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For a dimensional linear space $X$ with a Hamel basis $H$ the largest vector topology seems to coincide with the topology of free linear topological space over $H$ endowed with the discrete topology. 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.