1) This first link seems to provide a proof that seems to work for all normed vector spaces.
If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism?
2) This second link provides a counter example.
I can't seem to find any differences in the assumptions. Could someone explain whats the difference and if the statement is rue in general for all normed vector spaces? I know in the first link it mentions linear homeomorphisms and hilbert spaces but the proof provided seems to hold for any normed spaces(not just banach)and the the continuity of the inverse is not used in the proof.
Best Answer
In the first post the spaces are complete and in the second they are not. In the first post completeness is necessary. It is assumed that the inverse $S$ of $T$ is a bounded operator. (If it is just a linear map we cannot even define its adjoint). So Open Mapping Theorem is used here.