It is very well-known that two
real Banach spaces are isometric if and only if they are linearly-isometric or that two
uniform algebras are linearly-isometric if and only if they are isomorphic as algebras.
These and similar classical “isometric” results have been extended by E. Behrends,
M. Cambern, J. Gevirtz, R. Rochberg, the author and others to “almost
isometric” cases. Proofs of the extended results are usually quite technical. In this
note we show that using ultraproducts of Banach spaces we can in some
cases deduce an “almost isometric” result from the classical one in just a few
lines.
