Example

(1) If $G$ is the additive group of a complete normable space and $T_0(E)$ is canonically identified with $E$, $f^{-1}.df$ is the differential $df$ defined in Differentiable and Analytic Manifolds, R, 8.2.2.

(2) Suppose that $G$ is the multiplicative group $A^*$ associated with a complete normable algebra $A$. Then $f$ can be considered as a mapping of $M$ into $A$ and hence the differential $df$ in the sense of Differentiable and Analytic Manifolds, R, 8.2.2 is defined and the product $f^{-1}df$ in the sense of Differentiable and Analytic Manifolds, R, 8.3.2 is defined. Clearly the latter form is identical with the left differential of $f$.