Also, if is a topological vector space whose topology is induced by (possibly unknown) norm, then is a complete topological vector space if and only if may be assigned a norm that induces on the topology and also makes into a Banach space. If is a metrizable topological vector space (where note that every norm induced topology is metrizable), then is a complete TVS if and only if it is a complete TVS, meaning that it is enough to check that every Cauchy in converges in to some point of (i.e. there is no need to consider the more general notion of arbitrary Cauchy nets).