LB-space

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

If each of the bonding maps ${\displaystyle i_{nm}}${\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 ${\displaystyle X_{n}}${\displaystyle X_{n}} by ${\displaystyle X_{n+1}}${\displaystyle X_{n+1}} is identical to the original topology on ${\displaystyle X_{n}.}${\displaystyle X_{n}.}^[1] Some authors (e.g. Schaefer) define the term "LB-space" to mean "strict LB-space."

The topology on ${\displaystyle X}${\displaystyle X} can be described by specifying that an absolutely convex subset ${\displaystyle U}${\displaystyle U} is a neighborhood of ${\displaystyle 0}${\displaystyle 0} if and only if ${\displaystyle U\cap X_{n}}${\displaystyle U\cap X_{n}} is an absolutely convex neighborhood of ${\displaystyle 0}${\displaystyle 0} in ${\displaystyle X_{n}}${\displaystyle X_{n}} for every ${\displaystyle n.}${\displaystyle n.}

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

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

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

Let

${\displaystyle {\begin{alignedat}{4}\mathbb {R} ^{\infty }~&:=~\left\{\left(x_{1},x_{2},\ldots \right)\in \mathbb {R} ^{\mathbb {N} }~:~{\text{ all but finitely many }}x_{i}{\text{ are equal to 0 }}\right\},\end{alignedat}}}${\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 ${\displaystyle \mathbb {R} ^{\mathbb {N} }}${\displaystyle \mathbb {R} ^{\mathbb {N} }} denotes the space of all real sequences. For every natural number ${\displaystyle n\in \mathbb {N} ,}${\displaystyle n\in \mathbb {N} ,} let ${\displaystyle \mathbb {R} ^{n}}${\displaystyle \mathbb {R} ^{n}} denote the usual Euclidean space endowed with the Euclidean topology and let ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \mathbb {R} ^{\infty }}${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \mathbb {R} ^{\infty }} denote the canonical inclusion defined by ${\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}\left(x_{1},\ldots ,x_{n}\right):=\left(x_{1},\ldots ,x_{n},0,0,\ldots \right)}${\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

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

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

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

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

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

• 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 – Topological vector space

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• Topological vector spaces

• CS1 French-language sources (fr)