I'd like to prove that the composition of smooth functions between Banach spaces is smooth. What puzzles me a bit is notation, how do I write the chain rule in terms of functions without explicit evaluation? Like, one has $D(g\circ f)(x)=D(g)(f(x))\circ D(f)(x)$ but if I want to write it without the $x$ then I get $D(g\circ f)=(D(g)\circ f)\cdot D(f)$ where $\cdot$ denotes pointwise composition of (pointwise composable) functions into function spaces. Now I neither know the correct name resp. symbol for $\cdot$ nor do I know which differentiation rules apply to it...
2026-04-09 09:07:27.1775725647
Differentiation of pointwise composition operator
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Let us abbreviate $j:=D(g)\circ f$, $h:=D(f)$. We can then write $j\cdot h$ as $b\circ(j,h)$ with $(j,h):E\rightarrow L(F,G)\times L(E,F)$ where we understand $b$ as the bilinear map $(g,f)\mapsto g\circ f$, then the rest consists of applying the chain rule in combination with the well-known derivative of bilinear functions.