GENERALIZED 2-LOCAL ISOMETRIES OF SPACES OF CONTINUOUSLY DIFFERENTIABLE FUNCTIONS MALIHEH HOSSEINI

Let C-(n) (I) denote the Banach space of n-times continuously differentiable functions on I = [0, 1], equipped with the norm vertical bar vertical bar f vertical bar vertical bar(n) = max {vertical bar f(0)vertical bar, vertical bar f'(0)vertical bar,..., vertical bar f((n-1))(0)vertical bar, vertical bar vertical bar f(n)vertical bar vertical bar(infinity)} (f is an element of C-(n) (I)), where vertical bar vertical bar center dot vertical bar vertical bar(infinity) is the supremum norm. We call a map T : C-(n) (I) -> C-(n)(I) a 2-local real-linear isometry if for each pair f, g in C-(n)(I), there exists a surjective real-linear isometry T-f,T-g : C-(n)(I) -> C-(n)(I) such that T(f) = T-f,T-g(f) and T(g) = T-f,T-g(g). In this paper we show that every 2-local real-linear isometry of C-(n)(I) is a surjective real-linear isometry. Moreover, a complete description of such maps is presented.