triangle inequality for inner product spaces

Question

Show that $\left\lVert u+v \right\rVert = \left\lVert u \right\rVert + \left\lVert v \right\rVert$ if and only if $ \left\lVert \lambda u+ \alpha v \right\rVert = \lambda \left\lVert u\right\rVert + \alpha \left\lVert v \right\rVert$ for all $ \lambda$, $\alpha \geq 0$

We know $\left\lVert u+v \right\rVert \leq \left\lVert u \right\rVert + \left\lVert v \right\rVert$ is the triangle inequality for inner product

In the problem I understand if $u$ and $v$ are multiples and all elements of $u$, $v$ are positive or $\lambda$,$\alpha =0 $ the statement is true, but I don't know how to prove it.

In my attemp I made:

If assume $\left\lVert u+v \right\rVert = \left\lVert u \right\rVert + \left\lVert v \right\rVert$ (1)

by propierty $\left\lVert ku\right\rVert = k\left\lVert u\right\rVert $, we have in the other side $\left\lVert \lambda u+ \alpha v \right\rVert = \left\lVert \lambda u\right\rVert + \left\lVert \alpha v \right\rVert$, by (1) is true, but I'm no sure is that ok

with the other implication we assume $ \left\lVert \lambda u+ \alpha v \right\rVert = \lambda \left\lVert u\right\rVert + \alpha \left\lVert v \right\rVert$ and need to prove $\left\lVert u+v \right\rVert = \left\lVert u \right\rVert + \left\lVert v \right\rVert$

I transform $\left\lVert u+v \right\rVert$ to $\left\lVert u \right\rVert^2 +$ $\left\lVert u \right\rVert^2 +$ $2 \langle\ u,v \rangle$ then I don´t know what way take to proove the second implication.

Answer 1

Hint

$$\left\lVert u+v \right\rVert = \left\lVert u \right\rVert + \left\lVert v \right\rVert \Leftrightarrow \\ \left\lVert u+v \right\rVert^2 = (\left\lVert u \right\rVert + \left\lVert v \right\rVert)^2 \Leftrightarrow \\ \langle u+v , u+v \rangle = \left\lVert u \right\rVert^2+ 2\left\lVert u \right\rVert\left\lVert v \right\rVert + \left\lVert v \right\rVert^2 \Leftrightarrow \\ 2\langle u , v \rangle = \ 2\left\lVert u \right\rVert\left\lVert v \right\rVert $$

Now do the same steps for $$\left\lVert \lambda u+ \alpha v \right\rVert = \lambda \left\lVert u\right\rVert + \alpha \left\lVert v \right\rVert$$