---
tag: E8N3EC0YH5
kind: concept
name: Holomorphy on Banach spaces
msc: [46G20, 46T25]
sources:
  - url: https://en.wikipedia.org/wiki/Infinite-dimensional_holomorphy
same_as:
  wikidata: Q6029727
  wikipedia: Infinite-dimensional holomorphy
---

## Summary

A map between complex Banach spaces is holomorphic if it has a complex-linear Fréchet derivative at every point of its domain.
This extends holomorphic functions of one and several complex variables to infinite dimension.
Holomorphic maps are exactly the complex analytic maps, and they are also the continuous maps that are holomorphic on each complex line.
They appear in operator theory, in spectral theory and in the study of moduli spaces.

## Explanations

### L1 human

A holomorphic function of one complex variable is one that has a complex derivative at each point.
For a map between spaces of vectors, the derivative becomes a complex-linear map that approximates the map near the point.
A good test is to restrict the map to complex lines $z \mapsto a + z h$, where it must be an ordinary holomorphic function.
For example, $x \mapsto x^2$ in an algebra of complex matrices is holomorphic.