View source for LB-space

In [[mathematics]], an '''''LB''-space''', also written '''(''LB'')-space''', is a [[topological vector space]] <math>X</math> that is a locally convex [[inductive limit]] of a countable inductive system <math>(X_n, i_{nm})</math> of [[Banach space]]s. 
This means that <math>X</math> is a [[direct limit]] of a direct system <math>\left( X_n, i_{nm} \right)</math> in the category of [[Locally convex topological vector space|locally convex]] [[topological vector space]]s and each <math>X_n</math> is a Banach space.

If each of the bonding maps <math>i_{nm}</math> is an embedding of TVSs then the ''LB''-space is called a '''strict ''LB''-space'''. This means that the topology induced on <math>X_n</math> by <math>X_{n+1}</math> is identical to the original topology on <math>X_n.</math>{{sfn|Schaefer|Wolff|1999|pp=55-61}} 
Some authors (e.g. Schaefer) define the term "''LB''-space" to mean "strict ''LB''-space."

== Definition ==

The topology on <math>X</math> can be described by specifying that an absolutely convex subset <math>U</math> is a neighborhood of <math>0</math> if and only if <math>U \cap X_n</math> is an absolutely convex neighborhood of <math>0</math> in <math>X_n</math> for every <math>n.</math>
<!------------- Old definition: 
Some authors restrict the term ''LB''-space to mean that <math>X</math> is a strict locally convex inductive limit of Banach spaces, which means that <math>m \leq n</math> implies <math>X_m \subseteq X_n,</math> each mapping <math>i_{nm} : X_m \to X_n</math> is the natural inclusion, and '''strict''' means that each <math>X_m</math> has <math>X_n</math>'s [[subspace topology]] (where <math>m \leq n</math>).{{sfn|Schaefer|Wolff|1999|pp=55-61}} To distinguish this type of space from the more general definition, we will call this a '''{{em|strict ''LB''-space}}'''. 
--------->

== Properties ==

A strict ''LB''-space is [[Complete topological vector space|complete]],{{sfn|Schaefer|Wolff|1999|pp=60-63}} [[Barrelled space|barrelled]],{{sfn|Schaefer|Wolff|1999|pp=60-63}} and [[Bornological space|bornological]]{{sfn|Schaefer|Wolff|1999|pp=60-63}} (and thus [[Ultrabornological space|ultrabornological]]).

== Examples ==

If <math>D</math> is a locally compact [[topological space]] that is [[countable at infinity]] (that is, it is equal to a countable union of compact subspaces) then the space <math>C_c(D)</math> of all continuous, complex-valued functions on <math>D</math> with [[Support (mathematics)|compact support]] is a strict ''LB''-space.{{sfn|Schaefer|Wolff|1999|pp=57-58}} For any compact subset <math>K \subseteq D,</math> let <math>C_c(K)</math> denote the Banach space of complex-valued functions that are supported by <math>K</math> with the uniform norm and order the family of compact subsets of <math>D</math> by inclusion.{{sfn|Schaefer|Wolff|1999|pp=57-58}}

;Final topology on the direct limit of finite-dimensional Euclidean spaces

Let 
:<math>\begin{alignat}{4}
\R^{\infty} 
~&:=~ \left\{ \left(x_1, x_2, \ldots \right) \in \R^{\N} ~:~ \text{ all but finitely many } x_i \text{ are equal to 0 } \right\},
\end{alignat}
</math>

denote the '''{{em|[[space of finite sequences]]}}''', where <math>\R^{\N}</math> denotes the [[space of all real sequences]]. 
For every [[natural number]] <math>n \in \N,</math> let <math>\R^n</math> denote the usual [[Euclidean space]] endowed with the [[Euclidean topology]] and let <math>\operatorname{In}_{\R^n} : \R^n \to \R^{\infty}</math> denote the canonical inclusion defined by <math>\operatorname{In}_{\R^n}\left(x_1, \ldots, x_n\right) := \left(x_1, \ldots, x_n, 0, 0, \ldots \right)</math> so that its [[Image (mathematics)|image]] is
:<math>\operatorname{Im} \left( \operatorname{In}_{\R^n} \right) 
= \left\{ \left(x_1, \ldots, x_n, 0, 0, \ldots \right) ~:~ x_1, \ldots, x_n \in \R \right\} 
= \R^n \times \left\{ (0, 0, \ldots) \right\}</math>

and consequently,
:<math>\R^{\infty} = \bigcup_{n \in \N} \operatorname{Im} \left( \operatorname{In}_{\R^n} \right).</math> 

Endow the set <math>\R^{\infty}</math> with the [[final topology]] <math>\tau^{\infty}</math> induced by the family <math>\mathcal{F} := \left\{ \; \operatorname{In}_{\R^n} ~:~ n \in \N \; \right\}</math> of all canonical inclusions. 
With this topology, <math>\R^{\infty}</math> becomes a [[Complete topological vector space|complete]] [[Hausdorff space|Hausdorff]] [[Locally convex topological vector space|locally convex]] [[sequential space|sequential]] [[topological vector space]] that is {{em|not}} a [[Fréchet–Urysohn space]]. 
The topology <math>\tau^{\infty}</math> is [[Comparison of topologies|strictly finer]] than the [[subspace topology]] induced on <math>\R^{\infty}</math> by <math>\R^{\N},</math> where <math>\R^{\N}</math> is endowed with its usual [[product topology]]. 
Endow the image <math>\operatorname{Im} \left( \operatorname{In}_{\R^n} \right)</math> with the final topology induced on it by the [[bijection]] <math>\operatorname{In}_{\R^n} : \R^n \to \operatorname{Im} \left( \operatorname{In}_{\R^n} \right);</math> that is, it is endowed with the Euclidean topology transferred to it from <math>\R^n</math> via <math>\operatorname{In}_{\R^n}.</math> 
This topology on <math>\operatorname{Im} \left( \operatorname{In}_{\R^n} \right)</math> is equal to the subspace topology induced on it by <math>\left(\R^{\infty}, \tau^{\infty}\right).</math> 
A subset <math>S \subseteq \R^{\infty}</math> is open (resp. closed) in <math>\left(\R^{\infty}, \tau^{\infty}\right)</math> if and only if for every <math>n \in \N,</math> the set <math>S \cap \operatorname{Im} \left( \operatorname{In}_{\R^n} \right)</math> is an open (resp. closed) subset of <math>\operatorname{Im} \left( \operatorname{In}_{\R^n} \right).</math> 
The topology <math>\tau^{\infty}</math> is coherent with family of subspaces <math>\mathbb{S} := \left\{ \; \operatorname{Im} \left( \operatorname{In}_{\R^n} \right) ~:~ n \in \N \; \right\}.</math> 
This makes <math>\left(\R^{\infty}, \tau^{\infty}\right)</math> into an LB-space. 
Consequently, if <math>v \in \R^{\infty}</math> and <math>v_{\bull}</math> is a sequence in <math>\R^{\infty}</math> then <math>v_{\bull} \to v</math> in <math>\left(\R^{\infty}, \tau^{\infty}\right)</math> if and only if there exists some <math>n \in \N</math> such that both <math>v</math> and <math>v_{\bull}</math> are contained in <math>\operatorname{Im} \left( \operatorname{In}_{\R^n} \right)</math> and <math>v_{\bull} \to v</math> in <math>\operatorname{Im} \left( \operatorname{In}_{\R^n} \right).</math> 

Often, for every <math>n \in \N,</math> the canonical inclusion <math>\operatorname{In}_{\R^n}</math> is used to identify <math>\R^n</math> with its image <math>\operatorname{Im} \left( \operatorname{In}_{\R^n} \right)</math> in <math>\R^{\infty};</math> explicitly, the elements <math>\left( x_1, \ldots, x_n \right) \in \mathbb{R}^n</math> and <math>\left( x_1, \ldots, x_n, 0, 0, 0, \ldots \right)</math> are identified together. 
Under this identification, <math>\left( \left(\R^{\infty}, \tau^{\infty}\right), \left(\operatorname{In}_{\R^n}\right)_{n \in \N}\right)</math> becomes a [[direct limit]] of the direct system <math>\left( \left(\R^n\right)_{n \in \N}, \left(\operatorname{In}_{\R^m}^{\R^n}\right)_{m \leq n \text{ in } \N}, \N \right),</math> where for every <math>m \leq n,</math> the map <math>\operatorname{In}_{\R^m}^{\R^n} : \R^m \to \R^n</math> is the canonical inclusion defined by <math>\operatorname{In}_{\R^m}^{\R^n}\left(x_1, \ldots, x_m\right) := \left(x_1, \ldots, x_m, 0, \ldots, 0 \right),</math> where there are <math>n - m</math> trailing zeros.

=== Counter-examples ===

There exists a [[Bornological space|bornological]] LB-space whose strong bidual is {{em|not}} bornological.{{sfn|Khaleelulla|1982|pp=28-63}}  
There exists an LB-space that is not [[quasi-complete]].{{sfn|Khaleelulla|1982|pp=28-63}}

== See also ==

* {{annotated link|DF-space}}
* {{annotated link|Direct limit}}
* {{annotated link|Final topology}}
* {{annotated link|F-space}}
* {{annotated link|LF-space}}

== Citations ==

{{reflist|group=note}}
{{reflist}}

== References ==

* {{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 Produits Tensoriels Topologiques et Espaces Nucléaires}} <!-- {{sfn|Grothendieck|1955|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=}} -->
* {{Swartz An Introduction to Functional Analysis}} <!-- {{sfn|Swartz|1992|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->

{{Functional Analysis}}
{{TopologicalVectorSpaces}}

[[Category:Topological vector spaces]]