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
in the category of locally convex topological vector spaces and each
The name LF stands for Limit of Fréchet spaces.
closed) in (X, τf) if and only if fi- 1 (U) is open (resp.
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).
[3] Suppose that (I, ≤) is a directed set and that for all indices i ≤ j there are (continuous) morphisms in
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
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).
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
[4] (i.e. defined by x ↦ x) so that the subspace topology on Xi induced by Xj is weaker (i.e. coarser) than the original (i.e. given) topology on Xi.
In the category of locally convex topological vector spaces, the topology on the direct limit X of an injective directed inductive limit of locally convex spaces can be described by specifying that an absolutely convex subset U of X is a neighborhood of 0 if and only if U ∩ Xi is an absolutely convex neighborhood of 0 in Xi for every index i.
In the category of topological spaces, if every bonding map fij is/is a injective (resp.
surjective, bijective, homeomorphism, topological embedding, quotient map) then so is every fi : Xi → X.
[3] Direct limits in the categories of topological spaces, topological vector spaces (TVSs), and Hausdorff locally convex TVSs are "poorly behaved".
[4] For instance, the direct limit of a sequence (i.e. indexed by the natural numbers) of locally convex nuclear Fréchet spaces may fail 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.
[4] However, non-Hausdorff locally convex inductive limits do occur in natural questions of analysis.
is an embedding of TVSs onto proper vector subspaces and if the system is directed by
[1] In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces X can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if U ∩ Xn is an absolutely convex neighborhood of 0 in Xn for every n. An inductive limit in the category of locally convex TVSs of a family of bornological (resp.
[6] The strict inductive limit of a sequence of complete locally convex spaces (such as Fréchet spaces) is necessarily complete.
[8] LF spaces are distinguished and their strong duals are bornological and barrelled (a result due to Alexander Grothendieck).
If X is the strict inductive limit of an increasing sequence of Fréchet space Xn then a subset B of X is bounded in X if and only if there exists some n such that B is a bounded subset of Xn.
[9] A linear map from an LF-space X into a Fréchet space Y is continuous if and only if its graph is closed in X × Y.
[10] Every bounded linear operator from an LF-space into another TVS is continuous.
The LF-space structure is obtained by considering a sequence of compact sets
Such a sequence could be the balls of radius i centered at the origin.
The LF-space topology does not depend on the particular sequence of compact sets
is known as the space of test functions, of fundamental importance in the theory of distributions.
Also, the TVS inductive limit topology of X coincides with the topological inductive limit; that is, the direct limit of the finite dimensional spaces Xn in the category TOP and in the category TVS coincide.