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."

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{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} $$

denote the , where $$\R^{\N}$$ 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 $$\operatorname{In}_{\R^n} : \R^n \to \R^{\infty}$$ denote the canonical inclusion defined by $$\operatorname{In}_{\R^n}\left(x_1, \ldots, x_n\right) := \left(x_1, \ldots, x_n, 0, 0, \ldots \right)$$ so that its image is
 * $$\operatorname{Im} \left( \operatorname{In}_{\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\}$$

and consequently,
 * $$\R^{\infty} = \bigcup_{n \in \N} \operatorname{Im} \left( \operatorname{In}_{\R^n} \right).$$

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

Often, for every $$n \in \N,$$ the canonical inclusion $$\operatorname{In}_{\R^n}$$ is used to identify $$\R^n$$ with its image $$\operatorname{Im} \left( \operatorname{In}_{\R^n} \right)$$ in $$\R^{\infty};$$ 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, $$\left( \left(\R^{\infty}, \tau^{\infty}\right), \left(\operatorname{In}_{\R^n}\right)_{n \in \N}\right)$$ becomes a direct limit of the direct system $$\left( \left(\R^n\right)_{n \in \N}, \left(\operatorname{In}_{\R^m}^{\R^n}\right)_{m \leq n \text{ in } \N}, \N \right),$$ where for every $$m \leq n,$$ the map $$\operatorname{In}_{\R^m}^{\R^n} : \R^m \to \R^n$$ is the canonical inclusion defined by $$\operatorname{In}_{\R^m}^{\R^n}\left(x_1, \ldots, x_m\right) := \left(x_1, \ldots, x_m, 0, \ldots, 0 \right),$$ where there are $$n - m$$ trailing zeros.

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