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ON ISOMETRIC REPRESENTATION SUBSETS OF BANACH SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 3 / June 2016
- Published online by Cambridge University Press:
- 11 December 2015, pp. 486-496
- Print publication:
- June 2016
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- Article
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- You have access
- Export citation
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Let
$X,Y$ be two Banach spaces and 
$B_{X}$ the closed unit ball of 
$X$. We prove that if there is an isometry 
$f:B_{X}\rightarrow Y$ with 
$f(0)=0$, then there exists an isometry 
$F:X\rightarrow Y^{\ast \ast }$. If, in addition, 
$Y$ is weakly nearly strictly convex, then there is an isometry 
$F:X\rightarrow Y$. Making use of these results, we show that if 
$Y$ is weakly nearly strictly convex and there is an isometry 
$f:B_{X}\rightarrow Y$ with 
$f(0)=0$, then there exists a linear isometry 
$S:X\rightarrow Y$.
