Problem: Prove that if V is an inner product space, then $$\|x+y\|=\|x\|+\| y\|$$ if and only if one of the vectors $x$ or $y$ is a nonnegative scalar multiple of the other. Generalize it to the case of $n$ vectors.
Proof: $$\langle x+y,x+y\rangle=\langle x,x\rangle +2\|x\|\cdot\|y\|+\langle y,y\rangle $$
then, using linearity of the inner product I get
$$ \langle x,x\rangle +\langle y,y\rangle+\langle x,y\rangle+\langle y,x\rangle=\langle x,x\rangle +2\|x\|\cdot\|y\|+\langle y,y\rangle $$
After all the cancellation,
$$ \mathrm{Re}\langle x,y\rangle=\|x\|\cdot\|y|\ $$
By Cauchy Schwarz, we prove that equation is equal.
Question is how am I supposed to generalize to $n$ vectors?
Best Answer
Note that, if $\|x + y + x\| = \|x\| + \|y\| + \|z\|$, then $$\|x\| + \|y\| + \|z\| = \|x + y + x\| \le \|x + y\| + \|z\| \le \|x\| + \|y\| + \|z\|.$$ From this, we can conclude that $$\|x + y\| + \|z\| = \|x\| + \|y\| + \|z\| \implies \|x + y\| = \|x\| + \|y\|.$$ Similarly, we can deduce $\|y + z\| = \|y\| + \|z\|$ and $\|x + z\| = \|x\| + \|z\|$. Thus, $x, y, z$ are each positive multiples of each other.
Now extend via induction.