I know that the largest vector topology on countable dimensional vector space is sequential (i.e. every sequentially closed set is closed). Does it keep for the arbitrary vector space?


In countable dimensional case I can describe structure of largest vector topology (it coincides with the largest locally convex topology), but I don't know anything about uncountable dimensional case.


(Largest vector topology on a vector space is the supremum of set of all vector topologies on it.)



Sequentiality of largest vector topology
For a linear space $X$ with a Hamel basis $H$ the largest vector topology seems to coincide with the topology of free linear topological space over the discrete space $H$. If this is true, then we can apply known results on the sequentiality of free linear topological spaces, see e.g. <http://arxiv.org/pdf/1602.04857>
Theorem 10.12.4 of this paper implies that for a discrete space $X$ the free linear topological space $Lin(X)$ over $X$ is sequential iff $Lin(X)$ is a $k$-space iff $X$ is countable. 


