Derivative of Map of Banach algebra

Abstract

Let $A$ be Banach algebra over commutative ring $D$. The map $f:A\to A$ 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 $$\underset{a\to 0}{\lim }\frac{|o(a)|}{|a|}=0$$ 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 $$f(x)=\sum _{n=0}^{\infty }(n!{)}^{-1}{\partial }^{n}f(x_0)\circ (x-x_0{)}^n$$

English text - 27 pages; Russian text - 27 pages

References in corpus (1)

• Linear Map of $D$-Algebra