Let X be a Banach space. We know that X embeds into its double dual X** as a closed subspace isometrically via defined by and we say that X is reflexive if this embedding is a surjection. One may continue this process to obtain a chain of Banach spaces Here . We may ask whether the chain stabilizes. It turns out that only two extreme possbilities hold:

- If X is reflexive, then .
- If X is not reflexive, then .

The first part is easy. The second part amounts to the observation that closed subspace of a reflexive Banach space is reflexive (Check this post). If for some n, then is reflexive. Since X is a closed subspace of , it follows that X itself is reflexive.

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Are there any results comparing the “size” of X and X^{**} in a precise way?

What I mean is if X is infinite dimensional and X is non-reflexive, how “different” is X and X^{**}?

I guess many things can happen… I don’t have an idea about this matter. Perhaps I’ll look into this issue later.

Perhaps we can look at some examples first. Especially L^{1}, L^{\infty}

I looked into a textbook on Banach space theory, it said that Lindestrauss proved that every separable Banach space is isometrically isomorphic to X**/X for some Banach space X.