The weak topology on a Banach Space $E$ is defined to have sub-base consisting of open balls of the form $B_\alpha(x,r) = \lbrace y \in E : \vert \alpha(x-y) \vert < r\rbrace $ for each $x \in E$ and $r > 0$ and $\alpha \in E^*$ . This defines a uniform structure which is also a topological group.
Is there any reason we would want to define a similar topology, but only using a proper subset of the seminorms induced by dual space elements? That is the topology on $E$ with sub-base consisting of balls of the form $B_\beta(x,r) = \lbrace y \in E : \vert \beta(x-y)\vert < r\rbrace $ for each $x \in E$ and $r > 0$ and $\beta \in H \subset E^*$. Here $H$ is a proper subset of $E$. I think we can exclude the case where $E = F^*$ and $H = F$.
Looking at a silly example where $H$ contains only one element that returns 1 on a given Hamel basis element $e_0$, and 0 on all other elements, gives a topology containing only $E$, $E -$ span{$e_0$}, span{$e_0$}, and $\varnothing$. Clearly vector space operations are not continuous in this topology. Nevertheless it is still a uniform space. But do uniform spaces of this form come up anywhere?