LB-space explained

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} ~:~ allbutfinitelymany x _{i} areequalto0 \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} x \left\{ ( 0 , 0 , \ldots) \right\}

and consequently,

\R ^{infty} = c up _{n} \operatorname{Im} \left( \operatorname{In} _{\R ^{n}} \right).

Endow the set

\R ^{infty} with the final topology \tau ^{infty} induced by the family l{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 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 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} \right) becomes a direct limit of the direct system \left( \left(\R ^{n\right)} _{n \in \N} , \left(\operatorname{In} ^{\R ^{n}} _{\R ^{m}} \right) _{m} , \N \right), where for every m \leq n, the map \operatorname{In} ^{\R ^{n}} _{\R ^{m}} : \R ^{m} \to \R ^{n} is the canonical inclusion defined by \operatorname{In} ^{\R ^{n}} _{\R ^{m}} \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.