In mathematics , an LB -space , also written (LB )-space , is a topological vector space
X
{\displaystyle X}
that is a locally convex
inductive limit
of a countable inductive system
(
X
n
,
i
n
m
)
{\displaystyle (X_{n},i_{nm})}
of
Banach spaces .
This means that
X
{\displaystyle X}
is a
direct limit of a direct system
(
X
n
,
i
n
m
)
{\displaystyle \left(X_{n},i_{nm}\right)}
in the category of
locally convex topological vector spaces and each
X
n
{\displaystyle X_{n}}
is a Banach space.
If each of the bonding maps
i
n
m
{\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
X
n
{\displaystyle X_{n}}
by
X
n
+
1
{\displaystyle X_{n+1}}
is identical to the original topology on
X
n
.
{\displaystyle X_{n}.}
Some authors (e.g. Schaefer) define the term "LB -space" to mean "strict LB -space."
Definition
The topology on
X
{\displaystyle X}
can be described by specifying that an absolutely convex subset
U
{\displaystyle U}
is a neighborhood of
0
{\displaystyle 0}
if and only if
U
∩
X
n
{\displaystyle U\cap X_{n}}
is an absolutely convex neighborhood of
0
{\displaystyle 0}
in
X
n
{\displaystyle X_{n}}
for every
n
.
{\displaystyle n.}
Properties
A strict LB -space is complete , barrelled , and bornological (and thus ultrabornological ).
Examples
If
D
{\displaystyle D}
is a locally compact
countable at infinity
(that is, it is equal to a countable union of compact subspaces) then the space
C
c
(
D
)
{\displaystyle C_{c}(D)}
of all continuous, complex-valued functions on
D
{\displaystyle D}
with
compact support is a strict
LB -space. For any compact subset
K
⊆
D
,
{\displaystyle K\subseteq D,}
let
C
c
(
K
)
{\displaystyle C_{c}(K)}
denote the Banach space of complex-valued functions that are supported by
K
{\displaystyle K}
with the uniform norm and order the family of compact subsets of
D
{\displaystyle D}
by inclusion.
Final topology on the direct limit of finite-dimensional Euclidean spaces
Let
R
∞
:=
{
(
x
1
,
x
2
,
…
)
∈
R
N
:
all but finitely many
x
i
are equal to 0
}
,
{\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
R
N
{\displaystyle \mathbb {R} ^{\mathbb {N} }}
denotes the
space of all real sequences
.
For every
natural number
n
∈
N
,
{\displaystyle n\in \mathbb {N} ,}
let
R
n
{\displaystyle \mathbb {R} ^{n}}
denote the usual
Euclidean space endowed with the
Euclidean topology and let
In
R
n
:
R
n
→
R
∞
{\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}:\mathbb {R} ^{n}\to \mathbb {R} ^{\infty }}
denote the canonical inclusion defined by
In
R
n
(
x
1
,
…
,
x
n
)
:=
(
x
1
,
…
,
x
n
,
0
,
0
,
…
)
{\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
Im
(
In
R
n
)
=
{
(
x
1
,
…
,
x
n
,
0
,
0
,
…
)
:
x
1
,
…
,
x
n
∈
R
}
=
R
n
×
{
(
0
,
0
,
…
)
}
{\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,
R
∞
=
⋃
n
∈
N
Im
(
In
R
n
)
.
{\displaystyle \mathbb {R} ^{\infty }=\bigcup _{n\in \mathbb {N} }\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}
Endow the set
R
∞
{\displaystyle \mathbb {R} ^{\infty }}
with the final topology
τ
∞
{\displaystyle \tau ^{\infty }}
induced by the family
F
:=
{
In
R
n
:
n
∈
N
}
{\displaystyle {\mathcal {F}}:=\left\{\;\operatorname {In} _{\mathbb {R} ^{n}}~:~n\in \mathbb {N} \;\right\}}
of all canonical inclusions.
With this topology,
R
∞
{\displaystyle \mathbb {R} ^{\infty }}
becomes a complete Hausdorff locally convex sequential topological vector space that is not a Fréchet–Urysohn space .
The topology
τ
∞
{\displaystyle \tau ^{\infty }}
is strictly finer than the subspace topology induced on
R
∞
{\displaystyle \mathbb {R} ^{\infty }}
by
R
N
,
{\displaystyle \mathbb {R} ^{\mathbb {N} },}
where
R
N
{\displaystyle \mathbb {R} ^{\mathbb {N} }}
is endowed with its usual product topology .
Endow the image
Im
(
In
R
n
)
{\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}
with the final topology induced on it by the bijection
In
R
n
:
R
n
→
Im
(
In
R
n
)
;
{\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
R
n
{\displaystyle \mathbb {R} ^{n}}
via
In
R
n
.
{\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}.}
This topology on
Im
(
In
R
n
)
{\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}
is equal to the subspace topology induced on it by
(
R
∞
,
τ
∞
)
.
{\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right).}
A subset
S
⊆
R
∞
{\displaystyle S\subseteq \mathbb {R} ^{\infty }}
is open (resp. closed) in
(
R
∞
,
τ
∞
)
{\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}
if and only if for every
n
∈
N
,
{\displaystyle n\in \mathbb {N} ,}
the set
S
∩
Im
(
In
R
n
)
{\displaystyle S\cap \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}
is an open (resp. closed) subset of
Im
(
In
R
n
)
.
{\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}
The topology
τ
∞
{\displaystyle \tau ^{\infty }}
is coherent with family of subspaces
S
:=
{
Im
(
In
R
n
)
:
n
∈
N
}
.
{\displaystyle \mathbb {S} :=\left\{\;\operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)~:~n\in \mathbb {N} \;\right\}.}
This makes
(
R
∞
,
τ
∞
)
{\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}
into an LB-space.
Consequently, if
v
∈
R
∞
{\displaystyle v\in \mathbb {R} ^{\infty }}
and
v
∙
{\displaystyle v_{\bullet }}
is a sequence in
R
∞
{\displaystyle \mathbb {R} ^{\infty }}
then
v
∙
→
v
{\displaystyle v_{\bullet }\to v}
in
(
R
∞
,
τ
∞
)
{\displaystyle \left(\mathbb {R} ^{\infty },\tau ^{\infty }\right)}
if and only if there exists some
n
∈
N
{\displaystyle n\in \mathbb {N} }
such that both
v
{\displaystyle v}
and
v
∙
{\displaystyle v_{\bullet }}
are contained in
Im
(
In
R
n
)
{\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}
and
v
∙
→
v
{\displaystyle v_{\bullet }\to v}
in
Im
(
In
R
n
)
.
{\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right).}
Often, for every
n
∈
N
,
{\displaystyle n\in \mathbb {N} ,}
the canonical inclusion
In
R
n
{\displaystyle \operatorname {In} _{\mathbb {R} ^{n}}}
is used to identify
R
n
{\displaystyle \mathbb {R} ^{n}}
with its image
Im
(
In
R
n
)
{\displaystyle \operatorname {Im} \left(\operatorname {In} _{\mathbb {R} ^{n}}\right)}
in
R
∞
;
{\displaystyle \mathbb {R} ^{\infty };}
explicitly, the elements
(
x
1
,
…
,
x
n
)
∈
R
n
{\displaystyle \left(x_{1},\ldots ,x_{n}\right)\in \mathbb {R} ^{n}}
and
(
x
1
,
…
,
x
n
,
0
,
0
,
0
,
…
)
{\displaystyle \left(x_{1},\ldots ,x_{n},0,0,0,\ldots \right)}
are identified together.
Under this identification,
(
(
R
∞
,
τ
∞
)
,
(
In
R
n
)
n
∈
N
)
{\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
(
(
R
n
)
n
∈
N
,
(
In
R
m
R
n
)
m
≤
n
in
N
,
N
)
,
{\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
m
≤
n
,
{\displaystyle m\leq n,}
the map
In
R
m
R
n
:
R
m
→
R
n
{\displaystyle \operatorname {In} _{\mathbb {R} ^{m}}^{\mathbb {R} ^{n}}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}
is the canonical inclusion defined by
In
R
m
R
n
(
x
1
,
…
,
x
m
)
:=
(
x
1
,
…
,
x
m
,
0
,
…
,
0
)
,
{\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
n
−
m
{\displaystyle n-m}
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.
See also
DF-space – class of special local-convex spacePages displaying wikidata descriptions as a fallback
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
Citations
References
Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions . Lecture Notes in Mathematics. Vol. 639. Berlin New York: .
Bierstedt, Klaus-Dieter (1988). "An Introduction to Locally Convex Inductive Limits" . Functional Analysis and Applications . Singapore-New Jersey-Hong Kong: Universitätsbibliothek: 35–133. Retrieved 20 September 2020 .
.
.
Edwards, Robert E. (1995). Functional Analysis: Theory and Applications . New York: Dover Publications. .
.
.
Jarchow, Hans (1981). Locally convex spaces . Stuttgart: B.G. Teubner. .
Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces . .
.
.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces . Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. .
Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces . .
.
Swartz, Charles (1992). An introduction to Functional Analysis . New York: M. Dekker. .
.
.
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