Question

# if x, y belong to R^p, than is it true that the relation norm (x+y) = norm (x) + norm (y) holds if and only if x = cy or y = cx with c>0

Vectors and spaces

if x, y belong to $$R^p$$, than is it true that the relation norm $$(x+y) = norm (x) + norm (y)$$ holds if and only if $$x = cy\ or\ y = cx\ with\ c>0$$

This is not true. For example, if $$y=0$$, and $$\displaystyle{x}≠{0}$$ is some non-trivial vector, then $$\displaystyle{\left|{\left|{x}+{y}\right|}\right|}={\left|{\left|{x}+{0}\right|}\right|}={\left|{\left|{x}\right|}\right|}={\left|{\left|{x}\right|}\right|}+{\left|{\left|{y}\right|}\right|}\rbrace={0}$$
Now, if $$x=cy$$, then $$x=0$$, which is impossible. So suppose that there exists $$c>0$$ such that $$y=cx$$. However, $$||cx||=|c||x||>0$$, so $$\displaystyle{c}{x}≠{0}$$, but $$y=0$$. Therefore, there exists no $$c>0$$ such that $$x=cy\ or\ y=cx$$.