On the completeness of topologically isomorphic spaces

Let $ (E_1, tau_1) $ to be a locally convex space and let $ (E_2, tau_2) $ to be a complete space locally convex. Assume that $ T: (E_1, tau_1) longrightarrow (E_2, tau_2) $ is a topological isomorphism (i.e. $ T $ is linear, bijective, continuous and its inverse $ T ^ {- 1} $ is continuous too).

Is it true that space $ (E_1, tau_1) $ is necessarily complete?

Thank you for all the advice / comments.