In 1935, Jordan and von
Neumann proved that any Banach space which satisfies the parallelogram law
(1)
must be a Hilbert space.
Subsequent authors have found norm conditions weaker than (1) which require a
Banach space to be a Hilbert space. Notable examples include the results of Day,
Lorch, Senechalle and Carlsson.
In this paper, we study nontrivial linear identities such as
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