LB-space

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

If each of the bonding maps $i_{nm}$ is an embedding of TVSs then the LB-space is called a strict LB-space. This means that the topology induced on $X_{n}$ by $X_{n 1}$ is identical to the original topology on $X_{n}.$ ^[1] Some authors (e.g. Schaefer) define the term "LB-space" to mean "strict LB-space."

Definition

The topology on $X$ 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

A strict LB-space is complete,^[2] barrelled,^[2] and bornological^[2] (and thus ultrabornological).

Examples

If $D$ 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 $C_{c}(D)$ of all continuous, complex-valued functions on $D$ with compact support is a strict LB-space.^[3] For any compact subset $K\subseteq D,$ let $C_{c}(K)$ denote the Banach space of complex-valued functions that are supported by $K$ with the uniform norm and order the family of compact subsets of $D$ by inclusion.^[3]

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

Let

{\begin{alignedat}{4}\mathbb {R} ^{\infty }~&:=~\left\{\left(x_{1},x_{2},\ldots \right)\in \mathbb {R} ^{\mathbb {N} }~:~{\text{ all but finitely many }}x_{i}{\text{ are equal to 0 }}\right\},\end{alignedat}}

denote the space of finite sequences, where $\mathbb {R} ^{\mathbb {N} }$ denotes the space of all real sequences. For every natural number $n\in \mathbb {N} ,$ let $\mathbb {R} ^{n}$ denote the usual Euclidean space endowed with the Euclidean topology and let $\operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \mathbb {R} ^{\infty }$ denote the canonical inclusion defined by $\operatorname {In} _{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{n}\right):=\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)$ so that its image is

\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)=\left\{\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)~:~x_{1},\ldots ,x_{n}\in \mathbb {R} \right\}=\mathbb {R} ^{n}\times \left\{(0,0,\ldots )\right\}

and consequently,

\mathbb {R} ^{\infty }=\bigcup _{n\in \mathbb {N} }\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).

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

Often, for every $n\in \mathbb {N} ,$ the canonical inclusion $\operatorname {In} _{\mathbb {R} ^{n}}$ is used to identify $\mathbb {R} ^{n}$ with its image $\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)$ in $\mathbb {R} ^{\infty };$ explicitly, the elements $\left(x_{1},\ldots ,x_{n}\right)\in \mathbb {R} ^{n}$ and $\left(x_{1},\ldots ,x_{n},0,0,0,\ldots \right)$ are identified together. Under this identification, $\left(\left(\mathbb {R} ^{\infty },\tau ^{\infty }\right),\left(\operatorname {In} _{\mathbb {R} ^{n}}\right)_{n\in \mathbb {N} }\right)$ becomes a direct limit of the direct system $\left(\left(\mathbb {R} ^{n}\right)_{n\in \mathbb {N} },\left(\operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}\right)_{m\leq n{\text{ in }}\mathbb {N} },\mathbb {N} \right),$ where for every $m\leq n,$ the map $\operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}$ is the canonical inclusion defined by $\operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{m}\right):=\left(x_{1},\ldots ,x_{m},0,\ldots ,0\right),$ where there are $n-m$ trailing zeros.

Counter-examples

There exists a bornological LB-space whose strong bidual is not bornological.^[4] There exists an LB-space that is not quasi-complete.^[4]

Citations

^ Schaefer & Wolff 1999, pp. 55–61.
^ ^a ^b ^c Schaefer & Wolff 1999, pp. 60–63.
^ ^a ^b Schaefer & Wolff 1999, pp. 57–58.
^ ^a ^b Khaleelulla 1982, pp. 28–63.

References

Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
Bierstedt, Klaus-Dieter (1988). "An Introduction to Locally Convex Inductive Limits". Functional Analysis and Applications. Singapore-New Jersey-Hong Kong: Universitätsbibliothek: 35–133. Retrieved 20 September 2020.
Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). 16. Providence: American Mathematical Society. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
Horváth, John (1966). Topological Vector Spaces and Distributions. Addison-Wesley series in mathematics. Vol. 1. Reading, MA: Addison-Wesley Publishing Company. ISBN 978-0201029857.
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Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
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Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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[FOOTNOTESchaeferWolff199955–61-1] Schaefer & Wolff 1999, pp. 55–61.

[FOOTNOTESchaeferWolff199960–63-2] Schaefer & Wolff 1999, pp. 60–63.

[FOOTNOTESchaeferWolff199957–58-3] Schaefer & Wolff 1999, pp. 57–58.

[FOOTNOTEKhaleelulla198228–63-4] Khaleelulla 1982, pp. 28–63.

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v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category