LB-space

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

If each of the bonding maps [math]\displaystyle{ 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]\displaystyle{ X_n }[/math] by [math]\displaystyle{ X_{n+1} }[/math] is identical to the original topology on [math]\displaystyle{ X_n. }[/math] Some authors (e.g. Schaefer) define the term "LB-space" to mean "strict LB-space," so when reading mathematical literature, its recommended to always check how LB-space is defined.

Definition

The topology on [math]\displaystyle{ X }[/math] can be described by specifying that an absolutely convex subset [math]\displaystyle{ U }[/math] is a neighborhood of [math]\displaystyle{ 0 }[/math] if and only if [math]\displaystyle{ U \cap X_n }[/math] is an absolutely convex neighborhood of [math]\displaystyle{ 0 }[/math] in [math]\displaystyle{ X_n }[/math] for every [math]\displaystyle{ n. }[/math]

Properties

A strict LB-space is complete, barrelled, and bornological (and thus ultrabornological).

Examples

If [math]\displaystyle{ 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]\displaystyle{ C_c(D) }[/math] of all continuous, complex-valued functions on [math]\displaystyle{ D }[/math] with compact support is a strict LB-space. For any compact subset [math]\displaystyle{ K \subseteq D, }[/math] let [math]\displaystyle{ C_c(K) }[/math] denote the Banach space of complex-valued functions that are supported by [math]\displaystyle{ K }[/math] with the uniform norm and order the family of compact subsets of [math]\displaystyle{ D }[/math] by inclusion.

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

Let

[math]\displaystyle{ \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 space of finite sequences, where [math]\displaystyle{ \R^{\N} }[/math] denotes the space of all real sequences. For every natural number [math]\displaystyle{ n \in \N, }[/math] let [math]\displaystyle{ \R^n }[/math] denote the usual Euclidean space endowed with the Euclidean topology and let [math]\displaystyle{ \operatorname{In}_{\R^n} : \R^n \to \R^{\infty} }[/math] denote the canonical inclusion defined by [math]\displaystyle{ \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 is

[math]\displaystyle{ \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]\displaystyle{ \R^{\infty} = \bigcup_{n \in \N} \operatorname{Im} \left( \operatorname{In}_{\R^n} \right). }[/math]

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

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

Counter-examples

There exists a bornological LB-space whose strong bidual is not bornological. There exists an LB-space that is not quasi-complete.

See also

• DF-space
• Direct limit – Special case of colimit in category theory
• Final topology – Finest topology making some functions continuous
• F-space – Topological vector space with a complete translation-invariant metric
• LF-space

References

• Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. {3834. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. 
• Template:Bierstedt An Introduction to Locally Convex Inductive Limits
• Bourbaki, Nicolas (1987). Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. 2. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190. http://www.numdam.org/item?id=AIF_1950__2__5_0. 
• Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. 
• Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138. 
• Template:Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires
• Template:Horváth Topological Vector Spaces and Distributions Volume 1 1966
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. 
• Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. 
• Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. OCLC 840293704. 
• Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972. 
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. 
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.