Let $E$ a Banach set. We say that $E$ is reflexive if $$J: E\to E^{''}$$ where $$Jx(f)=f(x)$$ is bijective (where $E''$ is the topological bidual). Now, I know if there is a bijective isometry $E\to F$, then $E$ and $F$ are the same (in a sense that is still not clear... but anyway). Now in Brezis functional analysis book (analyse fonctionnelle published by Dunod) page 43 remark 13, they say that : "We must use $J$ in the definition of reflexivity because there are spaces $E$ such that there is an onto isometry $E\to E''$ but $E$ is not reflexive".
I thought that if there is a bijective isometry between two spaces $A$ and $B$, then $A$ and $B$ are the same... So why do we must talk $J$ in the definition of reflexivity ? (since if there is a bijective isometry between two spaces they are the same).
Reflexivity [in the $j:X\rightarrow X^{**}$ sense]is equivalent to weak compactness of the unit ball. The requirement that this map be onto is quite specific.