In this paper, we first derive the representation theorem of onto isometric mappings in the unit spheres of real l∞-type spaces, then we conclude that such mappings can be extended to the whole space as (real) linear isometries.

This article presents a novel method to prove that: let E be an AM-space and if dim E≥3, then there does not exist any odd subtractive isometric mapping from the unit sphere S(E) into S[L(Ω, μ)]. In particular, there does not exist any real linear isometry from E into L(Ω, μ).

In this paper, we study the extension of isometries between the unit spheres of ome Banach spaces E and the spaces C(Ω): We obtain that if the set sm:S1(E) of all smooth points of the unit sphere S1(E) is dense in S1(E), then under some condition, every surjective isometry V0 from S1(E) onto S1(C(Ω)) can be extended o be a real linearly isometric map V of E onto C(Ω): From this result we also obtain ome corollaries. This is the first time to study this problem on the differently typical paces, and the method of proof is also very different too.

In this survey article, we introduce the isometric extension problem of isometric mapping between the unit spheres or open domains also, the distance one preserving problem and some other problems such as the problem of the weak or strong perturbation of linear or nonlinear operators, the problem of the asymptotically isometric operator are also mentioned. Some important results in the development of the related problems are outlined in this paper and some recent advancement and open problems are repointed.

In this paper, we shall present a short and simple proof on the iso-metric linear extension problem of into-isometries between two unit spheres of atomic abstract Lp-spaces (0<p<∞).