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  "body": "\n## Summary\n\nA map between complex Banach spaces is holomorphic if it has a complex-linear Fréchet derivative at every point of its domain.\nThis extends holomorphic functions of one and several complex variables to infinite dimension.\nHolomorphic maps are exactly the complex analytic maps, and they are also the continuous maps that are holomorphic on each complex line.\nThey appear in operator theory, in spectral theory and in the study of moduli spaces.\n\n## Explanations\n\n### L1 human\n\nA holomorphic function of one complex variable is one that has a complex derivative at each point.\nFor a map between spaces of vectors, the derivative becomes a complex-linear map that approximates the map near the point.\nA good test is to restrict the map to complex lines $z \\mapsto a + z h$, where it must be an ordinary holomorphic function.\nFor example, $x \\mapsto x^2$ in an algebra of complex matrices is holomorphic.\n",
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