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 C √ nε for bounded subsets, where the constant C depends on the approximate linearity parameter K, which is defined later.
Loading....