Vol. 81, No. 2, 1979

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An implicit function theorem in Banach spaces

Iain Raeburn

Vol. 81 (1979), No. 2, 525–535

We prove the following theorem:

Theorem: Suppose X, Y , and Z are complex Banach spaces, U and V are open sets in X and Y respectively, and x U, y V . Suppose f : U V and k : V Z are holomorphic maps with f(x) = y, k f constant and range f(x) = ker k(y){0}. Let D be a domain in Cn, z D and g : D Y be a holomorphic map with g(z) = y and k g constant. Then there is an open neighborhood W of z and a holomorphic map h : W X such that h(z) = x and g|W = f h.

We use this result to prove an Oka principle for sections of a class of holomorphic fibre bundles on Stein manifolds whose fibres are orbits of actions of a Banach Lie group on a Banach space.

Mathematical Subject Classification 2000
Primary: 32A99
Secondary: 32E99, 58C15
Received: 10 January 1977
Published: 1 April 1979
Iain Raeburn