I need to show that if $p_i$ is a seminorm, then $\|\cdot\|$ is a norm s.t $\|x\|:=\max p_i(x)$, for every $x\in V$, where $V$ is a locally convex topological vector space.
I have shown that homogeneity and triangle inequality hold true, but I am unable to show that $||x|| = 0 \iff x=0$.
If $x=0$ then $\|x\| = 0$. But when $\|x\|=0$ $\Rightarrow max(p_i(x))=0$. For this I am only able to say since $p_i(x)$ is non negative that $p_i(x)=0$. And here is where I am stuck. As far as I know it is not true in seminorms that this implies $x=0$. What is it that I am missing?
Some assumptions are missing in your statement. If $(p_i)$ is just some family of semi-norm then the statement is obviously false. (Take a single semi-norm which is not a norm). So you have to assume that $p_i$'s generate the topology of $V$. This means that sets of the form $\{x:p_{i_1}(x) <r_1,p_{i_2}(x) <r_2,..,p_{i_N}(x) <r_N\}$ ( $N$, $i_j$'s arbitrary and $r_j$'s $>0$) form base for the neighborhoods of $0$. Under this assumption it follows that $p_i(x)=0$ for all $i$ implies that $x$ belongs to every neighborhood of $0$. Assuming that ths space is Hausdorff this implies $x=0$.