$$
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% GENERAL / UNIVERSAL
% Used throughout the documentation
% ============================================
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% ============================================
% CHAPTER 1: DIFFERENTIAL DUALITY
% ============================================
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% ============================================
% CHAPTER 2: DISCRETE DUALITY
% ============================================
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% ============================================
% CHAPTER 3: ULTRA CALCULUS
% ============================================
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% OPERATORS (DeclareMathOperator)
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% ============================================
% ENVIRONMENT SIDEBARS
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% marking structure without web-style color coding.
% ============================================
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bottomline=false,
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\surroundwithmdframed[style=envstyle]{example}
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$$
Polynomials on Banach spaces
(1) Definition (Homogeneous polynomials).
-
A map \(p: E \to F\) between Banach spaces is a
\(k\)-homogeneous polynomial if there exists a symmetric
bounded multilinear map
\(A \in \KL^s({}^k E, F)\) with
\[
p(x) = \frac{1}{k!}\, A[x^{\otimes k}].
\]
-
\(\KP^k(E, F)\) is the space of \(k\)-homogeneous polynomials.
- A polynomial map is a finite sum
\(p = \sum_{k=0}^n p_k\) with \(p_k \in \KP^k(E, F)\).
- \(\KP_{\leq n}(E, F)\) is the space of polynomials of degree
\(\leq n\).
(2) Lemma (Polarization). The top-degree multilinear form of
a polynomial \(p = \sum_{k=0}^n \frac{1}{k!} A_k[x^{\otimes k}]\)
is recovered by the forward difference E0004:
\[
A_n[v_1, \dots, v_n] = \Delta(p; 0; v_1, \dots, v_n).
\]
Every polynomial has a unique representation. The degree, the
degree-\(k\) part \(\pi_k(p)\), and the truncation
\(\pi_{\leq k}(p)\) are therefore well defined.
Proof.
By multilinearity,
\(p(\sum_{i \in I} v_i) = \sum_{J \subseteq I} A_{|J|}[v_J]\).
The alternating sum \(\sum_I (-1)^{n-|I|}\) inverts this by
Boolean Möbius inversion E0002: all terms with
\(J \subsetneq [n]\) cancel by \((1-1)^{n-|J|} = 0\), leaving
\(A_n[v_1, \dots, v_n]\). Uniqueness follows by induction:
recover \(A_n\) via polarization, subtract, and repeat.
(3) Lemma (Polynomial properties).
- Composition: \(\deg(q \circ p) \leq \deg(q) \cdot \deg(p)\).
- Lipschitz: \(\|\Delta(p; 0; x)\| \leq C \|x\|\) for small
\(\|x\|\).
- Upper vanishing:
\(\|\pi_{> k}(p)(x)\| / \|x\|^k \to 0\) as \(x \to 0\).
- Lower vanishing: If \(\|p(x)\| / \|x\|^k \to 0\) and
\(k \geq \deg(p)\), then \(p = 0\).
Proof.
Ad composition) Expanding \(q(p(x))\) by multilinearity: each
term \(B[A_1[x^{\otimes j_1}], \dots, A_\ell[x^{\otimes j_\ell}]]\)
is a bounded multilinear composition, hence a homogeneous
polynomial of degree \(j_1 + \cdots + j_\ell \leq \ell \cdot \deg(p)\).
Ad Lipschitz)
\(\|\Delta(p; 0; x)\| = \|\sum_{j=1}^n \frac{1}{j!} A_j[x^{\otimes j}]\|
\leq \sum_j \frac{\|A_j\|}{j!} \|x\|^j \leq C\|x\|\) for
\(\|x\| < 1\).
Ad upper vanishing) Each \(p_j\) with \(j > k\) satisfies
\(\|p_j(x)\| \leq C_j \|x\|^j\), so
\(\|p_j(x)\| / \|x\|^k = C_j \|x\|^{j-k} \to 0\).
Ad lower vanishing) If \(p \neq 0\), let \(j_0\) be the minimal
nonzero degree and choose \(v\) with \(p_{j_0}(v) \neq 0\). Then
\(p(tv) = t^{j_0} p_{j_0}(v) + O(t^{j_0+1})\): for
\(j_0 < k\) the ratio \(\|p(tv)\| / t^k\) diverges; for \(j_0 = k\)
it converges to \(\|p_{j_0}(v)\| \neq 0\). Either contradicts
\(p = o(\|x\|^k)\).