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- Title
On Non-linear Mappings Preserving the Semi-inner Product.
- Authors
Kobos, Tomasz; Wójcik, Paweł
- Abstract
We say that a smooth normed space X has a property (SL), if every mapping f : X → X preserving the semi-inner product on X is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. In this paper, we establish several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space ℓ p , but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach spaces in terms of subspaces of quotient spaces. As a consequence, we prove that the space ℓ p have the property (SL) for every 1 < p < ∞ . Finally, using a variant of the Gowers–Maurey space, we construct an infinite-dimensional uniformly smooth Banach space X such that every smooth Banach space isomorphic to X has the property (SL).
- Publication
Results in Mathematics / Resultate der Mathematik, 2023, Vol 78, Issue 6, p1
- ISSN
1422-6383
- Publication type
Article
- DOI
10.1007/s00025-023-01993-5