The vector space operation of addition is uniformly continuous and an open map. Scalar multiplication is Cauchy continuous but in general, it is almost never uniformly continuous. Because of this, every topological vector space can be completed and is thus a dense linear subspace of a complete topological vector space.
the vector space operation of addition is uniformly continuous and an open map scalar multiplication is cauchy continuous but in general it is almost never uniformly continuous because of this every topological vector space can be completed and is thus a dense linear subspace of a complete topological vector space