A linearly topologized vector space $V$ over a discrete field $F$ is a vector space equipped with a translation invariant Hausdorff topology $\Omega$ such that vector addition and scalar multiplication are (jointly) continuous.
Let $V$ and $W$ be two linearly topologized vector spaces with fundamental systems $\{N_i\}_{i\in I}$ and $\{M_i\}_{j \in J}$ of neighbourhoods of $0$ respectively. Their tensor product $V \otimes W$ can be equipped with the following topology: declare $\{N_i \otimes W + V \otimes M_j\}_{i \in I, j\in J}$ to be the fundamental system of neighbourhoods of $0$. The tensor product $V \otimes W$ together with this topology is again a linearly topologized vector space.
Question: what is the universal property of this tensor product topology?
Here (pg. 110) I found the following statement
It confuses me because the topology above is clearly not the finest (linear) topology making the tensor product map $\otimes \colon V \times W \to V \otimes W$ continuous (it is proven on pg. 109). And if I am not mistaken, any continuous linear map between two linearly topologized vector spaces is automatically uniformly continuous. Therefore the finest (linear) topology making $\otimes$ continuous should also be the finest (linear) topology making $\otimes$ uniformly continuous. What do I miss?
The map $\otimes:V\times W\to V\otimes W$ is bilinear, not linear. So continuity does not automatically imply uniform continuity for it.
That said, Lemma 24.16 is not quite correct as written, assuming "linear topology" is supposed to coincide with your definition of "linearly topologized vector space". It appears that the intended meaning of "linear topology" should additionally require that there is a neighborhood base at $0$ consisting of open sets that are additionally vector subspaces.