This is a problem 1a from section V in Reed & Simon's book on functional analysis. It states:
1a. Prove that a locally convex space has a topology given by a single norm if the topology is generated by finitely many seminorms
I have seen a few similar questions on here but they make the additional assumption that the space is Hausdorff or use the second part of this problem which states
1b. Prove that a locally convex space has a topology generated by a single norm if and only if 0 has a bounded neighborhood.
I would like to prove the problem without assuming the Hausdorff property and without using the (1b).
Let $\{\rho_i\}$ be a collection of seminorms that generate the topology where $i = 1, \ldots, n$. I am thinking of creating a norm using a linear combination of the $\{\rho_i\}$, or perhaps a directed family of them. However, I am not sure how any linear combination of the $\{\rho_i\}$ will necessarily satisfy the positive definite condition of a norm. Any advice on how to proceed?