Suppose that $X$ is a Banach space such that there exists a linear isometry $X \rightarrow X^*$. Must $X$ be reflexive?

Of course, this implies that $X$ is isometric with its second dual $X^{**}$. But with this alone it is not possible to conclude that $X$ is reflexive, James space is the famous counterexample for this. So a negative answer to my question should be at least as difficult as finding an example like the James space.. so probably not very easy.


No, consider $J\oplus_2 J^*$, where $J$ is a James space

  • $\begingroup$ Thanks, but what is $\oplus_2$? Does it mean that we take the "euclidean" norm for the product? $\endgroup$ – spin Nov 9 '13 at 21:32
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    $\begingroup$ This Hilbert sum of two Banach spaces. By definition $\Vert(x,y)\Vert_{X\oplus_2 Y}=(\Vert x\Vert_X^2+\Vert y\Vert_Y^2)^{1/2}$ $\endgroup$ – Norbert Nov 9 '13 at 21:34

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