On isometric representation subsets of Banach spaces (Q2814348)

Let \(X,Y\) be Banach spaces, \(0\in M_{1}\subset X\) and \(0\in M_{2}\subset Y\). Denote by \(B\left( X\right) \) and \(S\left( X\right) \) the unit ball and the unit sphere of \(X,\) respectively. A mapping \(f:M_{1}\rightarrow M_{2}\) is called a standard isometry if \(f\left( 0\right) =0\) and \(\left\| f\left( u\right) -f\left( v\right) \right\| _{Y}=\left\| u-v\right\| _{X}\) for all \(u,v\in M_{1}\). A Banach space \(Y\) is said to be weakly nearly strictly convex if \(\left\{ y\in S\left( Y\right) :y^{\ast }\left( y\right) =1\right\} \) is weakly compact for each norm-attaining functional \(y^{\ast }\in S\left( Y^{\ast }\right)\). The authors prove that, if there is a standard isometry \(f:B\left( X\right) \rightarrow Y\), then there is a standard isometry \(F:X\rightarrow Y^{\ast \ast }\) (Theorem 2.8). Furthermore, if \(Y\) is weakly nearly strictly convex and \(f:B\left( X\right) \rightarrow Y\) is a standard isometry, then there is a standard isometry \( F:X\rightarrow Y\) (Corollary 2.10). Finally, if \(Y\) is weakly nearly strictly convex and \(f:B\left( X\right) \rightarrow Y\) is a standard isometry, then there is a linear isometry \(F:X\rightarrow Y\) (Theorem 3.4).