we study the extension of isometries between the unit spheres S_1 (E) and S_1 (F),where E and F are both real normed spaces.We obtain that if the non-surjective isometry V_0 from S_1 (E) to S_1 (F) satisfy some properties,the V_0 can be extended to be real linearly isometric map V of E into F.This is the first time we study Tingley' problem without the condition of surjectivity.
Rui Dong WANG.
On Linear Extension of Isometries Between Unit Spheres. Acta Mathematica Sinica, Chinese Series, 2006, 49(6): 1335-133 https://doi.org/10.12386/A2006sxxb0168