Composition of $C^\infty$ maps between Banach spaces is $C^\infty$.

Question

Let $V$, $W$, and $X$ be Banach spaces, and let $A \subset V$ and $B \subset W$ be open. Suppose that $F \in C^\infty(A,W)$, $G \in C^\infty(B,X)$, and $F(A) \subset B$. Is there a non-combinatorial proof that $G \circ F \in C^\infty(A,X)$?

Answer 1

HINT: Use the chain rule and induction on $k$ to prove that $F,G\in C^k \implies G\circ F\in C^k$. (Remember that $F\in C^k \iff DF\in C^{k-1}$.)