Generalized Stability of Isometries

Let X and Y be real Banach spaces and let « , pG0. A mapping f : XªY is . < 5 . .5 5 5< 5 5 p called an « , p -isometry if f x yf y y xyy F« xyy holds for all x, ygX. A pair X, Y.is p-stable with respect to isometries if there exists a function d: w0,`.ªw0, `.with lim d «.s0 such that for every surjective «ª0 « , p.-isometry f : XªY there is a surjective isometry U: XªY satisfying the estimate 5 f x.yU x.5Fd «.5x5p, xgX. We show that every pair of Banach spaces X, Y.is p-stable for 0Fp-1. The pair R2, R2. is not 1-stable. When p)1 a superstability phenomenon occurs for finite-dimensional Banach spaces