If $X$ and $Y$ are Banach spaces and $T:X\to Y$ is an operator such that $T':Y'\to X'$ is an isomorphism, is $T$ an isomorphism? How would you prove this? Do you have any reference for this?
This is false if they fail to be complete. For example, if $X$ is a proper dense subspace of $Y$ and $T:X\to Y$ is the inclusion, $T$ is not an isomorphism but $T':Y'\to X'$ is the restriction and it is an isomorphism.
