In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system of Fréchet spaces. This means that X is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Fréchet space. The name LF stands for Limit of Fréchet spaces. If each of the bonding maps is an embedding of TVSs then the LF-space is called a strict LF-space. This means that the subspace topology induced on by is identical to the original topology on . Some authors (e.g. Schaefer) define the term "LF-space" to mean "strict LF-space," so when reading mathematical literature, it is recommended to always check how LF-space is defined. Definition Inductive/final/direct limit topology Throughout, it is assumed that is either the category of topological spaces or some subcategory of the category of topological vector spaces (TVSs); If all objects in the category have an algebraic structure, then all morphisms are assumed to be homomorphisms for that algebraic structure. is a non-empty directed set; is a family of objects in where is a topological space for every index ; To avoid potential confusion, should not be called 's "initial topology" since the term "initial topology" already has a well-known definition. The topology is called the original topology on or 's given topology. is a set (and if objects in also have algebraic structures, then is automatically assumed to have has whatever algebraic structure is needed); is a family of maps where for each index , the map has prototype . If all objects in the category have an algebraic structure, then these maps are also assumed to be homomorphisms for that algebraic structure. If it exists, then the final topology on in , also called the colimit or inductive topology in , and denoted by or , is the finest topology on such that is an object in , and for every index , the map is a continuous morphism in . In the category of topological spaces, the final topology always exists and moreover, a subset is open (resp. closed) in if and only if is open (resp. closed) in for every index . However, the final topology may not exist in the category of Hausdorff topological spaces due to the requirement that belong to the original category (i.e. belong to