Metrizable topological vector space

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS. A pseudometric on a set is a map satisfying the following properties: Symmetry: ; Subadditivity: A pseudometric is called a metric if it satisfies: Identity of indiscernibles: for all if then Ultrapseudometric A pseudometric on is called a ultrapseudometric or a strong pseudometric if it satisfies: Strong/Ultrametric triangle inequality: Pseudometric space A pseudometric space is a pair consisting of a set and a pseudometric on such that 's topology is identical to the topology on induced by We call a pseudometric space a metric space (resp. ultrapseudometric space) when is a metric (resp. ultrapseudometric). If is a pseudometric on a set then collection of open balls: as ranges over and ranges over the positive real numbers, forms a basis for a topology on that is called the -topology or the pseudometric topology on induced by If is a pseudometric space and is treated as a topological space, then unless indicated otherwise, it should be assumed that is endowed with the topology induced by Pseudometrizable space A topological space is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) on such that is equal to the topology induced by An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators. A topology on a real or complex vector space is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes into a topological vector space). Every topological vector space (TVS) is an additive commutative topological group but not all group topologies on are vector topologies.