In mathematics, an LB-space, also written (LB)-space, is a topological vector space
that is a locally convex inductive limit of a countable inductive system
in the category of locally convex topological vector spaces and each
is an embedding of TVSs then the LB-space is called a strict LB-space.
This means that the topology induced on
is identical to the original topology on
[1] Some authors (e.g. Schaefer) define the term "LB-space" to mean "strict LB-space."
can be described by specifying that an absolutely convex subset
is an absolutely convex neighborhood of
A strict LB-space is complete,[2] barrelled,[2] and bornological[2] (and thus ultrabornological).
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
of all continuous, complex-valued functions on
with compact support is a strict LB-space.
denote the Banach space of complex-valued functions that are supported by
with the uniform norm and order the family of compact subsets of
[3] Let denote the space of finite sequences, where
denotes the space of all real sequences.
denote the usual Euclidean space endowed with the Euclidean topology and let
denote the canonical inclusion defined by
so that its image is and consequently, Endow the set
becomes a complete Hausdorff locally convex sequential topological vector space that is not a Fréchet–Urysohn space.
is strictly finer than the subspace topology induced on
is endowed with its usual product topology.
with the final topology induced on it by the bijection
that is, it is endowed with the Euclidean topology transferred to it from
is equal to the subspace topology induced on it by
is coherent with family of subspaces
is the canonical inclusion defined by
There exists a bornological LB-space whose strong bidual is not bornological.
[4] There exists an LB-space that is not quasi-complete.