^\mathbb{R}$ is the set of all functions from $\mathbb R$ into itself.
