On linear isometries and ε-isometries between Banach spaces.

Let X , Y be two Banach spaces, and f : X → Y be a standard ε -isometry for some ε ≥ 0 . Recently, Cheng et al. showed that if co ‾ [ f ( X ) ∪ − f ( X ) ] = Y , then there exists a surjective linear operator T : Y → X with ‖ T ‖ = 1 such that the following sharp inequality holds: ‖ T f ( x ) − x ‖ ≤ 2 ε for all x ∈ X . Making use of the above result, we prove the following results: Suppose that co ‾ [ f ( X ) ∪ − f ( X ) ] = Y . Then (1) if there is a linear isometry S : X → Y such that T S = Id X , then T ⁎ S ⁎ : Y ⁎ → T ⁎ ( X ⁎ ) is a w ⁎ -to- w ⁎ continuous linear projection with ‖ T ⁎ S ⁎ ‖ = 1 , (2) if there exists a w ⁎ -to- w ⁎ continuous linear projection P : Y ⁎ → T ⁎ ( X ⁎ ) with ‖ P ‖ = 1 , then there is an unique linear isometry S ( P ) : X → Y such that T S ( P ) = Id X and P = T ⁎ S ( P ) ⁎ . Furthermore, if P 1 ≠ P 2 are two w ⁎ -to- w ⁎ continuous linear projection from Y ⁎ onto T ⁎ ( X ⁎ ) with ‖ P 1 ‖ = ‖ P 2 ‖ = 1 , then S ( P 1 ) ≠ S ( P 2 ) . We apply these results to provide an alternative proof of a recent theorem, which gives an affirmative answer of a question proposed by Vestfrid. We also unify several known theorems concerning the stability of ε -isometries. [ABSTRACT FROM AUTHOR]