LF-space

{{Main|Final topology}}

{{See also|Category (mathematics)}}

Throughout, it is assumed that

• $\backslash mathcal\{C\}$ 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.
• {{mvar|I}} is a non-empty directed set;
• {{math|1=X_• = ( X_i )_{i ∈ I}}} is a family of objects in $\backslash mathcal\{C\}$ where {{math|{{large|(}}X_i, τ_Xi{{large|)}}}} is a topological space for every index {{mvar|i}};
• To avoid potential confusion, {{math|τ_Xi}} should not be called {{math|X_i}}'s "initial topology" since the term "initial topology" already has a well-known definition. The topology {{math|τ_Xi}} is called the original topology on {{math|X_i}} or {{math|X_i}}'s given topology.
• {{mvar|X}} is a set (and if objects in $\backslash mathcal\{C\}$ also have algebraic structures, then {{mvar|X}} is automatically assumed to have whatever algebraic structure is needed);
• {{math|1=f_• = ( f_i )_{i ∈ I}}} is a family of maps where for each index {{mvar|i}}, the map has prototype {{math|f_i : {{large|(}}X_i, τ_Xi{{large|)}} → 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 {{mvar|X}} in $\backslash mathcal\{C\}$, also called the colimit or inductive topology in $\backslash mathcal\{C\}$, and denoted by {{math|τ_f•}} or {{math|τ_f}}, is the finest topology on {{mvar|X}} such that

1. {{math|{{large|(}}X, τ_f{{large|)}}}} is an object in $\backslash mathcal\{C\}$, and
2. for every index {{mvar|i}}, the map {{math|f_i : {{large|(}}X_i, τ_Xi{{large|)}} → {{large|(}}X, τ_f{{large|)}}}} is a continuous morphism in $\backslash mathcal\{C\}$.

In the category of topological spaces, the final topology always exists and moreover, a subset {{math|U ⊆ X}} is open (resp. closed) in {{math|{{large|(}}X, τ_f{{large|)}}}} if and only if {{math|f_i^- 1 (U)}} is open (resp. closed) in {{math|{{large|(}}X_i, τ_Xi{{large|)}}}} for every index {{mvar|i}}.

However, the final topology may not exist in the category of Hausdorff topological spaces due to the requirement that {{math|{{large|(}}X, τ_Xf{{large|)}}}} belong to the original category (i.e. belong to the category of Hausdorff topological spaces).{{sfn | Dugundji | 1966 | pp=420-435}}

{{Main|Direct limit}}

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

{{block indent|1= {{math|f_i^j : X_i → X_j}} }}

such that if {{math|1=i = j}} then {{math|f_i^j}} is the identity map on {{math|X_i}} and if {{math|i ≤ j ≤ k}} then the following compatibility condition is satisfied:

{{block indent|1= {{math|1=f_i^k = f_j^k ∘ f_i^j}}, }}

where this means that the composition

{{block indent|1= $X\_\{i\}\; \backslash xrightarrow\{f\_i^j\}\; X\_\{j\}\; \backslash xrightarrow\{f\_j^k\}\; X\_\{k\}\; \backslash ;\backslash ;\backslash ;\backslash ;\; \backslash text\{\; is\; equal\; to\; \}\; \backslash ;\backslash ;\backslash ;\backslash ;\; X\_\{i\}\; \backslash xrightarrow\{f\_i^k\}\; X\_\{k\}.$ }}

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

{{block indent|1= $\backslash left(\; X\_\{\backslash bull\},\; \backslash left\; \backslash \{f\_\{i\}^\{j\}\; \backslash ;:\backslash ;\; i,\; j\; \backslash in\; I\; \backslash ;\backslash text\{\; and\; \}\backslash ;\; i\; \backslash le\; j\; \backslash right\; \backslash \},\; I\; \backslash right)$ }}

is known as a direct system in the category $\backslash mathcal\{C\}$ that is directed (or indexed) by {{math|I}}. Since the indexing set {{mvar|I}} is a directed set, the direct system is said to be directed.{{sfn | Bierstedt | 1988 | pp=41-56}}

The maps {{math|f_i^j}} are called the bonding, connecting, or linking maps of the system.

If the indexing set {{mvar|I}} is understood then {{mvar|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 "{{math|X_•}} is a direct system" where "{{math|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 $f\_\{i\}^\{j\}$ is injective then the system is called injective.{{sfn | Bierstedt | 1988 | pp=41-56}}

{{block indent|1=Assumptions: In the case where the direct system is injective, it is often assumed without loss of generality that for all indices {{math|i ≤ j}}, each {{math|X_i}} is a vector subspace of {{math|X_j}} (in particular, {{math|X_i}} is identified with the range of $f\_\{i\}^\{j\}$) and that the bonding map $f\_\{i\}^\{j\}$ is the natural inclusion

{{block indent|1= {{math|In{{su|p=j|b=i}} : X_i → X_j}} }}

(i.e. defined by {{math|x ↦ x}}) so that the subspace topology on {{math|X_i}} induced by {{math|X_j}} is weaker (i.e. coarser) than the original (i.e. given) topology on {{math|X_i}}.

In this case, also take

{{block indent|1= {{math|1=X := {{underset|i ∈ I|{{big|∪}}}} X_i}}. }}

The limit maps are then the natural inclusions {{math|In_i : X_i → X}}. The direct limit topology on {{mvar|X}} is the final topology induced by these inclusion maps.

}}

If the {{math|X_i}}'s have an algebraic structure, say addition for example, then for any {{math|x, y ∈ X}}, we pick any index {{math|i}} such that {{math|x, y ∈ X_i}} and then define their sum using by using the addition operator of {{math|X_i}}. That is,

{{block indent|1= {{math|1=x + y := x +_i y}}, }}

where {{math|+_i}} is the addition operator of {{math|X_i}}. This sum is independent of the index {{mvar|i}} that is chosen.

In the category of locally convex topological vector spaces, the topology on the direct limit {{mvar|X}} of an injective directed inductive limit of locally convex spaces can be described by specifying that an absolutely convex subset {{math|U}} of {{mvar|X}} is a neighborhood of {{math|0}} if and only if {{math|U ∩ X_i}} is an absolutely convex neighborhood of {{math|0}} in {{math|X_i}} for every index {{mvar|i}}.{{sfn | Bierstedt | 1988 | pp=41-56}}

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 {{math|1=f_i^j}} is/is a injective (resp. surjective, bijective, homeomorphism, topological embedding, quotient map) then so is every {{math|f_i : X_i → X}}.{{sfn | Dugundji | 1966 | pp=420-435}}

Direct limits in the categories of topological spaces, topological vector spaces (TVSs), and Hausdorff locally convex TVSs are "poorly behaved".{{sfn | Bierstedt | 1988 | pp=41-56}}

For instance, the direct limit of a sequence (i.e. indexed by the natural numbers) of locally convex nuclear Fréchet spaces may {{strong|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.{{sfn | Bierstedt | 1988 | pp=41-56}}

However, non-Hausdorff locally convex inductive limits do occur in natural questions of analysis.{{sfn | Bierstedt | 1988 | pp=41-56}}

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 $\backslash mathbb\{N\}$ 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 {{math|X_i}} is a vector subspace of {{math|X_i+1}} and that the subspace topology induced on {{math|X_i}} by {{math|X_i+1}} is identical to the original topology on {{math|X_i}}.{{sfn | Schaefer | Wolff | 1999 | pp=55-61}}

In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces {{mvar|X}} can be described by specifying that an absolutely convex subset {{math|U}} is a neighborhood of {{math|0}} if and only if {{math|U ∩ X_n}} is an absolutely convex neighborhood of {{math|0}} in {{math|X_n}} for every {{mvar|n}}.

An inductive limit in the category of locally convex TVSs of a family of bornological (resp. barrelled, quasi-barrelled) spaces has this same property.{{sfn | Grothendieck | 1973 | pp=130-142}}

Every LF-space is a meager subset of itself.{{sfn | Narici | Beckenstein | 2011 | p=435}}

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.{{sfn | Schaefer | Wolff | 1999 | pp=59-61}}

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.{{sfn | Narici | Beckenstein | 2011 | p=436}}

LF spaces are distinguished and their strong duals are bornological and barrelled (a result due to Alexander Grothendieck).

If {{mvar|X}} is the strict inductive limit of an increasing sequence of Fréchet space {{math|X_n}} then a subset {{mvar|B}} of {{mvar|X}} is bounded in {{mvar|X}} if and only if there exists some {{mvar|n}} such that {{mvar|B}} is a bounded subset of {{math|X_n}}.{{sfn | Schaefer | Wolff | 1999 | pp=59-61}}

A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous.{{sfn | Trèves | 2006 | p=141}}

A linear map from an LF-space {{mvar|X}} into a Fréchet space {{mvar|Y}} is continuous if and only if its graph is closed in {{math|X × Y}}.{{sfn | Trèves | 2006 | p=173}}

Every bounded linear operator from an LF-space into another TVS is continuous.{{sfn | Trèves | 2006 | p=142}}

If {{mvar|X}} is an LF-space defined by a sequence $\backslash left(\; X\_i\; \backslash right)\_\{i=1\}^\{\backslash infty\}$ then the strong dual space $X^\{\backslash prime\}\_\{b\}$ of {{mvar|X}} is a Fréchet space if and only if all {{math|X_i}} are normable.{{sfn | Trèves | 2006 | p=201}}

Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an LB-space.

{{Main|Distribution (mathematics)}}

A typical example of an LF-space is, $C^\backslash infty\_c(\backslash mathbb\{R\}^n)$, the space of all infinitely differentiable functions on $\backslash mathbb\{R\}^n$ with compact support. The LF-space structure is obtained by considering a sequence of compact sets $K\_1\; \backslash subset\; K\_2\; \backslash subset\; \backslash ldots\; \backslash subset\; K\_i\; \backslash subset\; \backslash ldots\; \backslash subset\; \backslash mathbb\{R\}^n$ with $\backslash bigcup\_i\; K\_i\; =\; \backslash mathbb\{R\}^n$ 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 $C\_c^\backslash infty(K\_i)$ of infinitely differentiable functions on $\backslash mathbb\{R\}^n$ with compact support contained in $K\_i$ has a natural Fréchet space structure and $C^\backslash infty\_c(\backslash mathbb\{R\}^n)$ 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, $C^\backslash infty\_c(\backslash mathbb\{R\}^n)$ is known as the space of test functions, of fundamental importance in the theory of distributions.

Suppose that for every positive integer {{mvar|n}}, {{math|1=X_n := $\backslash mathbb\{R\}$ⁿ}} and for {{math|m < n}}, consider X_m as a vector subspace of {{math|X_n}} via the canonical embedding {{math|X_m → X_n}} defined by {{nowrap|1={{math|1=x := (x₁, ..., x_m) ↦ (x₁, ..., x_m, 0, ..., 0)}}}}.

Denote the resulting LF-space by {{mvar|X}}. Since any TVS topology on {{mvar|X}} makes continuous the inclusions of the X_m's into {{mvar|X}}, the latter space has the maximum among all TVS topologies on an $\backslash mathbb\{R\}$-vector space with countable Hamel dimension. It is a LC topology, associated with the family of all seminorms on {{mvar|X}}. Also, the TVS inductive limit topology of {{mvar|X}} coincides with the topological inductive limit; that is, the direct limit of the finite dimensional spaces {{math|X_n}} in the category TOP and in the category TVS coincide. The continuous dual space $X^\{\backslash prime\}$ of {{mvar|X}} is equal to the algebraic dual space of {{mvar|X}}, that is the space of all real valued sequences $\backslash mathbb\{R\}^\backslash mathbb\{N\}$ and the weak topology on $X^\{\backslash prime\}$ is equal to the strong topology on $X^\{\backslash prime\}$ (i.e. $X^\{\backslash prime\}\_\{\backslash sigma\}\; =\; X^\{\backslash prime\}\_\{b\}$).{{sfn | Schaefer | Wolff | 1999 | p=201}} In fact, it is the unique LC topology on $X^\{\backslash prime\}$ whose topological dual space is X.

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

{{reflist|group=note}}

{{reflist|group=proof}}

{{reflist}}

• {{Adasch Topological Vector Spaces|edition=2}}
• {{Bierstedt An Introduction to Locally Convex Inductive Limits}}
• {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}}
• {{Dugundji Topology}}
• {{Edwards Functional Analysis Theory and Applications}}
• {{Grothendieck Topological Vector Spaces}}
• {{Horváth Topological Vector Spaces and Distributions Volume 1 1966}}
• {{Jarchow Locally Convex Spaces}}
• {{Khaleelulla Counterexamples in Topological Vector Spaces}}
• {{Köthe Topological Vector Spaces I}}
• {{Köthe Topological Vector Spaces II}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Robertson Topological Vector Spaces}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Schechter Handbook of Analysis and Its Foundations}}
• {{Swartz An Introduction to Functional Analysis}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{Valdivia Topics in Locally Convex Spaces|edition=1}}
• {{Voigt A Course on Topological Vector Spaces|edition=1}}
• {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}

{{Functional Analysis}}

{{TopologicalVectorSpaces}}

Category:Topological vector spaces