I want to prove that: If $E$ and $F$ are two topological vector spaces Hausdorff, then there is a canonical isomorphism between $\hat{E} \times \hat{F}$ and $(E\times F)\hat{}$.
For this I thought about using Theorem 5.2 from, that is, by Theorem 5.2 of [1] there are completes Hausdorff topological vector spaces $ \hat{E} $ and $ \hat{F} $ such that the canonical injections $ i_E: E \longrightarrow \hat{E} $ and $ i_F: F \longrightarrow \hat{F} $ are isomorphisms.
On the other hand, since $ E $ and $ F $ are Hausdorff's topological vector spaces, then $ H: = E \times F $ is Hausdorff's topological vector space as well, with $ E \times F $ provided with the product topology. Thus, again by Theorem 5.2 of [1], there is a complete Hausdoff topological vector space $ \hat {H} = (E \times F) \hat{} $ such that the natural injection $ i_H: H \longrightarrow \hat{H} $ is isomorphism.
But from then on I was unable to proceed to prove what I desire.
See Theorem 5.2 in the page 41 of [1].
[1] Trèves, F.Topological vector spaces, distributions and kernels. Unabridged republication of the 1967 original.Dover Publications, Inc., Mineola, NY, 2006.