On the Complex Banach Conjecture

The complex conjecture of Stefan Banach states that if V is a Banach space over the complex numbers where for some n, 1 < n < dim(V) < ∞, all of its n-dimensional subspaces are isometric, then V is a Hilbert space. Mikhail Gromov proved it for n even in 1967. Here, we prove it for n = 1 mod 4.