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 $\displaystyle \left \| x+y \right \| = \left \| x\right \| + \left \| y\right \|$ then $\displaystyle \left \| ax+by \right \| = \left \| ax\right \| + \left \| by\right \|$ for all nonnegative $\displaystyle a, b|$

If $\displaystyle \left \| x+y\right \| = \left \| x\right \| + \left \| y\right \|$ if and only if $\displaystyle x=ky$ for some scalar $\displaystyle k$, then this problem can be solved easily. But I guess this property is not generally hold.

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