Is it possible for a Banach Space to be isomorphic to its double dual

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SUMMARY

R. C. James demonstrated in 1950 that a specific Banach space can be isometrically isomorphic to its bidual while not being reflexive. This indicates that the canonical injection from the Banach space to its double dual is not surjective, confirming that it does not represent an isomorphism. The proof of this construction is detailed in the book "Classical Banach Spaces" by Lindenstrauss and Tzafriri.

PREREQUISITES
  • Understanding of Banach spaces and their properties
  • Familiarity with dual spaces and biduals
  • Knowledge of isometric isomorphisms
  • Basic concepts of reflexivity in functional analysis
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  • Study the construction of Banach spaces in R. C. James's work
  • Explore the concepts of duality and reflexivity in functional analysis
  • Read "Classical Banach Spaces" by Lindenstrauss and Tzafriri for detailed proofs
  • Investigate examples of non-reflexive Banach spaces
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But the cannonical injection to not be an isomorphism?
 
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DeadWolfe said:
But the cannonical injection to not be an isomorphism?


In 1950, R. C. James constructed a Banach space which is isometrically isomorphic to its bidual but not reflexive. The natural injection is thus not surjective and therefore no isomorphism.

Reportedly, a proof can be found in Lindenstrauss/Tzafriri "Classical Banach Spaces".
 

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