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 $(X_{n},i_{nm})$ of Fréchet spaces.^[1] 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 $X n$ .^[1]^[2] 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

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

$(X, τ f)$ is an object in ${\mathcal {C}}$ , and
for every index $i$ , the map $f i : (X i, τ X i) \to (X, τ f)$ is a continuous morphism in ${\mathcal {C}}$ .

In the category of topological spaces, the final topology always exists and moreover, a subset $U \subseteq X$ is open (resp. closed) in $(X, τ f)$ if and only if $f i - 1 (U)$ is open (resp. closed) in $(X i, τ X i)$ for every index $i$ .

However, the final topology may not exist in the category of Hausdorff topological spaces due to the requirement that $(X, τ X f)$ belong to the original category (i.e. belong to the category of Hausdorff topological spaces).^[3]

Direct systems

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

f i j : X i \to X j

such that if $i = j$ then $f i j$ is the identity map on $X i$ and if $i \leq j \leq k$ then the following compatibility condition is satisfied:

f i k = f j k \circ f i j

,

where this means that the composition

X_{i}\xrightarrow {f_{i}^{j}} X_{j}\xrightarrow {f_{j}^{k}} X_{k}\;\;\;\;{\text{ is equal to }}\;\;\;\;X_{i}\xrightarrow {f_{i}^{k}} X_{k}.

If the above conditions are satisfied then the triple formed by the collections of these objects, morphisms, and the indexing set

\left(X_{\bullet },\left\{f_{i}^{j}\;:\;i,j\in I\;{\text{ and }}\;i\leq j\right\},I\right)

is known as a direct system in the category ${\mathcal {C}}$ that is directed (or indexed) by $I$ . Since the indexing set $I$ is a directed set, the direct system is said to be directed.^[4] The maps $f i j$ 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 $f_{i}^{j}$ is injective then the system is called injective.^[4]

Assumptions: In the case where the direct system is injective, it is often assumed without loss of generality that for all indices

i \leq j

, each

X i

is a vector subspace of

X j

(in particular,

X i

is identified with the range of

f_{i}^{j}

) and that the bonding map

f_{i}^{j}

is the natural inclusion

In j i : X i \to X j

(i.e. defined by $x \mapsto x$ ) so that the subspace topology on $X i$ induced by $X j$ is weaker (i.e. coarser) than the original (i.e. given) topology on $X i$ .

In this case, also take

X := \cup i \in I X i

.

The limit maps are then the natural inclusions

In i : X i \to X

. The direct limit topology on

X

is the final topology induced by these inclusion maps.

If the $X i$ 's have an algebraic structure, say addition for example, then for any $x, y \in X$ , we pick any index $i$ such that $x, y \in X i$ and then define their sum using by using the addition operator of $X i$ . That is,

x + y := x + i y

,

where $+ i$ is the addition operator of $X i$ . This sum is independent of the index $i$ that is chosen.

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 $U \cap X i$ is an absolutely convex neighborhood of $0$ in $X i$ for every index $i$ .^[4]

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 $f i : X i \to X$ .^[3]

Problem with direct limits

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. Such systems include LF-spaces.^[4] However, non-Hausdorff locally convex inductive limits do occur in natural questions of analysis.^[4]

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 $\mathbb {N}$ 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 $X i$ .^[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 $U \cap X n$ is an absolutely convex neighborhood of $0$ in $X n$ for every $n$ .

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.^[5]

LF-spaces

Every LF-space is a meager subset of itself.^[6] 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.^[7] 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.^[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 $X n$ 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 $X n$ .^[7]

A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous.^[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 $X \times Y$ .^[10] Every bounded linear operator from an LF-space into another TVS is continuous.^[11]

If $X$ is an LF-space defined by a sequence $\left(X_{i}\right)_{i=1}^{\infty }$ then the strong dual space $X_{b}^{\prime }$ of $X$ is a Fréchet space if and only if all $X i$ are normable.^[12] 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, $C_{c}^{\infty }(\mathbb {R} ^{n})$ , 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 $K_{1}\subset K_{2}\subset \ldots \subset K_{i}\subset \ldots \subset \mathbb {R} ^{n}$ 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 $C_{c}^{\infty }(K_{i})$ of infinitely differentiable functions on $\mathbb {R} ^{n}$ with compact support contained in $K_{i}$ has a natural Fréchet space structure and $C_{c}^{\infty }(\mathbb {R} ^{n})$ 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, $C_{c}^{\infty }(\mathbb {R} ^{n})$ 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 $n$ , $\mathbb {R}$ and for $m < n$ , consider X_m as a vector subspace of $X n$ via the canonical embedding $X m \to X n$ defined by $x := (x 1, ..., x m) \mapsto (x 1, ..., x m, 0, ..., 0)$ . Denote the resulting LF-space by $X$ . Since any TVS topology on $X$ makes continuous the inclusions of the X_m's into $X$ , 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 $X$ . Also, the TVS inductive limit topology of $X$ 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 $X$ is equal to the algebraic dual space of $X$ , that is the space of all real valued sequences $\mathbb {R} ^{\mathbb {N} }$ and the weak topology on $X^{\prime }$ is equal to the strong topology on $X^{\prime }$ (i.e. $X_{\sigma }^{\prime }=X_{b}^{\prime }$ ).^[13] In fact, it is the unique LC topology on $X^{\prime }$ whose topological dual space is X.

Citations

^ ^a ^b ^c Schaefer & Wolff 1999, pp. 55–61.
^ Helgason, Sigurdur (2000). Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions (Reprinted with corr. ed.). Providence, R.I: American Mathematical Society. p. 398. ISBN 0-8218-2673-5.
^ ^a ^b Dugundji 1966, pp. 420–435.
^ ^a ^b ^c ^d ^e ^f Bierstedt 1988, pp. 41–56.
^ Grothendieck 1973, pp. 130–142.
^ Narici & Beckenstein 2011, p. 435.
^ ^a ^b Schaefer & Wolff 1999, pp. 59–61.
^ Narici & Beckenstein 2011, p. 436.
^ Trèves 2006, p. 141.
^ Trèves 2006, p. 173.
^ Trèves 2006, p. 142.
^ Trèves 2006, p. 201.
^ Schaefer & Wolff 1999, p. 201.

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[FOOTNOTESchaeferWolff199955–61-1] Schaefer & Wolff 1999, pp. 55–61.

[2] Helgason, Sigurdur (2000). Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions (Reprinted with corr. ed.). Providence, R.I: American Mathematical Society. p. 398. ISBN 0-8218-2673-5.

[FOOTNOTEDugundji1966420–435-3] Dugundji 1966, pp. 420–435.

[FOOTNOTEBierstedt198841–56-4] ^ ^a ^b ^c ^d ^e ^f Bierstedt 1988, pp. 41–56.

[FOOTNOTEGrothendieck1973130–142-5] Grothendieck 1973, pp. 130–142.

[FOOTNOTENariciBeckenstein2011435-6] Narici & Beckenstein 2011, p. 435.

[FOOTNOTESchaeferWolff199959–61-7] Schaefer & Wolff 1999, pp. 59–61.

[FOOTNOTENariciBeckenstein2011436-8] Narici & Beckenstein 2011, p. 436.

[FOOTNOTETrèves2006141-9] Trèves 2006, p. 141.

[FOOTNOTETrèves2006173-10] Trèves 2006, p. 173.

[FOOTNOTETrèves2006142-11] Trèves 2006, p. 142.

[FOOTNOTETrèves2006201-12] Trèves 2006, p. 201.

[FOOTNOTESchaeferWolff1999201-13] Schaefer & Wolff 1999, p. 201.

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