If T is an isomorphism of（co）into C（Ω）（whereΩis a sequentially compact and paracompact space,or a compact metric space in particular）,which satisfies the con- dition‖T‖·‖T-1‖≤1+∈for some∈∈（0,1/5）,then T/（‖T‖）is close to an isometry with an error less than 9∈.The proof of this article is simple without using the dual space or adjoint operator.
Two kinds of convergent sequences on the real vector space m of all bounded sequences in a real normed space X were discussed in this paper,and we prove that they are equivalent,which improved the results of [1].
Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some condition,T0 can be extended to a linear isometry on the whole space.
The main result of this paper is to prove Fang and Wang's result by another method:Let E be any normed linear space and V0:S(E) → S(l1) be a surjective isometry.Then V0 can be linearly isometrically extended to E.
