norm induced by inner product and triangle inequality

Question

Let $\langle\cdot,\cdot\rangle$ be a scalar product on a space $X$, and let $\lVert \cdot\rVert$ denote the norm induced by this scalar product. I need to show that for $x,y\in X$, $\lVert x+y\rVert=\lVert x\rVert+\lVert y\rVert $ holds $\Longleftrightarrow$ $x$ and $y$ are linear transformations of each other, i.e. $\exists \alpha\geq 0$ with $x = \alpha y$. While one direction is clear ($\Longleftarrow$) I am struggling with $\Longrightarrow$. How can I construct such an $\alpha$?

Answer 1

Do you know that the Cauchy-Schwarz inequality $$\lvert \langle x, y \rangle \rvert \leq \lVert x \rVert \lVert y \rVert$$ is an equality if and only if $x = \alpha y$ for some $\alpha$? (If not, here's a proof.)

With that, you can prove the other direction by noting that $$\lVert x \rVert^2 +\lVert y \rVert^2 + 2\lVert x \rVert \lVert y \rVert = (\lVert x \rVert + \lVert y \rVert)^2 = \lVert x + y \rVert^2 =\langle x+y , x+y \rangle \\ = \langle x , x \rangle + \langle y , y \rangle + 2\langle x, y \rangle = \lVert x \rVert^2 + \lVert y \rVert^2 + 2 \langle x , y \rangle.$$ This implies that $\langle x , y \rangle = \lVert x \rVert \lVert y \rVert$, which by the first claim means that $x = \alpha y$ for some $\alpha$.