I have struggle with understanding details of the following construction: Let $M$ be a smooth manifold. consider $F:=C^{\infty}(M,\Bbb R)$, Now consider the following $f:M\to \Bbb R^F$!!!
What is going on? First we collect all real value and smooth maps to a set $F$, then we consider $\Bbb R^F$ that is $\{h|h:F\to \Bbb R\}$ then we consider $f:M\to \Bbb R^F$. What is happening here? What are these trying to explain?