Inner Product Norm is Norm

Theorem
Let $V$ be an inner product space over a subfield $\Bbb F$ of $\C$.

Let $\left\Vert{\cdot}\right\Vert$ denote the inner product norm on $V$.

Then $\left\Vert{\cdot}\right\Vert$ is a norm on $V$.

Proof
Take advantage of Properties of Semi-inner Product, the only thing that really remains is $\|x\|=0\implies x =0$.