Let $X$ be a Banach space. Then we say that $X$ is reflexive if the map $J:X\to X^{**}$ defined by $J(x)(f):=f(x)$ is an isometric isomorphism. Here $X^{**}$ is the double dual of $X$ and $f\in X^{*}$ which is the dual of $X$.
This is how a reflexive Banach space is defined. Now, let $X$ be a reflexive Banach space. Can we get a map $I$ not equal to $J$ above, but $I$ is an isometric isomorphism from $X\to X^{**}$? I hope it should exist since in the definition precisely $J$ is mentioned to define reflexivity.
I could not find an answer to it nor succeed to think of an example. Can someone please help me to understand this?
Thanks.
