The reflexivity of a Banach space is usually defined as having to be enforced by a particular isometric isomorphism. Namely the map that takes each element to the evaluation, which is already an injective linear isometry from a Banach space to its double dual. It just isn't necessarily surjective.
Are there examples of when a space is isometrically isomorphic to its double dual, but is not reflexive? If $X$ is reflexive, is the same true of the $k^{th}$ dual of $X$? (Or maybe this property moves backwards instead? As in maybe if the dual is reflexive then so is the original. Separability does this too, but I'm not sure how to intuit when a property should pushforward and when it should pull back under the operation of taking a dual.) Does being reflexive, or at least isometrically isomorphic to its double dual capture any information about the "size" of the space or any other intuitive concepts that are easy to picture?
Best Answer
Here are answers to your questions.
1) Robert C. James provided the first example of non-reflexive Banach space isomorphic to its double dual. Also, see this MO post. Thanks to David Mitra for the reference.
2) Banach space is reflexive iff $X^*$ is reflexive. See this post for details.
3) Reflexivity does not bound the size of Banach space. For example $\ell_p(S)$ with $1<p<+\infty$ is reflexive, $\dim \ell_p(S)>|S|$, though $|S|$ could be arbitrarily large.