In 1935, Jordan and von Neumann proved that any Banach space which satisfies the parallelogram law ‖x + y‖2 + ‖x - y‖2 = 2(‖x‖2 + ‖y‖2) for all elements x and y must be a Hubert space. Subsequent authors have found norm conditions weaker than (1) which require a Banach space to be a Hubert space. Notable examples include the results of Day, Lorch, Senechalle and Carlsson. In this paper, we study nontrivial linear identities such as (Formula Presented) for all elements xi on a Banach space X.
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