## Unconditionally converging holomorphic mappings between Banach spaces.(English)Zbl 0828.46043

Summary: It is proved that every holomorphic mapping between complex Banach spaces takes unconditionally convergent series into unconditionally convergent series, and (locally) weakly unconditionally Cauchy series into weakly unconditionally Cauchy series. The class of unconditionally convergent holomorphic mappings is introduced, as those mappings taking (locally) weakly unconditionally Cauchy series into unconditionally convergent series. It is shown that a holomorphic mapping is unconditionally converging if and only if all its derivatives at the origin are unconditionally converging polynomials. A characterization is given of the spaces $$E$$ such that the space $${\mathcal H}_b (E)$$ of holomorphic functions of bounded type on $$E$$ is reflexive. Other properties of unconditionally converging holomorphic mappings are investigated. The analogous properties for polynomials are surveyed.

### MSC:

 46G20 Infinite-dimensional holomorphy 46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
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