Suppose $X$ is a Hausdorff locally convex space, then we know its topology can be generated by a family of seminorms. Let us call $P$ to this defining family of seminorms on $X$. I need to show that X is normable iff $P$ is equivalent to a finite subfamily $P_0 \subset P$.
To prove it, we of course now that $P_0$ is dominated by $P$ (in the topological sense), i.e. that every open in $P_0$ is open in $P$. So we can reformulate our question by stating that $X$ is normable iff $P$ is dominated by a family subfamily of seminorms.
Using Kolmogorov's criterion I have shown the right to left implication. That is, we assume $P$ can be generated by a finite family of seminorms, and then we can prove that a convex bounded neighborhood of $0$ exists. We can pick and fix any $\epsilon>0$ and let $A$ be the convex hull of $\cup_{i=1}^n p_{\epsilon,i}$ (the union of the $\epsilon$ balls of the $n$ seminorms that generate the topology on $X$. Then it is by construction convex and around 0 and we just need to check it is bounded. I have done this.
Can you help me with the reverse implication? What is some clearer property that the seminorms must have because the space is normable? Maybe I can use this to prove that $P$ is dominated by $P_0$. Thank you.
Suppose $(X,\tau)$ is normable, let $p$ be the norm on $X$. Suppose $\tau$ is induced by a family of seminorms $(p_i)_{i\in I}$. Let $B$ be the open unit ball w.r.t $p$.
Basic neighborhoods of zero are described as $$ W(\varepsilon, i_1,\ldots, i_n) := \{x\in X \mid \max _{1\leqslant k\leqslant n}|p_{i_k}(x)| \leqslant \varepsilon\} \quad (i_k\in I,\ n\in\mathbb N, \quad\varepsilon>0) $$ Pick a basic NH of zero contained in $B$ and show that those seminorms induce $\tau$.