Let $(E_1,\tau_1)$ be a locally convex space and let $(E_2,\tau_2)$ be a complete locally convex space. Suppose that $T:(E_1,\tau_1) \longrightarrow (E_2,\tau_2)$ is a topological isomorphism (that is, $T$ is linear, bijective, continuous and its inverse $T^{-1}$ is continuous, too).
Is it true that the space $(E_1,\tau_1)$ is necessarily complete?
Thanks for any hints/comments.
Yes, of course (a property which is not stable under isomorphisms of a category is typically not very useful). The proof is as it should be: If $(x_i)_{i\in I}$ is a Cauchy net in $E_1$ then $T(x_i)$ is a Cauchy net in $E_2$ because linear continuous maps are uniformly continuous. If $y$ is a limit of $T(x_i)$ then $x_i$ converges to $T^{-1}(y)$ because of the continuity of $T^{-1}$.