Derivative for Functions $f : G \to H$, Where $G$ Is a Metric Divisible Group

TL;DR

The paper extends Carathéodory-style differentiability to functions $f: G \to H$ where $G$ is a metric divisible group and $H$ is an Abelian metric group with a group metric. It constructs the space $\tilde{\mathrm{Hom}}(G; H)$ of continuous group homomorphisms equipped with a metric $\tilde{d}$ and the binary operation $\oplus$, enabling a slope-based derivative defined via $f(x) f(a)^{-1} = \phi_f(x)[x a^{-1}]$ with $f'(a) = \phi_f(a)$. The main results include differentiability implying continuity, linearity of the derivative, and a chain rule for composition, together with a homogeneity discussion when $H$ is a topological vector space. An explicit example on $G = M_{n\times n}(\mathbb{R})$ with $f(X) = X^2$ shows $f'(A)[Y] = AY + YA$, matching the Fréchet derivative and illustrating the theory’s alignment with classical differentiability in familiar settings. Overall, the work generalizes derivative concepts to broader metric-group settings, enabling analysis and chain rule results beyond traditional vector-space frameworks.

Abstract

In this paper, a derivative for functions $f : G \to H$, where $G$ is any metric divisible group and $H$ is a metric Abelian group with a group metric, is defined. Basic differentiation theorems are stated and demonstrated. In particular, we obtain the Chain Rule

Theorems & Definitions (28)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• Proposition 2.6
• Theorem 2.7
• Definition 3.1: Definition 4
• Definition 3.2: Definition 5