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 formula_1 of Fréchet spaces. 
This means that "X" is a direct limit of a direct system formula_1 in the category of locally convex topological vector spaces and each formula_3 is a Fréchet space. 
If each of the bonding maps formula_4 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 
If it exists, then the final topology on in formula_5, also called the colimit or inductive topology in formula_5, and denoted by or , is the finest topology on such that 
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 the category of Hausdorff topological spaces).
Direct systems.
Suppose that is a directed set and that for all indices there are (continuous) morphisms in formula_5
such that if then is the identity map on and if 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 formula_5 that is directed (or indexed) by . Since the indexing set is a directed set, the direct system is said to be directed. 
The maps are called the bonding, connecting, or linking maps of the system. 
If the indexing set is understood then 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 " is a direct system" where "" 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 formula_14 is injective then the system is called injective. 
If the 's have an algebraic structure, say addition for example, then for any , we pick any index such that and then define their sum using by using the addition operator of . That is, 
where is the addition operator of . This sum is independent of the index that is chosen. 
In the category of locally convex topological vector spaces, the topology on the direct limit of an injective directed inductive limit of locally convex spaces can be described by specifying that an absolutely convex subset of is a neighborhood of if and only if is an absolutely convex neighborhood of in for every index . 
Direct limits in Top
Direct limits of directed direct systems always exist in the categories of sets, topological spaces, groups, and locally convex TVSs. 
In the category of topological spaces, if every bonding map is/is a injective (resp. surjective, bijective, homeomorphism, topological embedding, quotient map) then so is every . 
Problem with direct limits.
Direct limits in the categories of topological spaces, topological vector spaces (TVSs), and Hausdorff locally convex TVSs are "poorly behaved". 
For instance, the direct limit of a sequence (i.e. indexed by the natural numbers) of locally convex nuclear Fréchet spaces may to be Hausdorff (in which case the direct limit does not exist in the category of Hausdorff TVSs). 
For this reason, only certain "well-behaved" direct systems are usually studied in functional analysis. Such systems include "LF"-spaces. 
However, non-Hausdorff locally convex inductive limits do occur in natural questions of analysis.
Strict inductive limit.
If each of the bonding maps formula_14 is an embedding of TVSs onto proper vector subspaces and if the system is directed by with its natural ordering, then the resulting limit is called a strict (countable) direct limit. In such a situation we may assume without loss of generality that each is a vector subspace of and that the subspace topology induced on by is identical to the original topology on . 
In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces can be described by specifying that an absolutely convex subset is a neighborhood of if and only if is an absolutely convex neighborhood of in for every .
Properties.
An inductive limit in the category of locally convex TVSs of a family of bornological (resp. barrelled, quasi-barrelled) spaces has this same property. 
LF-spaces.
Every LF-space is a meager subset of itself.
The strict inductive limit of a sequence of complete locally convex spaces (such as Fréchet spaces) is necessarily complete. 
In particular, every LF-space is complete. 
Every "LF"-space is barrelled and bornological, which together with completeness implies that every LF-space is ultrabornological. 
An LF-space that is the inductive limit of a countable sequence of separable spaces is separable. 
LF spaces are distinguished and their strong duals are bornological and barrelled (a result due to Alexander Grothendieck).
If is the strict inductive limit of an increasing sequence of Fréchet space then a subset of is bounded in if and only if there exists some such that is a bounded subset of . 
A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous. 
A linear map from an LF-space into a Fréchet space is continuous if and only if its graph is closed in .
Every bounded linear operator from an LF-space into another TVS is continuous.
If is an LF-space defined by a sequence formula_16 then the strong dual space formula_17 of is a Fréchet space if and only if all are normable. 
Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an LB-space.
Examples.
Space of smooth compactly supported functions.
A typical example of an "LF"-space is, formula_18, the space of all infinitely differentiable functions on formula_19 with compact support. The "LF"-space structure is obtained by considering a sequence of compact sets formula_20 with formula_21 and for all i, formula_22 is a subset of the interior of formula_23. Such a sequence could be the balls of radius "i" centered at the origin. The space formula_24 of infinitely differentiable functions on formula_19 with compact support contained in formula_22 has a natural Fréchet space structure and formula_18 inherits its "LF"-space structure as described above. The "LF"-space topology does not depend on the particular sequence of compact sets formula_22.
With this "LF"-space structure, formula_18 is known as the space of test functions, of fundamental importance in the theory of distributions.
Direct limit of finite-dimensional spaces.
Suppose that for every positive integer , and for , consider "X""m" as a vector subspace of via the canonical embedding defined by . 
Denote the resulting LF-space by . 
The continuous dual space formula_30 of is equal to the algebraic dual space of and the weak topology on formula_30 is equal to the strong topology on formula_30 (i.e. formula_33). 
Furthermore, the canonical map of into the continuous dual space of formula_34 is surjective.