Introduction into Calculus over Banach algebra

Abstract

Let $A$, $B$ be Banach $D$-algebras. The map $f:A\to B$ 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 $$f(x+dx)-f(x)=\frac{df(x)}{dx}\circ dx+o(dx)$$ where $$\frac{df(x)}{dx}:A\to B$$ is linear map and $o:A\to B$ is such continuous map that $$\underset{a\to 0}{\lim }\frac{\Vert o(a){\Vert }_B}{\Vert a{\Vert }_A}=0$$ Linear map $\frac{df(x)}{dx}$ is called derivative of map $f$. I considered differential forms in Banach Algebra. Differential form $\omega \in \mathcal{L}\mathcal{A}(D;A\to B)$ is defined by map $g:A\to B\otimes B$, $\omega =g\circ dx$. If the map $g$, is derivative of the map $f:A\to B$, 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 $b$.

English text - 139 pages; Russian text - 144 pages. arXiv admin note: substantial text overlap with arXiv:1505.03625, arXiv:1006.2597, arXiv:0812.4763

References in corpus (1)

• Doing Algebra over an Associative Algebra

Cited by in corpus (1)

• Differential Equation over Banach Algebra