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.