Let $E$ be a locally convex Hausdorff topological vector space. Show that $E$ is isomorphic to a subspace of a product of normed spaces.
All I know is that, if $E$ is locally convex Hausdorff, then there is a family of separated semi-norms. The problem at this point is, I have no idea how to construct(if necessary) such a product of normed space and what this space looks like. Furthermore, I think maybe this has something to do with the family of separated semi-norms, but I don't know how to connect the separated pieces.
Any hints are welcomed.