{"paper":{"title":"Derivative of Map of Banach algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Aleks Kleyn","submitted_at":"2015-05-14T05:42:10Z","abstract_excerpt":"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 $$ \\lim_{a\\rightarrow 0}\\frac{|o(a)|}{|a|}=0 $$\n  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 ord"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03625","kind":"arxiv","version":1},"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"}