Faà di Bruno is Taylor Composition

We approach Faà di Bruno as a composition theorem for Taylor polynomials.
For $C^k$ maps $\phi: E \to F$ and $\psi: F \to G$ between Banach spaces,
let $T^k_\ast(\phi; x)$ denote the reduced Taylor polynomial of $\phi$ at $x$, obtained by removing the constant term.
We show that

$$T^k_\ast(\psi \circ \phi; x) = \pi_{\le k}(T^k_\ast(\psi; \phi(x)) \circ T^k_\ast(\phi; x)).$$

The proof is an elementary estimate of the Peano remainder and does not use partitions or combinatorial enumeration.

Expanding this composition identity recovers the classical Faà di Bruno formulas.
Polarization gives the multivariate partition formula (Lévy 2006),
while coefficient extraction gives the multi-index formula (Constantine and Savits 1996).
Our approach separates the functorial nature of Taylor approximation from the combinatorial bookkeeping of polarization and coefficient extraction.

As an application, we give a general higher-order product rule.

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