{"paper":{"title":"Introduction into Calculus over Banach algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Aleks Kleyn","submitted_at":"2016-01-11T05:43:09Z","abstract_excerpt":"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 $f$.\n  I considered differential forms in Banach Algebra. Differential form $\\omega\\in\\mathcal{LA}(D;A\\rightarrow B)$ is def"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03259","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}