The Stability of Isometry by Singular Value Decomposition

Hyers and Ulam considered the problem of whether there is a true isometry that approximates the ε-isometry defined on a Hilbert space with a stability constant 10ε. Subsequently, Fickett considered the same question on a bounded subset of the n-dimensional Euclidean space Rn with a stability constant of 27ε1/2n. And Vestfrid gave a stability constant of 27nε as the answer for bounded subsets. In this paper, by applying singular value decomposition, we improve the previous stability constants by Cnε for bounded subsets, where the constant C depends on the approximate linearity parameter K, which is defined later.