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
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asked 2021-06-14

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 \(x = cy\ or\ y = cx\ with\ c>0\)

Answers (1)

2021-06-15

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 \(x=cy\ or\ y=cx\).

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