Generalized Stability of Isometries

✍ Scribed by Gregor Dolinar

• Book ID: 102596641
• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 127 KB
• Volume: 242
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1999.6649

✦ Synopsis

Let X and Y be real Banach spaces and let , p G 0. A mapping f : X ª Y is Ž . < 5 Ž . Ž .5 5 5 < 5 5 p called an , p -isometry if f x y f y y x y y F x y y holds for all Ž . x, y g X. A pair X, Y is p-stable with respect to isometries if there exists a w . w . Ž. function ␦ : 0,ϱ ª 0, ϱ with lim ␦ s 0 such that for every surjective

We show that every pair of Banach Ž . Ž 2 2 . spaces X, Y is p-stable for 0 F p -1. The pair ‫ޒ‬ , ‫ޒ‬ is not 1-stable. When p ) 1 a superstability phenomenon occurs for finite-dimensional Banach spaces.