Introduction into Calculus over Banach algebra

Abstract

Let $A$, $B$ be Banach $D$-algebras. The map $f:A\rightarrow B$ is called differentiable on the set $U\subset A$, if at every point $x\in U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x) =\frac{d f(x)}{d x}\circ dx +o(dx)$$ where $$\frac{d f(x)}{d x}:A\rightarrow B$$ is linear map and $o:A\rightarrow B$ is such continuous map that $$\lim_{a\rightarrow 0}\frac{\|o(a)\|_B}{\|a\|_A}=0$$ Linear map $\displaystyle\frac{d f(x)}{d x}$ is called derivative of map $f$. I considered differential forms in Banach Algebra. Differential form $\omega\in\mathcal{LA}(D;A\rightarrow B)$ is defined by map $g:A\rightarrow B\otimes B$, $\omega=g\circ dx$. If the map $g$, is derivative of the map $f:A\rightarrow B$, then the map $f$ is called indefinite integral of the map $g$ $$f(x)=\int g(x)\circ dx=\int\omega$$ Then, for any $A$-numbers $a$, $b$, we define definite integral by the equality $$\int_a^b\omega=\int_{\gamma}\omega$$ for any path $\gamma$ from $a$ to $b$.