Result of Turpin leads to a contradiction

[Justin Smith]

There's an celebrated result of Philippe Turpin (Philippe Turpin,
Produits tensoriels d‰??espaces vectoriels topologiques, Bulletin de la
Soci?t? Math?matique de France 110 (1982), 3‰??13.) that asserts:

If A and B are F-spaces then there's an explicit formula for an F-norm on
the algebraic tensor product A\otimes B (not the completed tensor
product). In addition, the topology induced by this F-norm is
topologically complete.

This would appear to lead to a contradiction: If A\otimes B is
topologically complete in the topology induced by the completion A\hat
{\otimes} B, then it would be a countable intersection of dense open
sets. The Baire category theorem implies that

A\otimes B = A\hat{\otimes}B, which is patently false. (Let x\in A\hat
{\otimes} B be an arbitrary element. Then x+A\otimes B is also a
countable intersection of dense open sets, so the Baire Category Theorem
implies that the intersection

(A\otimes B) \cap (x+A\otimes B) is nonempty, which implies that x is in A
\otimes B.)

Have I misunderstood Turpin's result or is it well-known to be wrong?