A continuous form of Schwarz’s lemma in normed linear spaces

Lawrence Albert Harris

Vol. 38 (1971), No. 3, 635–639

DOI: 10.2140/pjm.1971.38.635

Abstract

Our main result is an inequality which shows that a holomorphic function mapping the open unit ball of one normed linear space into the closed unit ball of another is close to being a linear map when the Fréchet derivative of the function at 0 is close to being a surjective isometry. We deduce this result as a corollary of a kind of uniform rotundity at the identity of the sup norm on bounded holomorphic functions mapping the open unit ball of a normed linear space into the same space.

Mathematical Subject Classification

Primary: 46B05

Milestones

Received: 16 October 1970

Published: 1 September 1971