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 ( X n , i n m ) {\displaystyle (X_{n},i_{nm})} of Fréchet spaces. This means that X is a direct limit of a direct system ( X n , i n m ) {\displaystyle (X_{n},i_{nm})} in the category of locally convex topological vector spaces and each X n {\displaystyle X_{n}} is a Fréchet space. The name LF stands for Limit of Fréchet spaces. If each of the bonding maps i n m {\displaystyle i_{nm}} 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.
Definition
Inductive/final/direct limit topology
Throughout, it is assumed that
C {\displaystyle {\mathcal {C}}} 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. I is a non-empty directed set; X• = ( Xi )i ∈ I is a family of objects in C {\displaystyle {\mathcal {C}}} 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 C {\displaystyle {\mathcal {C}}} also have algebraic structures, then X is automatically assumed to have 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 C {\displaystyle {\mathcal {C}}} , also called the colimit or inductive topology in C {\displaystyle {\mathcal {C}}} , and denoted by τf• or τf, is the finest topology on X such that
(X, τf) is an object in C {\displaystyle {\mathcal {C}}} , and for every index i, the map fi : (Xi, τXi) → (X, τf) is a continuous morphism in C {\displaystyle {\mathcal {C}}} . In the category of topological spaces, the final topology always exists and moreover, a subset U ⊆ X is open (resp. closed) in (X, τf) if and only if fi- 1 (U) is open (resp. closed) in (Xi, τXi) for every index i. However, the final topology may not exist in the category of Hausdorff topological spaces due to the requirement that (X, τXf) belong to the original category (i.e. belong to the category of Hausdorff topological spaces).
Direct systems
Suppose that (I, ≤) is a directed set and that for all indices i ≤ j there are (continuous) morphisms in C {\displaystyle {\mathcal {C}}}
such that if i = j then fij is the identity map on Xi and if i ≤ j ≤ k then the following compatibility condition is satisfied:
where this means that the composition
If the above conditions are satisfied then the triple formed by the collections of these objects, morphisms, and the indexing set
is known as a direct system in the category C {\displaystyle {\mathcal {C}}} that is directed (or indexed) by I. Since the indexing set I is a directed set, the direct system is said to be directed. The maps fij are called the bonding, connecting, or linking maps of the system. If the indexing set I is understood then I is often omitted from the above tuple (i.e. not written); the same is true for the bonding maps if they are understood. Consequently, one often sees written "X• is a direct system" where "X•" actually represents a triple with the bonding maps and indexing set either defined elsewhere (e.g. canonical bonding maps, such as natural inclusions) or else the bonding maps are merely assumed to exist but there is no need to assign symbols to them (e.g. the bonding maps are not needed to state a theorem).
Direct limit of a direct system For the construction of a direct limit of a general inductive system, please see the article: direct limit. Direct limits of injective systems If each of the bonding maps f i j {\displaystyle f_{i}^{j}} is injective then the system is called injective.
If the Xi's have an algebraic structure, say addition for example, then for any x, y ∈ X, we pick any index i such that x, y ∈ Xi and then define their sum using by using the addition operator of Xi. That is,
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