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](a230x.png) | (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
![∑m p
ak∥ck(0)x0 + ⋅⋅⋅+ ck(n)xn∥ = 0 for all elements xi
k=0](a231x.png) | (2) |
on a Banach space X.
|