Examples
If
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. For any compact subset
let
denote the Banach space of complex-valued functions that are supported by
with the uniform norm and order the family of compact subsets of
by inclusion.
- 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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e159b9881b289c5ebd1beac792ba345e0533a79a)
denote the space of finite sequences, where
denotes the space of all real sequences.
For every natural number
let
denote the usual Euclidean space endowed with the Euclidean topology and let
denote the canonical inclusion defined by
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\))](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4dbf5cf90b70a8b4b14f2138af74c00ed6e5e90)
and consequently,
![{\displaystyle \mathbb {R} ^{\infty }=\bigcup _{n\in \mathbb {N} }\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n))\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6165abb0a1836d72dca3f32633b858d4adafed11)
Endow the set
with the final topology
induced by the family
of all canonical inclusions.
With this topology,
becomes a complete Hausdorff locally convex sequential topological vector space that is not a Fréchet–Urysohn space.
The topology
is strictly finer than the subspace topology induced on
by
where
is endowed with its usual product topology.
Endow the image
with the final topology induced on it by the bijection
that is, it is endowed with the Euclidean topology transferred to it from
via
This topology on
is equal to the subspace topology induced on it by
A subset
is open (resp. closed) in
if and only if for every
the set
is an open (resp. closed) subset of
The topology
is coherent with family of subspaces
This makes
into an LB-space.
Consequently, if
and
is a sequence in
then
in
if and only if there exists some
such that both
and
are contained in
and
in
Often, for every
the canonical inclusion
is used to identify
with its image
in
explicitly, the elements
and
are identified together.
Under this identification,
becomes a direct limit of the direct system
where for every
the map
is the canonical inclusion defined by
where there are
trailing zeros.
Counter-examples
There exists a bornological LB-space whose strong bidual is not bornological.
There exists an LB-space that is not quasi-complete.