Linearization of isometric embeddings between Banach spaces: An elementary approach

Abstract

Mazur-Ulam’s classical theorem states that any isometric onto map between normed spaces is linear. This result has been generalized by T. Figiel [F] who showed that ifΦ is an isometric embedding from a Banach space X to a Banach space Y such that Φ(0) = 0 and vect [φ(X )] = Y , there exists a linear quotient map Q such that ‖Q‖ = 1 and Q ◦Φ= I dX . The third chapter of this short story is [GK] where it is shown that if a quotient map Q from a Banach space Y onto a separable Banach space X has a Lipschitz lifting, then it actually has a continuous linear lifting. Combining this statement with Figiel’s theorem provides another result from [GK] : if a separable Banach space X isometrically embeds into a Banach space Y , there exists an isometric linear embedding from X into Y . Nigel Kalton had an outstanding ability to set theorems and proofs in their proper frame. His articles are therefore fountains of ideas, irrigating each of the many fields to which he contributed : a non exhaustive survey on these contributions is [G]. The paper [GK] is no exception to this rule, and among other things it prepared the ground for far-reaching extensions, where e.g. the Lipschitz assumption is weakened to the Holder condition, leading to very different conclusions ( [K1], [K2], [K3] ). However, some readers could find daunting some arguments from [GK], whichwould probably turn down undergraduate students. Since the extension from [GK] of Mazur-Ulam’s theorem which is recalled above has a statement which is understandable to any student in mathematics, it seems appropriate to provide an elementary proof. This is the purpose of the present short note, where only basic functional analysis (elementary duality theory) and calculus (culminating at Fubini’s theorem for continuous functions on Rn) are used. Moreover this note is fully self-contained : even the generic smoothness of convex fuctions on Rn is shown, and a detailed proof of Figiel’s theorem is provided (following [BL], although Lemma 2 is not stated there). Themain result of the note is Theorem 5, whose proof follows the strategy of the proof of ( [GK], Corollary 3.2) but in such an elementary way that diagrams, free spaces and infinitedimensional integration or differentiation arguments are avoided. Thus our approach makes it clear that the core of the proof consists of finite-dimensional considerations. We can therefore teach at an undergraduate level Mazur-Ulam’s theorem and its natural extensions from [F] and [GK].