What does $\mathbb{R}^{[0,1]}$ stand for? What does $\mathbb{R}^{[0,1]}$ stand for in the following expression?  $\Phi: C[0,1] \to \mathbb{R}^{[0,1]}$, where $C$ is the space of all the continuous function in $[0,1]$ and $\Phi$ is an operator.
EDIT 1
Considering the notation $\mathbb{R}^{[0,1]}$ that means the collection of functions from $[0,1] \to \mathbb{R}$.
If, 
$$\Phi: C[0,1] \to \mathbb{R}$$
is the collection of continuous functions from $[0,1] \to \mathbb{R}$. 
What is the meaning of this one?
$$\Phi: C[0,1] \to \mathbb{R}^{[0,1]}$$

It denotes the collection of functions $[0, 1] \to \mathbb{R}$. In general, $Y^X$ denotes the collection of functions $X \to Y$.

Regarding your edit, $\Phi : C([0, 1]) \to \mathbb{R}$ is not the collection of continuous functions $[0, 1] \to \mathbb{R}$. It is a function from the set of continuous functions $[0, 1] \to \mathbb{R}$, to the set of real numbers $\mathbb{R}$. So, if $f : [0, 1] \to \mathbb{R}$ is a continuous function (i.e. $f \in C([0, 1])$), then $\Phi(f)$ is a real number (i.e. $\Phi(f) \in \mathbb{R}$).
If instead $\Phi : C([0, 1]) \to \mathbb{R}^{[0, 1]}$, then for any $f \in C([0, 1])$, $\Phi(f)$ is a function $[0, 1] \to \mathbb{R}$.
