A Note on Weak Stability of Ε-Sometries on Certain Banach Spaces

Abstract

In this paper, we will discuss the weak stability of epsilon-isometries on certain Banach spaces. Let f: X -> Y be a standard epsilon-isometry. If Y* is strictly convex, then for any x* is an element of X*, there is phi is an element of Y* that satisfies parallel to phi parallel to r = parallel to x*parallel to, such that vertical bar < x*, x > - <phi, f(x)>vertical bar <= 2r epsilon, x is an element of X. Also, we show that if X and Y are both L-P spaces (1 < p < infinity), f: X -> Y is a standard epsilon-isometry, then there exists a linear operator T:Y -> X with norm 1 such that parallel to Tf(x) - x parallel to <= 2 epsilon, for all x is an element of X.