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 $X_n.$ 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

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

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

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

and consequently,

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

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

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

Counter-examples

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

See also

• DF-space
• Direct limit – Special case of colimit in category theory
• Final topology – Finest topology making some functions continuous
• F-space – Topological vector space with a complete translation-invariant metric
• LF-space

References

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