Holomorphy on Banach spaces

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.