E0023 theorem notation E0020 → E0001

Fréchet product rule

The discrete product rule E0014 passes to Fréchet derivatives: ordered coverings collapse to ordered partitions, since overlapping subsets produce zero in the symmetric derivative.

(1) Theorem. Let \(X\) be a Banach space, \(A\) a Banach algebra, and \(f_1, \dots, f_r: X \to A\) be \(C^n\) near \(x\), with \(n \geq k\). For \(v_1, \dots, v_k \in X\):

  • Boolean product rule:

    \[ D(f_1 \cdots f_r; x; v_{[k]}) = \sum_{J_1 \sqcup \cdots \sqcup J_r = [k]} D(f_1; x; v_{J_1}) \cdots D(f_r; x; v_{J_r}). \]

  • Binomial product rule. For \(\gamma \in \IN_0^k\) with \(|\gamma| \leq n\):

    \[ D(f_1 \cdots f_r; x; v_\bullet^{\times \gamma}) = \sum_{\alpha_1 + \cdots + \alpha_r = \gamma} \frac{\gamma!}{\alpha_1! \cdots \alpha_r!}\, D(f_1; x; v_\bullet^{\times \alpha_1}) \cdots D(f_r; x; v_\bullet^{\times \alpha_r}). \]

    The Boolean sum runs over ordered partitions \((J_1, \dots, J_r)\) of \([k]\): the \(J_a\) are pairwise disjoint and empty \(J_a\) are allowed. This is the discrete product rule E0014 with coverings replaced by partitions.

Proof. The coefficients of \(f_1 \cdots f_r\) up to order \(n\) agree with those of the product \(P_1 \cdots P_r\) of the Taylor polynomials \(P_a = T^n(f_a; x)\): multiply the Peano expansions (the multiplication of \(A\) is bounded bilinear) and use uniqueness of the Taylor polynomial E0024. To the polynomial maps the discrete product rule E0014 applies: \(\Delta(P_1 \cdots P_r)\) is the sum over ordered coverings \((J_1, \dots, J_r)\) of \([k]\) of \(\prod_a \Delta(P_a; x; v_{J_a})\).

The derivative \(D(f_1 \cdots f_r; x; v_{[k]})\) is the part of this difference that is multilinear in \(v_1, \dots, v_k\), as in the collapse argument of E0022. Every term of \(\prod_a \Delta(P_a; x; v_{J_a})\) has slot degree \(\geq \sum_a |J_a| \geq k\), with equality iff the \(J_a\) are pairwise disjoint, so only ordered partitions contribute multilinear terms; for those, the multilinear part of each factor is the polarization \(D(f_a; x; v_{J_a})\).

The binomial form follows by applying the Boolean form to the slot set \(S(\gamma)\) with directions \(v \circ \pi\) and grouping by the profiles \(\alpha_a = \nu(J_a)\): there are \(\gamma!/(\alpha_1! \cdots \alpha_r!)\) ordered partitions of \(S(\gamma)\) with given fiberwise profiles summing to \(\gamma\).

(2) Validation (AI review, 2026-07-19, claude-fable-5, pass). Statement verified (Leibniz \(r = 2\), \(k = 1\); ordered partitions at \(k = 2\); binomial multinomial count). The previous collapse justification (vanishing "after symmetrization and truncation") was invalid — replaced with the multilinear extraction argument of E0022; corrected the Taylor-product citation to Peano + uniqueness E0024; added the missing proof of the binomial form. Statement unchanged.