EDB — 0ZY

Exercises

1. [0ZY] Note that if $v,w\in X$ are linearly dependent and have the same direction (i.e. you can write $v=𝜆w$ or $w=𝜆v$, for $𝜆\ge 0$), then you have

   $$\Vert v+w\Vert =\Vert v\Vert +\Vert w\Vert .$$

   In particular, a norm is not a strictly convex function, because

   $$\Vert \frac{v}{2}+\frac{v}{2}\Vert =\frac{1}{2}\Vert v\Vert +\frac{1}{2}\Vert v\Vert .$$