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 $(X_n,i_{nm})$ 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 $X_n.$Template:Sfn 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,Template:Sfn barrelled,Template:Sfn and bornologicalTemplate:Sfn (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.Template:Sfn 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.Template:Sfn

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

Let

$\begin{array}{cc}{ℝ}^{\mathrm{\infty }}& :=\{(x_1,x_2,\dots )\in {ℝ}^{\mathbb{N}}:\text{ all but finitely many }{x}_i\text{ are equal to 0 }\},\end{array}$

denote the Template:Em, where ${ℝ}^{\mathbb{N}}$ denotes the space of all real sequences. For every natural number $n\in \mathbb{N},$ let ${ℝ}^n$ denote the usual Euclidean space endowed with the Euclidean topology and let ${\mathrm{In}}_{{ℝ}^n}:{ℝ}^n\to {ℝ}^{\mathrm{\infty }}$ denote the canonical inclusion defined by ${\mathrm{In}}_{{ℝ}^n}(x_1,\dots ,x_n):=(x_1,\dots ,x_n,0,0,\dots )$ so that its image is

$\mathrm{Im}\left({\mathrm{In}}_{{ℝ}^n}\right)=\{(x_1,\dots ,x_n,0,0,\dots ):x_1,\dots ,x_n\in ℝ\}={ℝ}^n\times \{(0,0,\dots )\}$

and consequently,

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

Endow the set ${ℝ}^{\mathrm{\infty }}$ with the final topology ${\tau }^{\mathrm{\infty }}$ induced by the family $ℱ:=\{{\mathrm{In}}_{{ℝ}^n}:n\in \mathbb{N}\}$ of all canonical inclusions. With this topology, ${ℝ}^{\mathrm{\infty }}$ becomes a complete Hausdorff locally convex sequential topological vector space that is Template:Em a Fréchet–Urysohn space. The topology ${\tau }^{\mathrm{\infty }}$ is strictly finer than the subspace topology induced on ${ℝ}^{\mathrm{\infty }}$ by ${ℝ}^{\mathbb{N}},$ where ${ℝ}^{\mathbb{N}}$ is endowed with its usual product topology. Endow the image $\mathrm{Im}\left({\mathrm{In}}_{{ℝ}^n}\right)$ with the final topology induced on it by the bijection ${\mathrm{In}}_{{ℝ}^n}:{ℝ}^n\to \mathrm{Im}\left({\mathrm{In}}_{{ℝ}^n}\right);$ that is, it is endowed with the Euclidean topology transferred to it from ${ℝ}^n$ via ${\mathrm{In}}_{{ℝ}^n}.$ This topology on $\mathrm{Im}\left({\mathrm{In}}_{{ℝ}^n}\right)$ is equal to the subspace topology induced on it by $({ℝ}^{\mathrm{\infty }},{\tau }^{\mathrm{\infty }}).$ A subset $S\subseteq {ℝ}^{\mathrm{\infty }}$ is open (resp. closed) in $({ℝ}^{\mathrm{\infty }},{\tau }^{\mathrm{\infty }})$ if and only if for every $n\in \mathbb{N},$ the set $S\cap \mathrm{Im}\left({\mathrm{In}}_{{ℝ}^n}\right)$ is an open (resp. closed) subset of $\mathrm{Im}\left({\mathrm{In}}_{{ℝ}^n}\right).$ The topology ${\tau }^{\mathrm{\infty }}$ is coherent with family of subspaces $𝕊:=\{\mathrm{Im}\left({\mathrm{In}}_{{ℝ}^n}\right):n\in \mathbb{N}\}.$ This makes $({ℝ}^{\mathrm{\infty }},{\tau }^{\mathrm{\infty }})$ into an LB-space. Consequently, if $v\in {ℝ}^{\mathrm{\infty }}$ and $v_{\bullet }$ is a sequence in ${ℝ}^{\mathrm{\infty }}$ then $v_{\bullet }\to v$ in $({ℝ}^{\mathrm{\infty }},{\tau }^{\mathrm{\infty }})$ if and only if there exists some $n\in \mathbb{N}$ such that both $v$ and $v_{\bullet }$ are contained in $\mathrm{Im}\left({\mathrm{In}}_{{ℝ}^n}\right)$ and $v_{\bullet }\to v$ in $\mathrm{Im}\left({\mathrm{In}}_{{ℝ}^n}\right).$

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

Counter-examples

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

See also

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References

• Template:Adasch Topological Vector Spaces
• Template:Bierstedt An Introduction to Locally Convex Inductive Limits
• Template:Bourbaki Topological Vector Spaces Part 1 Chapters 1–5
• Template:Dugundji Topology
• Template:Edwards Functional Analysis Theory and Applications
• Template:Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires
• Template:Horváth Topological Vector Spaces and Distributions Volume 1 1966
• Template:Jarchow Locally Convex Spaces
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Köthe Topological Vector Spaces I
• Template:Köthe Topological Vector Spaces II
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Robertson Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Swartz An Introduction to Functional Analysis
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Wilansky Modern Methods in Topological Vector Spaces