Seminorm $p$ induces norm on $V/\text{ker}(p)$

Question

I am stuck in the following problem:

Let $V$ be a vector space and $p$ seminorm in $V$. Show that $p$ induces norm on quotient space $V/\text{ker}(p)$ defined as $\lVert v+\text{ker}(p)\rVert=p(v)$.

It's easy to show the axioms of norm, except the case $$\lVert v+\text{ker}(p)\rVert=p(v)=0 \Rightarrow v+\text{ker}(p)=0\in V/\text{ker}(p) .$$ So we should show that $p(v)=0$ implies $v=0$. But because $p$ is seminorm, there can be non-zero vectors which norms are zero. I don't know how to proceed, any help or hints would be appreciated.

Answer 1

We denote by $ \hat 0$ the null in $V/\text{ker}(p)$. Then:

$||v +\ker(p)||=0 \iff v \in \ker(p) \iff v+ \ker(p)= \hat 0$