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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\\mathbb{K}$.\nFor $n \\ge 1$, let $L^n(X, Y)$ be the space of continuous [[multilinear maps|0443XNARDV]] $A \\colon X^n \\to Y$, with the norm $\\norm{A} = \\sup \\set{\\norm{A(h_1, \\ldots, h_n)} : \\norm{h_1}, \\ldots, \\norm{h_n} \\le 1}$.\nLet $L^0(X, Y) = Y$.\nLet $U \\subseteq X$ be open, let $f \\colon U \\to Y$ and let $a \\in U$.\nThe map $f$ is *analytic at $a$* if there are $A_n \\in L^n(X, Y)$ for $n \\ge 0$ and a real $r \u003e 0$ such that:\n\n1. $\\sup_n \\norm{A_n} r^n \u003c \\infty$.\n2. $f(a + h) = \\sum_{n=0}^{\\infty} A_n(h, \\ldots, h)$ for all $h \\in X$ with $\\norm{h} \u003c r$.\n\nThe map $f$ is *analytic on $U$* if it is analytic at each point of $U$.\n\n## Remarks\n\nClause 1 makes the series in clause 2 converge absolutely, since $\\norm{A_n(h, \\ldots, h)} \\le \\norm{A_n} \\norm{h}^n$.\nThe maps $h \\mapsto A_n(h, \\ldots, h)$ are the continuous homogeneous polynomials of degree $n$; one may take each $A_n$ symmetric.\nFor $X = Y = \\R$, this is the usual [[analytic function|081WKEDK6C]].\nAn analytic map is $C^\\infty$ in the sense of the Fréchet derivative.\n\nExample: let $B$ be a Banach algebra with unit $1$ and $\\norm{xy} \\le \\norm{x} \\norm{y}$, and let $f(x) = (1 - x)^{-1}$ for $\\norm{x} \u003c 1$.\nTake $A_n(h_1, \\ldots, h_n) = h_1 h_2 \\cdots h_n$, so $\\norm{A_n} \\le 1$ for $n \\ge 1$, and $r = 1$.\nThe Neumann series $\\sum_n x^n$ converges and equals $(1 - x)^{-1}$, so $f$ is analytic at $0$.\n\nNon-example: the [[smooth non-analytic function|09RTYYHXKA]] $f(t) = e^{-1/t^2}$, $f(0) = 0$, on $\\R$ is not analytic at $0$.\nAll its derivatives at $0$ vanish, so any series as in clause 2 would be $0$, but $f(t) \\neq 0$ for $t \\neq 0$.\n",
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