I'm reading about Topological Vector Spaces (using Treves's book) and I failed to answer the question on title. I'll be more precise:
1) We say that a TVS $E$ is complete if every Cauchy filter in $E$ is convergent.
2) Given $E_1$ and $E_2$ two TVS, we consider $E=E_1 \times E_2$ with the product topology. With this topology and the natural structure of vector space, $E$ is a TVS.
In the Bourbaki's book is proved that it's true when we use the notion of uniform structure. But looking at page 140, ex. 5 of Topological Vector Spaces and Distributions (by John Horvath, where he uses just the definition 1), he states just the converse.
So, is it true that $E$ is complete when $E_1$ and $E_2$ are complete? If it's not true, are there some example?