LF-space

Template:Short description 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 $(X_n,i_{nm})$ of Fréchet spaces.Template:Sfn This means that X is a direct limit of a direct system $(X_n,i_{nm})$ in the category of locally convex topological vector spaces and each $X_n$ is a Fréchet space. The name LF stands for Limit of Fréchet spaces.

If each of the bonding maps $i_{nm}$ is an embedding of TVSs then the LF-space is called a strict LF-space. This means that the subspace topology induced on X_n by X_n+1 is identical to the original topology on X_n.Template:Sfn 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

• ${𝒞}$ 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.
• Template:Mvar is a non-empty directed set;
• X_• = ( X_i )_{i ∈ I} is a family of objects in ${𝒞}$ where (X_i, τ_Xi) is a topological space for every index Template:Mvar;
  • To avoid potential confusion, τ_Xi should not be called X_i's "initial topology" since the term "initial topology" already has a well-known definition. The topology τ_Xi is called the original topology on X_i or X_i's given topology.
• Template:Mvar is a set (and if objects in ${𝒞}$ also have algebraic structures, then Template:Mvar is automatically assumed to have whatever algebraic structure is needed);
• f_• = ( f_i )_{i ∈ I} is a family of maps where for each index Template:Mvar, the map has prototype f_i : (X_i, τ_Xi) → 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 Template:Mvar in ${𝒞}$, also called the colimit or inductive topology in ${𝒞}$, and denoted by τ_f• or τ_f, is the finest topology on Template:Mvar such that

1. (X, τ_f) is an object in ${𝒞}$, and
2. for every index Template:Mvar, the map f_i : (X_i, τ_Xi) → (X, τ_f) is a continuous morphism in ${𝒞}$.

In the category of topological spaces, the final topology always exists and moreover, a subset U ⊆ X is open (resp. closed) in (X, τ_f) if and only if f_i^- 1 (U) is open (resp. closed) in (X_i, τ_Xi) for every index Template:Mvar.

However, the final topology may not exist in the category of Hausdorff topological spaces due to the requirement that (X, τ_Xf) belong to the original category (i.e. belong to the category of Hausdorff topological spaces).Template:Sfn

Direct systems

Suppose that (I, ≤) is a directed set and that for all indices i ≤ j there are (continuous) morphisms in ${𝒞}$

such that if i = j then f_i^j is the identity map on X_i 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 ${𝒞}$ that is directed (or indexed) by I. Since the indexing set Template:Mvar is a directed set, the direct system is said to be directed.Template:Sfn The maps f_i^j are called the bonding, connecting, or linking maps of the system.

If the indexing set Template:Mvar is understood then Template:Mvar 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 $f_i^j$ is injective then the system is called injective.Template:Sfn

If the X_i's have an algebraic structure, say addition for example, then for any x, y ∈ X, we pick any index i such that x, y ∈ X_i and then define their sum using by using the addition operator of X_i. That is, Template:Block indent where +_i is the addition operator of X_i. This sum is independent of the index Template:Mvar that is chosen.

In the category of locally convex topological vector spaces, the topology on the direct limit Template:Mvar of an injective directed inductive limit of locally convex spaces can be described by specifying that an absolutely convex subset U of Template:Mvar is a neighborhood of 0 if and only if U ∩ X_i is an absolutely convex neighborhood of 0 in X_i for every index Template:Mvar.Template:Sfn

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 f_i^j is/is a injective (resp. surjective, bijective, homeomorphism, topological embedding, quotient map) then so is every f_i : X_i → X.Template:Sfn

Problem with direct limits

Direct limits in the categories of topological spaces, topological vector spaces (TVSs), and Hausdorff locally convex TVSs are "poorly behaved".Template:Sfn 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. Such systems include LF-spaces.Template:Sfn However, non-Hausdorff locally convex inductive limits do occur in natural questions of analysis.Template:Sfn

Strict inductive limit

If each of the bonding maps $f_i^j$ is an embedding of TVSs onto proper vector subspaces and if the system is directed by $\mathbb{\mathbb{N}}$ 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 X_i is a vector subspace of X_i+1 and that the subspace topology induced on X_i by X_i+1 is identical to the original topology on X_i.Template:Sfn

In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces Template:Mvar can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if U ∩ X_n is an absolutely convex neighborhood of 0 in X_n for every Template:Mvar.

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.Template:Sfn

LF-spaces

Every LF-space is a meager subset of itself.Template:Sfn 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.Template:Sfn 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.Template:Sfn LF spaces are distinguished and their strong duals are bornological and barrelled (a result due to Alexander Grothendieck).

If Template:Mvar is the strict inductive limit of an increasing sequence of Fréchet space X_n then a subset Template:Mvar of Template:Mvar is bounded in Template:Mvar if and only if there exists some Template:Mvar such that Template:Mvar is a bounded subset of X_n.Template:Sfn

A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous.Template:Sfn A linear map from an LF-space Template:Mvar into a Fréchet space Template:Mvar is continuous if and only if its graph is closed in X × Y.Template:Sfn Every bounded linear operator from an LF-space into another TVS is continuous.Template:Sfn

If Template:Mvar is an LF-space defined by a sequence ${\left(X_i\right)}_{i=1}^{\mathrm{\infty }}$ then the strong dual space $X_b^{\prime }$ of Template:Mvar is a Fréchet space if and only if all X_i are normable.Template:Sfn 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, $C_c^{\mathrm{\infty }}({\mathbb{ℝ}}^n)$, the space of all infinitely differentiable functions on ${\mathbb{ℝ}}^n$ with compact support. The LF-space structure is obtained by considering a sequence of compact sets $K_1\subset K_2\subset \dots \subset K_i\subset \dots \subset {\mathbb{ℝ}}^n$ with $\bigcup _i{K}_i={\mathbb{ℝ}}^n$ and for all i, $K_i$ is a subset of the interior of $K_{i+1}$. Such a sequence could be the balls of radius i centered at the origin. The space $C_c^{\mathrm{\infty }}(K_i)$ of infinitely differentiable functions on ${\mathbb{ℝ}}^n$ with compact support contained in $K_i$ has a natural Fréchet space structure and $C_c^{\mathrm{\infty }}({\mathbb{ℝ}}^n)$ inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets $K_i$.

With this LF-space structure, $C_c^{\mathrm{\infty }}({\mathbb{ℝ}}^n)$ 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 Template:Mvar, X_n := $\mathbb{ℝ}$ⁿ and for m < n, consider X_m as a vector subspace of X_n via the canonical embedding X_m → X_n defined by x := (x₁, ..., x_m) ↦ (x₁, ..., x_m, 0, ..., 0). Denote the resulting LF-space by Template:Mvar. Since any TVS topology on Template:Mvar makes continuous the inclusions of the X_m's into Template:Mvar, the latter space has the maximum among all TVS topologies on an $\mathbb{ℝ}$-vector space with countable Hamel dimension. It is a LC topology, associated with the family of all seminorms on Template:Mvar. Also, the TVS inductive limit topology of Template:Mvar coincides with the topological inductive limit; that is, the direct limit of the finite dimensional spaces X_n in the category TOP and in the category TVS coincide. The continuous dual space $X^{\prime }$ of Template:Mvar is equal to the algebraic dual space of Template:Mvar, that is the space of all real valued sequences ${\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}$ and the weak topology on $X^{\prime }$ is equal to the strong topology on $X^{\prime }$ (i.e. $X_{\sigma }^{\prime }=X_b^{\prime }$).Template:Sfn In fact, it is the unique LC topology on $X^{\prime }$ whose topological dual space is X.

See also

• DF-space
• Direct limit
• Final topology
• F-space
• LB-space

Bibliography

• Template:Adasch Topological Vector Spaces
• Template:Bierstedt An Introduction to Locally Convex Inductive Limits
• Template:Bourbaki Topological Vector Spaces Part 1 Chapters 1–5
• Template:Dugundji Topology
• Template:Edwards Functional Analysis Theory and Applications
• Template:Grothendieck Topological Vector Spaces
• Template:Horváth Topological Vector Spaces and Distributions Volume 1 1966
• Template:Jarchow Locally Convex Spaces
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Köthe Topological Vector Spaces I
• Template:Köthe Topological Vector Spaces II
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Robertson Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Schechter Handbook of Analysis and Its Foundations
• Template:Swartz An Introduction to Functional Analysis
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