Definition 85.4.7. Let $R$ be a topological ring. Let $M$ and $N$ be linearly topologized $R$-modules. The tensor product of $M$ and $N$ is the (usual) tensor product $M \otimes _ R N$ endowed with the linear topology defined by declaring
to be a fundamental system of open submodules, where $M_\mu \subset M$ and $N_\nu \subset N$ run through fundamental systems of open submodules in $M$ and $N$. The completed tensor product
is the completion of the tensor product.