1
$\begingroup$

So I have this problem set from my analysis lecture and I'm having some bother. We have been asked to show that the following are equivalent:

(i) $\|x + y\| =\|x\| + \|y\|$

(ii) $\operatorname{Re} \langle x,y\rangle = \|x\| \cdot \|y\|$

(iii) $\langle x, y\rangle = \|x\|\cdot \|y\|$

(iv) $\|y\| x = \|x\| y$

So far I have assumed (i) to be true and proved (i) to (ii) and (ii) to (iii).

However, I am stumped as to how to show (iii) to (iv) and then to show (iv) to (i). I have been randomly using different inequalities but to no avail. Any hints or answers greatly appreciated.

  • 0
    To show that $(iii)\Rightarrow (iv)$ it is enough to prove it when $\|x\|=\|y\|=1$, by homogeneity. It is even enough to prove it for $2$-dimensional spaces, since all the action takes place in $\text{span}(x, y)$. – 2017-02-10
  • 1
    Do you know the corollary to Cauchy Schwarz, that $\langle x, y \rangle = \|x\| \|y\|$ holds iff $x$ and $y$ are linearly dependent? – 2017-02-10

0 Answers 0