LB-space

In mathematics, an "LB"-space, also written ("LB")-space, is a topological vector space formula_1 that is a locally convex inductive limit of a countable inductive system formula_2 of Banach spaces. 
This means that formula_1 is a direct limit of a direct system formula_4 in the category of locally convex topological vector spaces and each formula_5 is a Banach space.
If each of the bonding maps formula_6 is an embedding of TVSs then the "LB"-space is called a strict "LB"-space. This means that the topology induced on formula_5 by formula_8 is identical to the original topology on formula_9 
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 formula_1 can be described by specifying that an absolutely convex subset formula_11 is a neighborhood of formula_12 if and only if formula_13 is an absolutely convex neighborhood of formula_12 in formula_5 for every formula_16
Properties.
A strict "LB"-space is complete, barrelled, and bornological (and thus ultrabornological).
Examples.
If formula_17 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 formula_18 of all continuous, complex-valued functions on formula_17 with compact support is a strict "LB"-space. For any compact subset formula_20 let formula_21 denote the Banach space of complex-valued functions that are supported by formula_22 with the uniform norm and order the family of compact subsets of formula_17 by inclusion.
Let 
denote the , where formula_25 denotes the space of all real sequences. 
For every natural number formula_26 let formula_27 denote the usual Euclidean space endowed with the Euclidean topology and let formula_28 denote the canonical inclusion defined by formula_29 so that its image is
and consequently,
Endow the set formula_32 with the final topology formula_33 induced by the family formula_34 of all canonical inclusions. 
With this topology, formula_32 becomes a complete Hausdorff locally convex sequential topological vector space that is a Fréchet–Urysohn space. 
The topology formula_33 is strictly finer than the subspace topology induced on formula_32 by formula_38 where formula_25 is endowed with its usual product topology. 
Endow the image formula_40 with the final topology induced on it by the bijection formula_41 that is, it is endowed with the Euclidean topology transferred to it from formula_27 via formula_43 
This topology on formula_40 is equal to the subspace topology induced on it by formula_45 
A subset formula_46 is open (resp. closed) in formula_47 if and only if for every formula_26 the set formula_49 is an open (resp. closed) subset of formula_50 
The topology formula_33 is coherent with family of subspaces formula_52 
This makes formula_47 into an LB-space. 
Consequently, if formula_54 and formula_55 is a sequence in formula_32 then formula_57 in formula_47 if and only if there exists some formula_59 such that both formula_60 and formula_55 are contained in formula_40 and formula_57 in formula_50 
Often, for every formula_26 the canonical inclusion formula_66 is used to identify formula_27 with its image formula_40 in formula_69 explicitly, the elements formula_70 and formula_71 are identified together. 
Under this identification, formula_72 becomes a direct limit of the direct system formula_73 where for every formula_74 the map formula_75 is the canonical inclusion defined by formula_76 where there are formula_77 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.