Approximate isometries on finite dimensional Banach spaces

Author:
Richard D. Bourgin

Journal:
Trans. Amer. Math. Soc. **207** (1975), 309-328

MSC:
Primary 46B05

MathSciNet review:
0370137

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Abstract: A map ( Banach spaces) is an -isometry if whenever . The problem of uniformly approximating such maps by isometries was first raised by Hyers and Ulam in 1945 and subsequently studied for special infinite dimensional Banach spaces. This question is here broached for the class of finite dimensional Banach spaces. The only positive homogeneous candidate isometry approximating a given -isometry is defined by the formal limit . It is shown that, whenever is a surjective -isometry and is a finite dimensional Banach space for which the set of extreme points of the unit ball is totally disconnected, then this limit exists. When - dimensional a uniform bound of uniform approximation is obtained for surjective -isometries by isometries; this bound varies linearly in and with .

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DOI:
https://doi.org/10.1090/S0002-9947-1975-0370137-4

Keywords:
Approximate isometry,
isometry,
finite dimensional Banach space

Article copyright:
© Copyright 1975
American Mathematical Society