Generalized 2-local isometries of spaces of continuously differentiable functions

2-local isometries on function spaces

“…There exists an extensive literature on 2-local isometries in Iso C (M 1 , M 2 ) and 2-iso-reflexivity of Iso C (M 1 , M 2 ) (see, for example, [1,2,4,7,11,12,15,20,21]). Note that Hosseini showed that a 2-local real-linear isometry is in fact a surjective real-linear isometry on the algebra of n-times continuously differentiable functions on the interval [0, 1] with a certain norm [9,Theorem 3.1]. She described that a 2local real-linear isometry defined on the Banach algebra C(X) of all complex-valued continuous functions on a compact Hausdorff space X which is separable and first countable is in fact a surjective real-linear isometry on C(X) [9,Proposition 3.2].…”

“…Note that Hosseini showed that a 2-local real-linear isometry is in fact a surjective real-linear isometry on the algebra of n-times continuously differentiable functions on the interval [0, 1] with a certain norm [9,Theorem 3.1]. She described that a 2local real-linear isometry defined on the Banach algebra C(X) of all complex-valued continuous functions on a compact Hausdorff space X which is separable and first countable is in fact a surjective real-linear isometry on C(X) [9,Proposition 3.2]. At this point we emphasize that the situation is very different from that for the problem of 2-local isometry.…”