Derivative of Map of Banach algebra

Abstract

Let $A$ be Banach algebra over commutative ring $D$. The map $f:A\rightarrow A\ $ is called differentiable in the Gateaux sense, if $$f(x+a)-f(x)=\partial f(x)\circ a+o(a)$$ where the Gateaux derivative $\partial f(x)$ of map $f$ is linear map of increment $a$ and $o$ is such continuous map that $$ \lim_{a\rightarrow 0}\frac{|o(a)|}{|a|}=0 $$ Assuming that we defined the Gateaux derivative $\partial^{n-1} f(x)$ of order $n-1$, we define $$ \partial^n f(x)\circ(a_1\otimes...\otimes a_n) =\partial(\partial^{n-1} f(x)\circ(a_1\otimes...\otimes a_{n-1}))\circ a_n $$ the Gateaux derivative of order $n$ of map $f$. Since the map $f(x)$ has all derivatives, then the map $f(x)$ has Taylor series expansion $$ f(x)=\sum_{n=0}^{\infty}(n!)^{-1}\partial^n f(x_0)\circ(x-x_0)^n $$