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We prove a certain non-linear version of the Levi extension theorem formeromorphic functions. This means that the meromorphic function in question issupposed to be extendable along a sequence of complex curves, which arearbitrary, not necessarily straight lines. Moreover, these curves are notsupposed to belong to any finite dimensional analytic family. The conclusion ofour theorem is that nevertheless the function in question meromorphicallyextends along an (infinite dimensional) analytic family of complex curves andits domain of existence is a pinched domain filled in by this analytic family.

Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem