Sorry, I made some errors on the post and I don't know how to edit it.

This is the edited version:

This is from problem 1.1 of the book "Analysis for Applied Mathematics" written by Ward Cheney.

17. Prove that in a normed linear space, if ||x+y|| = ||x|| + ||y|| then ||ax+by|| = ||ax|| + ||by|| for all nonnegative a, b

If ||x+y|| = ||x|| + ||y|| implies x = ky for some scalar k, then this problem can be solved easily. But I guess this property does not hold generally.

I cannot find a counter example and cannot prove it too. Anyone has an idea?