LB-space
# 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 $X_{n}.$ 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 $X$ 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, barrelled, and bornological (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. 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.

### 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.

There exists an LB-space that is not quasi-complete.

