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  "body": "\n## Definition\n\nLet $X$ and $Y$ be complex [[Banach spaces|06GZ956657]], let $U \\subseteq X$ be open and let $f \\colon U \\to Y$.\nThe map $f$ is *holomorphic* on $U$ if, for each $a \\in U$, there is a continuous complex-linear map $L \\colon X \\to Y$ with\n$$\\lim_{h \\to 0} \\frac{\\norm{f(a + h) - f(a) - L h}}{\\norm{h}} = 0.$$\nThat is, $f$ has a complex [[Fréchet derivative|06EH97V2A7]] $Df(a) = L$ at each point $a$ of $U$.\n\n## Remarks\n\nThe following conditions on $f \\colon U \\to Y$ are equivalent:\n\n1. $f$ is holomorphic.\n2. $f$ is [[analytic|H1KHRWNH8B]] on $U$ over $\\C$.\n3. $f$ is continuous, and for all $a \\in U$ and $h \\in X$, the map $z \\mapsto f(a + z h)$ is a [[holomorphic function|0NAHXR1FRH]] on the open set of $z \\in \\C$ with $a + z h \\in U$.\n\nA map with the property on complex lines in 3, without continuity, is called Gateaux holomorphic, after the [[Gateaux derivative|EAZF9NJW12]].\n\nExample: let $B$ be a complex Banach algebra and $f(x) = x^2$.\nThen $f(a + h) - f(a) = a h + h a + h^2$, and $\\norm{h^2} \\le \\norm{h}^2$, so $Df(a) h = a h + h a$, which is complex-linear.\n\nNon-example: on $\\C$, the map $f(z) = \\overline{z}$ has the real derivative $h \\mapsto \\overline{h}$, which is not complex-linear.\nSo $f$ is not holomorphic, although it is real Fréchet differentiable.\n\nA discontinuous linear form on an infinite-dimensional complex Banach space is holomorphic on every complex line but is not holomorphic, since it is not continuous.\nThis shows that continuity in clause 3 cannot be dropped.\n",
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