LF-space

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 Xn by Xn+1 is identical to the original topology on Xn. 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. Final topology Category (mathematics) Throughout, it is assumed that is either the or some subcategory of the 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. I is a non-empty directed set; X• = ( Xi )i ∈ I is a family of objects in where Xi, τXi is a topological space for every index i; To avoid potential confusion, τXi should not be called Xi's "initial topology" since the term "initial topology" already has a well-known definition. The topology τXi is called the original topology on Xi or Xi's given topology. X is a set (and if objects in also have algebraic structures, then X is automatically assumed to have has whatever algebraic structure is needed); f• = ( fi )i ∈ I is a family of maps where for each index i, the map has prototype fi : Xi, τXi → X. 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 X in , also called the colimit or inductive topology in , and denoted by τf• or τf, is the finest topology on X such that X, τf is an object in , and for every index i, the map fi : Xi, τXi → X, τf is a continuous morphism in .