I am having a little trouble understanding something my lecturer wrote about Reflexivity and isometric isomorphisms
We have an isomorphism which happens to be isometric, and lecturer says:
“This (reflexivity is not quite the same as saying that $X$ and $X^{∗∗}$ are isometrically isomorphic. In this case, the isometric isomorphism must be given by the specific operator Jx”, where Jx is the canonical embedding
It is my understanding that for two things to be isomorphic, all that matters is that there exists an isomorphism between the two things. And to be isometrically isomorphic just means that that isomorphism is an isometry? So why is it the case that in this example the two spaces aren’t isometrically isomorphic???
Many thanks.
SOLVED: Reflexive does indeed imply isometrically isomorphic, but the reverse implication isn’t true, hence “not the same”
Best Answer
[I am writing here because comment is not sufficient for this. You may take it as an answer.]
Well, I think your lecturer is trying to say that the spaces $X$ and $X^{**}$ are isometrically isomorphic does not mean that the space $X$ is reflexive.
$X$ is reflexive if the canonical embedding $C:X\to X^{**}$ is surjective.
During calculation this canonical map may appear as an isometric isomorphism in that case $X$ and $X^{**}$ are isometrically isomorphic, but for any other map $T:X\to X^{**}$ which is an isometry and an isomorphism may not imply that $T$ is a canonical embedding i.e. $X$ may not be Reflexive.