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.

  • 1
    $\begingroup$ The $\Leftarrow$ direction is trivial: let $\lambda = \alpha = 1$. $\endgroup$ – Michael Lee May 19 '18 at 23:55


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

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.