An implicit function theorem in Banach spaces

Iain Raeburn

Vol. 81 (1979), No. 2, 525–535

DOI: 10.2140/pjm.1979.81.525

Abstract

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

Milestones

Received: 10 January 1977

Published: 1 April 1979