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Let ## v_{1}, v_{2} ... v_{n} ## be vectors in a complex inner product space ##V##. Suppose that ## | v_{1} + v_{2} +...+ v_{n}| = |v_{1}| + |v_{2}| +...+ |v_{n}| ##. Then is it necessarily the case that ##v_{1},v_{2}...v_{n}## are all non-negative scalar multiples of some nonzero vector ##v## ?

It seems much easier to prove for a real inner product space, but I'm not even sure if this is true for complex inner product spaces. I'm trying an induction, first using the simple case where ## n = 2 ## but I can't seem to prove that the scaling factor must be a non-negative real number.

BiP