Vol. 81, No. 2, 1979

Recent Issues
Vol. 329: 1
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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