LB-space

In mathematics, an LB-space, also written (LB)-space, is a topological vector space ${\displaystyle X}$ that is a locally convex inductive limit of a countable inductive system ${\displaystyle (X_{n},i_{nm})}$ of Banach spaces. This means that ${\displaystyle X}$ is a direct limit of a direct system ${\displaystyle \left(X_{n},i_{nm}\right)}$ in the category of locally convex topological vector spaces and each ${\displaystyle X_{n}}$ is a Banach space.

If each of the bonding maps ${\displaystyle i_{nm}}$ is an embedding of TVSs then the LB-space is called a strict LB-space. This means that the topology induced on ${\displaystyle X_{n}}$ by ${\displaystyle X_{n+1}}$ is identical to the original topology on ${\displaystyle X_{n}.}$ ^[1] 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 ${\displaystyle X}$ can be described by specifying that an absolutely convex subset ${\displaystyle U}$ is a neighborhood of ${\displaystyle 0}$ if and only if ${\displaystyle U\cap X_{n}}$ is an absolutely convex neighborhood of ${\displaystyle 0}$ in ${\displaystyle X_{n}}$ for every ${\displaystyle n.}$

Properties

A strict LB-space is complete, ^[2] barrelled, ^[2] and bornological ^[2] (and thus ultrabornological).

Examples

If ${\displaystyle D}$ 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 ${\displaystyle C_{c}(D)}$ of all continuous, complex-valued functions on ${\displaystyle D}$ with compact support is a strict LB-space. ^[3] For any compact subset ${\displaystyle K\subseteq D,}$ let ${\displaystyle C_{c}(K)}$ denote the Banach space of complex-valued functions that are supported by ${\displaystyle K}$ with the uniform norm and order the family of compact subsets of ${\displaystyle D}$ by inclusion. ^[3]

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

Let

${\displaystyle {\begin{alignedat}{4}\mathbb {R} ^{\infty }~&:=~\left\{\left(x_{1},x_{2},\ldots \right)\in \mathbb {R} ^{\mathbb {N} }~:~{\text{ all but finitely many }}x_{i}{\text{ are equal to 0 }}\right\},\end{alignedat}}}$

denote the space of finite sequences , where ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$ denotes the space of all real sequences. For every natural number ${\displaystyle n\in \mathbb {N} ,}$ let ${\displaystyle \mathbb {R} ^{n}}$ denote the usual Euclidean space endowed with the Euclidean topology and let ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \mathbb {R} ^{\infty }}$ denote the canonical inclusion defined by ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{n}\right):=\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)}$ so that its image is

${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)=\left\{\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)~:~x_{1},\ldots ,x_{n}\in \mathbb {R} \right\}=\mathbb {R} ^{n}\times \left\{(0,0,\ldots )\right\}}$

and consequently,

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

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

Often, for every ${\displaystyle n\in \mathbb {N} ,}$ the canonical inclusion ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}}$ is used to identify ${\displaystyle \mathbb {R} ^{n}}$ with its image ${\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}$ in ${\displaystyle \mathbb {R} ^{\infty };}$ explicitly, the elements ${\displaystyle \left(x_{1},\ldots ,x_{n}\right)\in \mathbb {R} ^{n}}$ and ${\displaystyle \left(x_{1},\ldots ,x_{n},0,0,0,\ldots \right)}$ are identified together. Under this identification, ${\displaystyle \left(\left(\mathbb {R} ^{\infty },\tau ^{\infty }\right),\left(\operatorname {In} _{\mathbb {R} ^{n}}\right)_{n\in \mathbb {N} }\right)}$ becomes a direct limit of the direct system ${\displaystyle \left(\left(\mathbb {R} ^{n}\right)_{n\in \mathbb {N} },\left(\operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}\right)_{m\leq n{\text{ in }}\mathbb {N} },\mathbb {N} \right),}$ where for every ${\displaystyle m\leq n,}$ the map ${\displaystyle \operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}$ is the canonical inclusion defined by ${\displaystyle \operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{m}\right):=\left(x_{1},\ldots ,x_{m},0,\ldots ,0\right),}$ where there are ${\displaystyle n-m}$ trailing zeros.

Counter-examples

There exists a bornological LB-space whose strong bidual is not bornological. ^[4] There exists an LB-space that is not quasi-complete. ^[4]

See also

• DF-space
• Direct limit
• Final topology  – finest topology making some functions continuous
• F-space  – Topological vector space with a complete translation-invariant metric
• LF-space

Citations

1. Schaefer & Wolff 1999, pp. 55–61.
2. Schaefer & Wolff 1999, pp. 60–63.
3. Schaefer & Wolff 1999, pp. 57–58.
4. Khaleelulla 1982, pp. 28–63.

References

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