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 $(X_n, i_)$ of Banach spaces.
This means that $X$ is a direct limit of a direct system $\left( X_n, i_ \right)$ in the category of locally convex topological vector spaces and each $X_n$ is a Banach space.

If each of the bonding maps $i_$ 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 $X_$ 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
:$\begin \R^ ~&:=~ \left\, \end$

denote the , where $\R^$ denotes the space of all real sequences.
For every natural number $n \in \N,$ let $\R^n$ denote the usual Euclidean space endowed with the Euclidean topology and let $\operatorname_ : \R^n \to \R^$ denote the canonical inclusion defined by $\operatorname_\left(x_1, \ldots, x_n\right) := \left(x_1, \ldots, x_n, 0, 0, \ldots \right)$ so that its image is
:$\operatorname \left( \operatorname_ \right) = \left\ = \R^n \times \left\$

and consequently,
:$\R^ = \bigcup_ \operatorname \left( \operatorname_ \right).$

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

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

Counter-examples

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

See also

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Citations

References

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{{TopologicalVectorSpaces

Topological vector spaces