Any linear map, , between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) is sequentially continuous in the sense of the product topology, then the map is continuous and its graph, , is necessarily closed. Conversely, if is such a linear map with, in place of (1a), the graph of is (1b) known to be closed in the Cartesian product space , then is continuous and therefore necessarily sequentially continuous.