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( T o p )
1
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2
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3
E x a m p l e s
T o g g l e E x a m p l e s s u b s e c t i o n
3 . 1
C o u n t e r - e x a m p l e s
4
S e e a l s o
5
C i t a t i o n s
6
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
L B - s p a c e
2 l a n g u a g e s
● D e u t s c h
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● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Definition [ edit ]
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 [ edit ]
A strict LB -space is complete , barrelled , and bornological (and thus ultrabornological ).
Examples [ edit ]
If
D
{\displaystyle 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
)
{\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 [ edit ]
There exists a bornological LB-space whose strong bidual is not bornological.
There exists an LB-space that is not quasi-complete .
See also [ edit ]
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 [ edit ]
References [ edit ]
Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions . Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag . ISBN 978-3-540-08662-8 . OCLC 297140003 .
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 .
Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 . Éléments de mathématique . Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4 . OCLC 17499190 .
Dugundji, James (1966). Topology . Boston: Allyn and Bacon. ISBN 978-0-697-06889-7 . OCLC 395340485 .
Edwards, Robert E. (1995). Functional Analysis: Theory and Applications . New York: Dover Publications. ISBN 978-0-486-68143-6 . OCLC 30593138 .
Grothendieck, Alexander (1955).『Produits Tensoriels Topologiques et Espaces Nucléaires』[Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). 16 . Providence: American Mathematical Society. ISBN 978-0-8218-1216-7 . MR 0075539 . OCLC 1315788 .
Horváth, John (1966). Topological Vector Spaces and Distributions . Addison-Wesley series in mathematics. Vol. 1. Reading, MA: Addison-Wesley Publishing Company. ISBN 978-0201029857 .
Jarchow, Hans (1981). Locally convex spaces . Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4 . OCLC 8210342 .
Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces . Lecture Notes in Mathematics . Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag . ISBN 978-3-540-11565-6 . OCLC 8588370 .
Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I . Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2 . MR 0248498 . OCLC 840293704 .
Köthe, Gottfried (1979). Topological Vector Spaces II . Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9 . OCLC 180577972 .
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces . Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666 . OCLC 144216834 .
Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces . Cambridge Tracts in Mathematics . Vol. 53. Cambridge England: Cambridge University Press . ISBN 978-0-521-29882-7 . OCLC 589250 .
Schaefer, Helmut H. ; Wolff, Manfred P. (1999). Topological Vector Spaces . GTM . Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0 . OCLC 840278135 .
Swartz, Charles (1992). An introduction to Functional Analysis . New York: M. Dekker. ISBN 978-0-8247-8643-4 . OCLC 24909067 .
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels . Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1 . OCLC 853623322 .
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces . Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4 . OCLC 849801114 .
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C a t e g o r y :
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H i d d e n c a t e g o r i e s :
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● T h i s p a g e w a s l a s t e d i t e d o n 1 4 F e b r u a r y 2 0 2 4 , a t 0 1 : 1 7 ( U T C ) .
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a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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