E0020 notation

Fréchet setting

Standing notation for the smooth calculus. Extends the cubical setting E0001 with Banach spaces and Fréchet derivatives.

(1) Notation (Banach spaces).

  • A Banach space \(E\) is a complete normed vector space over \(\IR\).
  • \(\KL(E, F)\) is the Banach space of bounded linear maps \(E \to F\) with operator norm \(\|A\| = \sup_{\|v\| \leq 1} \|A(v)\|\).
  • \(\KL({}^k E, F)\) is the Banach space of bounded \(k\)-fold multilinear maps \(E^{\times k} \to F\) with norm \(\|A_k\| = \sup_{\|v_i\| \leq 1} \|A_k(v_1, \dots, v_k)\|\).
  • \(\KL^s({}^k E, F) \subset \KL({}^k E, F)\) is the subspace of symmetric multilinear maps.

(2) Definition (Fréchet differentiability).

  • A map \(g: U \to F\) on an open \(U \subset E\) is Fréchet differentiable at \(x \in U\) if there exists \(L \in \KL(E, F)\) such that

    \[ \lim_{v \to 0} \frac{1}{\|v\|}\, \|\Delta(g; x; v) - L(v)\| = 0. \]

    The map \(L\) is unique and written \(D(g; x)\), so that \(D(g; x; v) = L(v)\).

  • A map \(g\) is \(C^1\) on \(U\) if \(D(g; x)\) exists for every \(x \in U\) and \(x \mapsto D(g; x) \in \KL(E, F)\) is continuous.

  • A map \(g\) is \(C^k\) on \(U\) if \(g\) is \(C^1\) and \(x \mapsto D(g; x)\) is \(C^{k-1}\). The \(k\)-th Fréchet differential is

    \[ D^k(g; x; v_1, \dots, v_k) := D(z \mapsto D^{k-1}(g; z; v_1, \dots, v_{k-1}); \, x;\, v_k). \]

    This is a symmetric \(k\)-linear form: \(D^k(g; x) \in \KL^s({}^k E, F)\).

  • For a multi-index \(\alpha \in \IN_0^k\), write \(D^\alpha(g; x; v_\bullet) := D^{|\alpha|}(g; x; v_\bullet^{\times \alpha})\), where \(v_\bullet^{\times \alpha}\) repeats \(v_i\) a total of \(\alpha_i\) times. By convention \(D^0(g; x) := g(x)\).

(3) Definition (Taylor polynomial).

  • The Taylor polynomial of order \(n\) of \(g \in C^n(U, F)\) at \(x\) is

    \[ T^n(g; x; v) := \sum_{\ell=0}^n \frac{1}{\ell!}\, D^\ell(g; x; v, \dots, v). \]

  • The reduced Taylor polynomial is \(T_*^n(g; x; v) := T^n(g; x; v) - g(x)\), so that \(T_*^n(g; x; 0) = 0\).

  • The Taylor remainder in Peano form is \(R^n(g; x; v) := \Delta(g; x; v) - T_*^n(g; x; v)\); for \(g \in C^n\) it satisfies \(\|R^n(g; x; v)\| / \|v\|^n \to 0\) as \(v \to 0\) E0024.

(4) Convention (Maps and points).

  • \(X, Y, Z\) denote Banach spaces; points are \(x \in X\), \(y \in Y\), \(z \in Z\).
  • \(g: X \to Y\) and \(f: Y \to Z\) are \(C^n\) maps with \(g(x) = y\), \(f(y) = z\).
  • Directions are \(v_1, \dots, v_k \in X\).
  • Explicit arguments over subscript notation: \(D(g; x; v_1, \dots, v_k)\) not \(D^k g(x)[v_1, \dots, v_k]\).
  • Semicolons separate arguments of different kinds.

(5) Validation (AI review, 2026-07-19, claude-fable-5, pass). Definitions checked against standard Fréchet calculus and for consistency with E0001/E0004 (\(\Delta\) reused, Banach spaces are abelian groups). Fixed \(D^{|\alpha|}\) in the multi-index bullet, added the \(D^0\) convention, attributed the Peano decay to E0024. Symmetry of \(D^k\) is classical (accepted). Caveat: \(T^n\) (Taylor polynomial) overloads the translation \(T^\alpha\) E0004 across chapters.

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