This paper considers the isometric extension problem concerning the mapping from the unit sphere S 1( E) of the normed space E into the unit sphere S 1( l ∞(Γ)). We find a condition under which an isometry from S 1( E) into S 1( l ∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V 0: S 1( E) → S 1( F) is a surjective isometry, where c 00(Γ) ⊆ F ⊆ l ∞(Γ), then V 0 can be extended to be an isometric operator defined on the whole space. [ABSTRACT FROM AUTHOR]
In this paper, we .rst derive the representation theorem of onto isometric mappings in the unit spheres ofl1(G) type spaces, and then conclude that such mappings can be extended to the whole space as real linear isometries by using a previous result of the author. [ABSTRACT FROM AUTHOR]