How can I prove that every inner product space is strictly convex normed space?

Question

Let we have $(X,\langle,\rangle)$ inner product space , we know that we can define a norm on $X$ by $\|x\|=\sqrt{\langle x,x\rangle}$ how can I prove that $(X,\|\cdot\|)$ is strictly convex normed space?

By using the following definition of strictly convex normed space : If $x$ does not equal to $0$ and $y$ does not equal to $0$ and $\|x+y\|=\|x\|+\|y\|$ then $y=ax$ where $a>0$

Answer 1

Let us say that we have two distinct points $x,y$ on the unit sphere. Then

$$\|\tfrac{1}{2}(x+y)\|^2=\tfrac{1}{2}\|x\|^2+\tfrac{1}{2}\|y\|^2-\tfrac{1}{4}\underbrace{\|x-y\|^2}_{>0}< 1.$$

Consequently, $\|x-y\|<2$.