Is the product of two complete TVS a complete TVS? 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?
Yes the product of tow complete topological vector space is complete. The very simple example is considering $\mathbb R $ and $\mathbb R^2$
Yes the product of tow complete topological vector space is complete. The very simple example is considering $\mathbb R $ and $\mathbb R^2$