Given a vector space and a family of semi-norms defined on it, I have to prove that it becomes a locally convex topological vector space.
To prove that it becomes a locally convex space I have to show that every open set containing zero contains a convex balanced absorbing open set.
But I can show only that such a convex balanced absorbing neighbourhood exist. How to show that such an open set exist ?
If you start with a family $\{ |\cdot|_{\alpha} \}_{\alpha \in \Lambda}$ of seminorms, then you can define the topology on the vector space $V$ as the weakest topology for which each seminorm $|\cdot|_{\alpha}$ is continuous. Equivalently, $B$ is a neighborhood of $0$ iff there exists a finite subset $\{ \alpha_{1},\cdots,\alpha_{n}\} \subseteq \Lambda$ and positive radii $\{ r_{1},\cdots,r_{n}\}$ such that $$ \bigcap_{j=1}^{n}\{ x : |x|_{\alpha_{j}} < r_{j} \} \subseteq B. $$ This is a locally convex topology on $V$ with a base of neighborhoods defined as such intersections. Every set $\{ x : |x|_{\alpha} < r \}$ is balanced, convex and absorbing, and is open in this topology. And every open neighborhood $B$ of $0$ contains such a finite intersection, by the definition of the topology.