Banach Space Isometric And Reflexive Space Characterizations

The metric and linear structure are two of the most important structures in normed linear space. Mazur-Ulam had shown that any surjective isometry between two real normed spaces must be an affine map. Therefore, the metric structure determines the linear structure.D. Tingley raised a general problem: Let E and F be normed spaces with unit spheres S(E) and S(F). Suppose that V₀:S(E)→S(F) is an onto isometric mapping, is there a linear isometric mapping V : E→F such that V|_s(e)=V₀?In chapter one, we study the extension of isometric mapping between the unit spheres S(E) and S(F) of normed space E and F respectively. And we get the following results:In section 1.2, we give a problem which is equivalent to Tingley's problem.In section 1.3, we prove that for any n∈N an into isometry form S(l_(n)^∞) to S(E), which under some conditions, can be extended to be a linear isometry defined on the whole space.In section 1.4, we study the Tingley's problem between the unit spheres of normed space E and l^p(p＞1).In section 1.5, We obtain that if T^-1 is a 1-Lipschtiz mapping between the unit sphere of l¹(Γ) and the unit sphere of l¹(Δ), moreover (?), thenT can be extended to be a real linear isometry from l¹(Γ) to l¹(Δ).In section 1.6, we proved that an into 1-Lipschitz mapping from the unit sphere of an Hilbert space to the unit sphere of an arbitrary normed space, which under some conditions, can be extended to be a linear isometry on the whole space.In section 1.7, we study the linear extension of the mapping between unit spheres of general normed spaces.In chapter 2, we give a small but interesting result which is a new character of reflexive.