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In [[Mathematics|mathematics]], an '''''LF''-space''', also written '''(''LF'')-space''', is a [[Topological vector space|topological vector space]] (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system <math>(X_n, i_{nm})</math> of [[Fréchet space]]s.{{sfn | Schaefer | Wolff | 1999 | pp=55-61}}  
This means that ''X'' is a [[Direct limit|direct limit]] of a direct system <math>(X_n, i_{nm})</math> in the category of locally convex [[Topological vector space|topological vector space]]s and each <math>X_n</math> is a Fréchet space. The name ''LF'' stands for '''L'''imit of '''F'''réchet spaces.

If each of the bonding maps <math>i_{nm}</math> is an embedding of TVSs then the ''LF''-space is called a '''strict ''LF''-space'''. This means that the subspace topology induced on {{math|''X''<sub>''n''</sub>}} by {{math|''X''<sub>''n''+1</sub>}} is identical to the original topology on {{math|''X''<sub>''n''</sub>}}.{{sfn | Schaefer | Wolff | 1999 | pp=55-61}}<ref>{{cite book|last1=Helgason|first1=Sigurdur|title=Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions|date=2000|publisher=American Mathematical Society|location=Providence, R.I|isbn=0-8218-2673-5|page=398|edition=Reprinted with corr.}}</ref>
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 ==

<!--------- START: Assumptions -----------
Suppose that {{mvar|X}} is a set and {{math|1=''f''<sub>•</sub> = (&thinsp;''f''<sub>''i''</sub>&thinsp;)<sub>''i'' ∈ ''I''</sub>}} is a family of maps where for each index {{mvar|i}}, &thinsp;{{math|''f''<sub>''i''</sub> &thinsp;:&thinsp; {{large|(}}''X''<sub>''i''</sub>, τ<sub>''X''<sub>''i''</sub></sub>{{large|)}} &thinsp;→&thinsp; ''X''}} is a map. 
----------- END:   Assumptions ----------->

=== Inductive/final/direct limit topology ===
{{Main|Final topology}}

Throughout, it is assumed that 
* <math>\mathcal{C}</math> is either the [[Category of topological spaces|category of topological spaces]] or some subcategory of the [[Category (mathematics)|category]] of [[Topological vector space|topological vector space]]s (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|directed set]];
* {{math|1=''X''<sub>•</sub> = ( ''X''<sub>''i''</sub> )<sub>''i'' ∈ ''I''</sub>}} is a family of objects in <math>\mathcal{C}</math> where {{math|{{large|(}}''X''<sub>''i''</sub>, τ<sub>''X''<sub>''i''</sub></sub>{{large|)}}}} is a topological space for every index {{mvar|i}};
** To avoid potential confusion, {{math|τ<sub>''X''<sub>''i''</sub></sub>}} should '''''not''''' be called {{math|''X''<sub>''i''</sub>}}'s "initial topology" since the term "[[Initial topology|initial topology]]" already has a well-known definition. The topology {{math|τ<sub>''X''<sub>''i''</sub></sub>}} is called the '''original''' topology on {{math|''X''<sub>''i''</sub>}} or {{math|''X''<sub>''i''</sub>}}'s '''given topology'''. 
* {{mvar|X}} is a set (and if objects in <math>\mathcal{C}</math> also have algebraic structures, then {{mvar|X}} is automatically assumed to have has whatever algebraic structure is needed); 
* {{math|1=''f''<sub>•</sub> = ( ''f''<sub>''i''</sub> )<sub>''i'' ∈ ''I''</sub>}} is a family of maps where for each index {{mvar|i}}, the map has prototype  {{math|''f''<sub>''i''</sub> : {{large|(}}''X''<sub>''i''</sub>, τ<sub>''X''<sub>''i''</sub></sub>{{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|final topology]] on {{mvar|X}} in <math>\mathcal{C}</math>''', also called the '''colimit''' or '''inductive topology''' in <math>\mathcal{C}</math>, and denoted by {{math|τ<sub>''f''<sub>•</sub></sub>}} or {{math|τ<sub>f</sub>}}, is the [[Comparison of topologies|finest topology]] on {{mvar|X}} such that 

# {{math|{{large|(}}''X'', τ<sub>f</sub>{{large|)}}}} is an object in <math>\mathcal{C}</math>, and
# for every index {{mvar|i}}, the map {{math|''f''<sub>''i''</sub> : {{large|(}}''X''<sub>''i''</sub>, τ<sub>''X''<sub>''i''</sub></sub>{{large|)}} → {{large|(}}''X'', τ<sub>f</sub>{{large|)}}}} is a continuous morphism in <math>\mathcal{C}</math>.

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<sub>f</sub>{{large|)}}}} [[If and only if|if and only if]] {{math|''f''<sub>''i''</sub><sup>- 1</sup> (''U'')}} is open (resp. closed) in {{math|{{large|(}}''X''<sub>''i''</sub>, τ<sub>''X''<sub>''i''</sub></sub>{{large|)}}}} for every index {{mvar|i}}. 

However, the final topology may <em>not</em> exist in the category of [[Hausdorff space|Hausdorff topological spaces]] due to the requirement that {{math|{{large|(}}''X'', τ<sub>''X''<sub>f</sub></sub>{{large|)}}}} belong to the original category (i.e. belong to the category of Hausdorff topological spaces).{{sfn | Dugundji | 1966 | pp=420-435}}

=== Direct systems ===
{{Main|Direct limit}}

Suppose that {{math|(''I'', ≤)}} is a [[Directed set|directed set]] and that for all indices {{math|''i'' ≤ ''j''}} there are (continuous) morphisms in <math>\mathcal{C}</math>

{{block indent|1= {{math|''f''<sub>''i''</sub><sup>''j''</sup> : ''X''<sub>''i''</sub> → ''X''<sub>''j''</sub>}} }}

such that if {{math|1=''i'' = ''j''}} then {{math|''f''<sub>''i''</sub><sup>''j''</sup>}} is the identity map on {{math|''X''<sub>''i''</sub>}} and if {{math|''i'' ≤ ''j'' ≤ ''k''}} then the following '''compatibility condition''' is satisfied: 

{{block indent|1= {{math|1=''f''<sub>''i''</sub><sup>''k''</sup> = ''f''<sub>''j''</sub><sup>''k''</sup>  ∘  ''f''<sub>''i''</sub><sup>''j''</sup>}}, }}

where this means that the composition 

{{block indent|1= <math>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}.</math> }}

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

{{block indent|1= <math>\left( X_{\bull}, \left \{f_{i}^{j} \;:\; i, j \in I \;\text{ and }\; i \le j \right \}, I \right)</math> }}

is known as a '''[[Direct limit|direct system]]''' in the category <math>\mathcal{C}</math> that is [[Directed set|'''directed''']] (or '''indexed''') by {{math|''I''}}. Since the indexing set {{mvar|I}} is a [[Directed set|directed set]], the direct system is said to be '''directed'''.{{sfn | Bierstedt | 1988 | pp=41-56}} 
The maps {{math|''f''<sub>''i''</sub><sup>''j''</sup>}} 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''<sub>•</sub>}} is a direct system" where "{{math|''X''<sub>•</sub>}}" 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 limit]]. 
<!-------------- START: Removed definition of the direct limit of a general direct system. ------------------
;Direct limit of a general inductive system

We now describe the direct limit of a directed direct system that is not assumed to be injective (i.e. the bonding maps are not necessarily injective functions). 
The case where the bonding maps are injections is simpler and defined later. 

Let {{math|1=∐''X''<sub>•</sub>}} denote the disjoint union of all {{math|''X''<sub>''i''</sub>}} (this can also be defined as the direct sum of these {{math|''X''<sub>''i''</sub>}}) and define an equivalence relation {{mvar|R}} on this set by declaring two elements {{math|''x''<sub>''i''</sub> ∈ ''X''<sub>''i''</sub>}} and {{math|''x''<sub>''j''</sub> ∈ ''X''<sub>''j''</sub>}} '''equivalent''' if and only if there exists an index {{mvar|k}} such that {{math|''i'' ≤ ''k''}}, {{math|''j'' ≤ ''k''}}, and {{math|1=''f''&thinsp;<sub>''i''</sub><sup>''k''</sup>(''x''<sub>''i''</sub>) = ''f''&thinsp;<sub>''j''</sub><sup>''k''</sup>(''x''<sub>''j''</sub>)}}.
Denote the resulting quotient by {{math|1=''X'' := ∐''X''<sub>•</sub> / ''R''}} (note that if all {{math|''X''<sub>''i''</sub>}} are Hausdorff locally convex spaces, then it is not known whether {{mvar|R}} is necessarily a closed subspace of the locally convex direct sum {{math|1=∐''X''<sub>•</sub>}}{{sfn | Schaefer | Wolff | 1999 | pp=55-61}}). 
For every index {{mvar|i}}, let {{math|1=''f''<sub>''i''</sub> : ''X''<sub>''i''</sub> → ''X''}} denote the restriction to {{math|''X''<sub>''i''</sub>}} of the natural projection map {{math|∐''X''<sub>•</sub> → ''X''}}.{{sfn | Dugundji | 1966 | pp=420-435}} 
These maps satisfy {{math|1=''f''&thinsp;<sub>''i''</sub> &thinsp;=&thinsp; ''f''&thinsp;<sub>''j''</sub> &thinsp;∘&thinsp; ''f''&thinsp;<sub>''i''</sub><sup>''j''</sup>}} for all indices {{math|''i'' ≤ ''j'' ≤ ''k''}}. 

The pair {{math|(''X'', ''f''<sub>•</sub>&thinsp;)}} is the direct limit of the above direct system in the category of sets. 
In the category of topological spaces, if {{mvar|X}} is endowed with the inductive limit topology in <math>\mathcal{C}</math> induced by {{math|1=''f''<sub>•</sub> = (&thinsp;''f''<sub>''i''</sub>&thinsp;)<sub>''i'' ∈ ''I''</sub>}} then {{math|(''X'', ''f''<sub>•</sub>&thinsp;)}} is the direct limit in the category of topological spaces of the above direct system. 

More generally, if the system is in the category <math>\mathcal{C}</math> (of say, TVSs, locally convex TVSs, Hausdorff locally convex TVS, [[Metrizable topological vector space|metrizable TVSs]], etc), then this direct system's [[Direct limit|direct limit]] in <math>\mathcal{C}</math>, if it exists, is the pair {{math|(''X'', ''f''<sub>•</sub>)}} where {{mvar|X}} is endowed with the inductive limit topology in <math>\mathcal{C}</math> induced by {{math|1=''f''<sub>•</sub> = (&thinsp;''f''<sub>''i''</sub>&thinsp;)<sub>''i'' ∈ ''I''</sub>}}. 
---------------- END:   Removed definition of the direct limit of a general direct system. ------------------>

'''Direct limits of injective systems'''

If each of the bonding maps <math>f_{i}^{j}</math> is [[Injective function|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''<sub>''i''</sub>}} is a vector subspace of {{math|''X''<sub>''j''</sub>}} (in particular, {{math|''X''<sub>''i''</sub>}} is identified with the range of <math>f_{i}^{j}</math>) and that the bonding map <math>f_{i}^{j}</math> is the natural inclusion 
{{block indent|1= {{math|In{{su|p=''j''|b=''i''}} : ''X''<sub>''i''</sub> → ''X''<sub>''j''</sub>}} }}
(i.e. defined by {{math|''x'' ↦ ''x''}}) so that the subspace topology on {{math|''X''<sub>''i''</sub>}} induced by {{math|''X''<sub>''j''</sub>}} is [[Comparison of topologies|weaker (i.e. coarser)]] than the original (i.e. given) topology on {{math|''X''<sub>''i''</sub>}}.

In this case, also take 

{{block indent|1= {{math|1=''X'' := {{underset|''i'' ∈ ''I''|{{big|∪}}}}  ''X''<sub>''i''</sub>}}. }}

The limit maps are then the natural inclusions {{math|In<sub>''i''</sub> : ''X''<sub>''i''</sub> → ''X''}}.  The direct limit topology on {{mvar|X}} is the final topology induced by these inclusion maps. 
}}

If the {{math|''X''<sub>''i''</sub>}}'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''<sub>''i''</sub>}} and then define their sum using by using the addition operator of {{math|''X''<sub>''i''</sub>}}. That is, 
{{block indent|1= {{math|1=''x'' + ''y'' := ''x'' +<sub>''i''</sub>  ''y''}}, }}
where {{math|+<sub>''i''</sub>}} is the addition operator of {{math|''X''<sub>''i''</sub>}}. 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 set|absolutely convex]] subset {{math|U}} of {{mvar|X}} is a neighborhood of {{math|0}} if and only if {{math|''U'' ∩ ''X''<sub>''i''</sub>}} is an absolutely convex neighborhood of {{math|0}} in {{math|''X''<sub>''i''</sub>}} 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 topological vector space|locally convex]] TVSs.  
In the category of topological spaces, if every bonding map {{math|1=''f''<sub>''i''</sub><sup>''j''</sup>}} is/is a injective (resp. surjective, bijective, [[Homeomorphism|homeomorphism]], topological embedding, quotient map) then so is every {{math|''f''<sub>''i''</sub> : ''X''<sub>''i''</sub> → ''X''}}.{{sfn | Dugundji | 1966 | pp=420-435}}  

==== Problem with direct limits ====

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 space|nuclear]] [[Fréchet space]]s  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|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}}

=== Strict inductive limit ===

If each of the bonding maps <math>f_{i}^{j}</math> is an embedding of TVSs onto proper vector subspaces and if the system is directed by <math>\mathbb{N}</math> 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''<sub>''i''</sub>}} is a vector subspace of {{math|''X''<sub>''i''+1</sub>}} and that the subspace topology induced on {{math|''X''<sub>''i''</sub>}} by {{math|''X''<sub>''i''+1</sub>}} is identical to the original topology on {{math|''X''<sub>''i''</sub>}}.{{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''<sub>''n''</sub>}} is an absolutely convex neighborhood of {{math|0}} in {{math|''X''<sub>''n''</sub>}} for every {{mvar|n}}.

== Properties ==

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

=== LF-spaces ===

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 space|barrelled]] and [[Bornological space|bornological]], which together with completeness implies that every LF-space is [[Ultrabornological space|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 space|distinguished]] and their strong duals are [[Bornological space|bornological]] and [[Barrelled space|barrelled]] (a result due to [[Biography:Alexander Grothendieck|Alexander Grothendieck]]).

If {{mvar|X}} is the strict inductive limit of an increasing sequence of [[Fréchet space]] {{math|''X''<sub>''n''</sub>}} 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''<sub>''n''</sub>}}.{{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 <math>\left( X_i \right)_{i=1}^{\infty}</math> then the strong dual space <math>X^{\prime}_{b}</math> of {{mvar|X}} is a Fréchet space if and only if all {{math|''X''<sub>''i''</sub>}} 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]].

== Examples ==

=== Space of smooth compactly supported functions ===
{{Main|Distribution (mathematics)}}

A typical example of an ''LF''-space is, <math>C^\infty_c(\mathbb{R}^n)</math>, the space of all infinitely differentiable functions on <math>\mathbb{R}^n</math> with compact support. The ''LF''-space structure is obtained by considering a sequence of compact sets <math>K_1 \subset K_2 \subset \ldots \subset K_i \subset \ldots \subset \mathbb{R}^n</math> with <math>\bigcup_i K_i = \mathbb{R}^n</math> and for all i,  <math>K_i</math> is a subset of the interior of <math>K_{i+1}</math>. Such a sequence could be the balls of radius ''i'' centered at the origin.  The space <math>C_c^\infty(K_i)</math> of infinitely differentiable functions on <math>\mathbb{R}^n</math> with compact support contained in <math>K_i</math> has a natural [[Fréchet space]] structure and  <math>C^\infty_c(\mathbb{R}^n)</math> inherits its ''LF''-space structure as described above. The ''LF''-space topology does not depend on the particular sequence of compact sets <math>K_i</math>.

With this ''LF''-space structure, <math>C^\infty_c(\mathbb{R}^n)</math> is known as the space of test functions, of fundamental importance in the [[Distribution (mathematics)|theory of distributions]].

=== Direct limit of finite-dimensional spaces ===

Suppose that for every positive integer {{mvar|n}}, {{math|1=''X''<sub>''n''</sub> := <math>\mathbb{R}</math><sup>''n''</sup>}} and for {{math|''m'' < ''n''}}, consider ''X''<sub>''m''</sub> as a vector subspace of {{math|''X''<sub>''n''</sub>}} via the canonical embedding {{math|''X''<sub>''m''</sub> → ''X''<sub>''n''</sub>}} defined by {{nowrap|1={{math|1=''x'' := (''x''<sub>1</sub>, ..., ''x''<sub>''m''</sub>) ↦ (''x''<sub>1</sub>, ..., ''x''<sub>''m''</sub>, 0, ..., 0)}}}}. 
Denote the resulting LF-space by {{mvar|X}}. Since any TVS topology on {{mvar|X}} makes continuous the inclusions of the ''X''<sub>''m''</sub>'s into  {{mvar|X}}, the latter space has the maximum among all TVS topologies on an <math>\mathbb{R}</math>-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''<sub>''n''</sub>}} in the category TOP and in the category TVS coincide. The continuous dual space <math>X^{\prime}</math> of {{mvar|X}} is equal to the algebraic dual space of {{mvar|X}}, that is the space of all real valued sequences <math>\mathbb{R}^\mathbb{N}</math>  and the weak topology on <math>X^{\prime}</math> is equal to the [[Strong topology|strong topology]] on <math>X^{\prime}</math> (i.e. <math>X^{\prime}_{\sigma} = X^{\prime}_{b}</math>).{{sfn | Schaefer | Wolff | 1999 | p=201}}. In fact, it is the unique  LC topology on <math>X^{\prime}</math> whose topological dual space is X.   
Furthermore, the canonical map of {{mvar|X}} into the continuous dual space of <math>X^{\prime}_{\sigma}</math> is surjective.{{sfn | Schaefer | Wolff | 1999 | p=201}}

== See also ==

* [[DF-space]]
* [[Direct limit]]
* [[Final topology]]
* [[F-space]]
* [[LB-space]]

== Citations ==
{{reflist|group=note}}
{{reflist|group=proof}}
{{reflist}}

== Bibliography ==

* {{Adasch Topological Vector Spaces|edition=2}} <!-- {{sfn | Adasch | Ernst | Keim | 1978 | p=}} -->
* {{Bierstedt An Introduction to Locally Convex Inductive Limits}} <!-- {{sfn | Bierstedt | 1988 | p=}} -->
* {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}} <!-- {{sfn | Bourbaki | 1987 | p=}} -->
* {{Dugundji Topology}} <!-- {{sfn | Dugundji | 1966 | p=}} -->
* {{Edwards Functional Analysis Theory and Applications}} <!-- {{sfn | Edwards | 1995 | p=}} -->
* {{Grothendieck Topological Vector Spaces}} <!-- {{sfn | Grothendieck | 1973 | p=}} -->
* {{Horváth Topological Vector Spaces and Distributions Volume 1 1966}} <!-- {{sfn | Horváth | 1966 | p=}} -->
* {{Jarchow Locally Convex Spaces}} <!-- {{sfn | Jarchow | 1981 | p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | 1982 | p=}} -->
* {{Köthe Topological Vector Spaces I}} <!-- {{sfn | Köthe | 1969 | p=}} -->
* {{Köthe Topological Vector Spaces II}} <!-- {{sfn | Köthe | 1979 | p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | Beckenstein | 2011 | p=}} -->
* {{Robertson Topological Vector Spaces}} <!-- {{sfn | Robertson | Robertson | 1980 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | Wolff | 1999 | p=}} -->
* {{Schechter Handbook of Analysis and Its Foundations}} <!-- {{sfn | Schechter | 1996 | p=}} -->
* {{Swartz An Introduction to Functional Analysis}} <!-- {{sfn | Swartz | 1992 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->
* {{Valdivia Topics in Locally Convex Spaces|edition=1}} <!-- {{sfn | Valdivia | 1982 | p=}} -->
* {{Voigt A Course on Topological Vector Spaces|edition=1}} <!-- {{sfn | Voigt | 2020 | p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn | Wilansky | 2013 | p=}} -->

{{Functional Analysis}}
{{TopologicalVectorSpaces}}

[[Category:Topological vector spaces]]

{{Sourceattribution|LF-space}}