• Source: Vector bornology

In mathematics, especially functional analysis, a bornology





B




{\displaystyle {\mathcal {B}}}

on a vector space



X


{\displaystyle X}

over a field




K

,


{\displaystyle \mathbb {K} ,}

where




K



{\displaystyle \mathbb {K} }

has a bornology ℬ




F



{\displaystyle \mathbb {F} }

, is called a vector bornology if





B




{\displaystyle {\mathcal {B}}}

makes the vector space operations into bounded maps.




Definitions






= Prerequisits

=

A bornology on a set



X


{\displaystyle X}

is a collection





B




{\displaystyle {\mathcal {B}}}

of subsets of



X


{\displaystyle X}

that satisfy all the following conditions:






B




{\displaystyle {\mathcal {B}}}

covers



X
;


{\displaystyle X;}

that is,



X
=
∪


B




{\displaystyle X=\cup {\mathcal {B}}}







B




{\displaystyle {\mathcal {B}}}

is stable under inclusions; that is, if



B
∈


B




{\displaystyle B\in {\mathcal {B}}}

and



A
⊆
B
,


{\displaystyle A\subseteq B,}

then



A
∈


B




{\displaystyle A\in {\mathcal {B}}}







B




{\displaystyle {\mathcal {B}}}

is stable under finite unions; that is, if




B

1


,
…
,

B

n


∈


B




{\displaystyle B_{1},\ldots ,B_{n}\in {\mathcal {B}}}

then




B

1


∪
⋯
∪

B

n


∈


B




{\displaystyle B_{1}\cup \cdots \cup B_{n}\in {\mathcal {B}}}


Elements of the collection





B




{\displaystyle {\mathcal {B}}}

are called





B




{\displaystyle {\mathcal {B}}}

-bounded or simply bounded sets if





B




{\displaystyle {\mathcal {B}}}

is understood.
The pair



(
X
,


B


)


{\displaystyle (X,{\mathcal {B}})}

is called a bounded structure or a bornological set.
A base or fundamental system of a bornology





B




{\displaystyle {\mathcal {B}}}

is a subset






B



0




{\displaystyle {\mathcal {B}}_{0}}

of





B




{\displaystyle {\mathcal {B}}}

such that each element of





B




{\displaystyle {\mathcal {B}}}

is a subset of some element of






B



0


.


{\displaystyle {\mathcal {B}}_{0}.}

Given a collection





S




{\displaystyle {\mathcal {S}}}

of subsets of



X
,


{\displaystyle X,}

the smallest bornology containing





S




{\displaystyle {\mathcal {S}}}

is called the bornology generated by





S


.


{\displaystyle {\mathcal {S}}.}


If



(
X
,


B


)


{\displaystyle (X,{\mathcal {B}})}

and



(
Y
,


C


)


{\displaystyle (Y,{\mathcal {C}})}

are bornological sets then their product bornology on



X
×
Y


{\displaystyle X\times Y}

is the bornology having as a base the collection of all sets of the form



B
×
C
,


{\displaystyle B\times C,}

where



B
∈


B




{\displaystyle B\in {\mathcal {B}}}

and



C
∈


C


.


{\displaystyle C\in {\mathcal {C}}.}


A subset of



X
×
Y


{\displaystyle X\times Y}

is bounded in the product bornology if and only if its image under the canonical projections onto



X


{\displaystyle X}

and



Y


{\displaystyle Y}

are both bounded.
If



(
X
,


B


)


{\displaystyle (X,{\mathcal {B}})}

and



(
Y
,


C


)


{\displaystyle (Y,{\mathcal {C}})}

are bornological sets then a function



f
:
X
→
Y


{\displaystyle f:X\to Y}

is said to be a locally bounded map or a bounded map (with respect to these bornologies) if it maps





B




{\displaystyle {\mathcal {B}}}

-bounded subsets of



X


{\displaystyle X}

to





C




{\displaystyle {\mathcal {C}}}

-bounded subsets of



Y
;


{\displaystyle Y;}

that is, if



f

(


B


)

⊆


C


.


{\displaystyle f\left({\mathcal {B}}\right)\subseteq {\mathcal {C}}.}


If in addition



f


{\displaystyle f}

is a bijection and




f

−
1




{\displaystyle f^{-1}}

is also bounded then



f


{\displaystyle f}

is called a bornological isomorphism.




= Vector bornology

=
Let



X


{\displaystyle X}

be a vector space over a field




K



{\displaystyle \mathbb {K} }

where




K



{\displaystyle \mathbb {K} }

has a bornology






B




K



.


{\displaystyle {\mathcal {B}}_{\mathbb {K} }.}


A bornology





B




{\displaystyle {\mathcal {B}}}

on



X


{\displaystyle X}

is called a vector bornology on



X


{\displaystyle X}

if it is stable under vector addition, scalar multiplication, and the formation of balanced hulls (i.e. if the sum of two bounded sets is bounded, etc.).
If



X


{\displaystyle X}

is a vector space and





B




{\displaystyle {\mathcal {B}}}

is a bornology on



X
,


{\displaystyle X,}

then the following are equivalent:






B




{\displaystyle {\mathcal {B}}}

is a vector bornology
Finite sums and balanced hulls of





B




{\displaystyle {\mathcal {B}}}

-bounded sets are





B




{\displaystyle {\mathcal {B}}}

-bounded
The scalar multiplication map




K

×
X
→
X


{\displaystyle \mathbb {K} \times X\to X}

defined by



(
s
,
x
)
↦
s
x


{\displaystyle (s,x)\mapsto sx}

and the addition map



X
×
X
→
X


{\displaystyle X\times X\to X}

defined by



(
x
,
y
)
↦
x
+
y
,


{\displaystyle (x,y)\mapsto x+y,}

are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets)
A vector bornology





B




{\displaystyle {\mathcal {B}}}

is called a convex vector bornology if it is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then





B


.


{\displaystyle {\mathcal {B}}.}


And a vector bornology





B




{\displaystyle {\mathcal {B}}}

is called separated if the only bounded vector subspace of



X


{\displaystyle X}

is the 0-dimensional trivial space



{
0
}
.


{\displaystyle \{0\}.}


Usually,




K



{\displaystyle \mathbb {K} }

is either the real or complex numbers, in which case a vector bornology





B




{\displaystyle {\mathcal {B}}}

on



X


{\displaystyle X}

will be called a convex vector bornology if





B




{\displaystyle {\mathcal {B}}}

has a base consisting of convex sets.




Characterizations


Suppose that



X


{\displaystyle X}

is a vector space over the field




F



{\displaystyle \mathbb {F} }

of real or complex numbers and





B




{\displaystyle {\mathcal {B}}}

is a bornology on



X
.


{\displaystyle X.}


Then the following are equivalent:






B




{\displaystyle {\mathcal {B}}}

is a vector bornology
addition and scalar multiplication are bounded maps
the balanced hull of every element of





B




{\displaystyle {\mathcal {B}}}

is an element of





B




{\displaystyle {\mathcal {B}}}

and the sum of any two elements of





B




{\displaystyle {\mathcal {B}}}

is again an element of





B




{\displaystyle {\mathcal {B}}}





Bornology on a topological vector space


If



X


{\displaystyle X}

is a topological vector space then the set of all bounded subsets of



X


{\displaystyle X}

from a vector bornology on



X


{\displaystyle X}

called the von Neumann bornology of



X


{\displaystyle X}

, the usual bornology, or simply the bornology of



X


{\displaystyle X}

and is referred to as natural boundedness.
In any locally convex topological vector space



X
,


{\displaystyle X,}

the set of all closed bounded disks form a base for the usual bornology of



X
.


{\displaystyle X.}


Unless indicated otherwise, it is always assumed that the real or complex numbers are endowed with the usual bornology.




Topology induced by a vector bornology


Suppose that



X


{\displaystyle X}

is a vector space over the field




K



{\displaystyle \mathbb {K} }

of real or complex numbers and





B




{\displaystyle {\mathcal {B}}}

is a vector bornology on



X
.


{\displaystyle X.}


Let





N




{\displaystyle {\mathcal {N}}}

denote all those subsets



N


{\displaystyle N}

of



X


{\displaystyle X}

that are convex, balanced, and bornivorous.
Then





N




{\displaystyle {\mathcal {N}}}

forms a neighborhood basis at the origin for a locally convex topological vector space topology.




Examples






= Locally convex space of bounded functions

=
Let




K



{\displaystyle \mathbb {K} }

be the real or complex numbers (endowed with their usual bornologies), let



(
T
,


B


)


{\displaystyle (T,{\mathcal {B}})}

be a bounded structure, and let



L
B
(
T
,

K

)


{\displaystyle LB(T,\mathbb {K} )}

denote the vector space of all locally bounded




K



{\displaystyle \mathbb {K} }

-valued maps on



T
.


{\displaystyle T.}


For every



B
∈


B


,


{\displaystyle B\in {\mathcal {B}},}

let




p

B


(
f
)
:=
sup

|

f
(
B
)

|



{\displaystyle p_{B}(f):=\sup \left|f(B)\right|}

for all



f
∈
L
B
(
T
,

K

)
,


{\displaystyle f\in LB(T,\mathbb {K} ),}

where this defines a seminorm on



X
.


{\displaystyle X.}


The locally convex topological vector space topology on



L
B
(
T
,

K

)


{\displaystyle LB(T,\mathbb {K} )}

defined by the family of seminorms




{


p

B


:
B
∈


B



}



{\displaystyle \left\{p_{B}:B\in {\mathcal {B}}\right\}}

is called the topology of uniform convergence on bounded set.
This topology makes



L
B
(
T
,

K

)


{\displaystyle LB(T,\mathbb {K} )}

into a complete space.




= Bornology of equicontinuity

=
Let



T


{\displaystyle T}

be a topological space,




K



{\displaystyle \mathbb {K} }

be the real or complex numbers, and let



C
(
T
,

K

)


{\displaystyle C(T,\mathbb {K} )}

denote the vector space of all continuous




K



{\displaystyle \mathbb {K} }

-valued maps on



T
.


{\displaystyle T.}


The set of all equicontinuous subsets of



C
(
T
,

K

)


{\displaystyle C(T,\mathbb {K} )}

forms a vector bornology on



C
(
T
,

K

)
.


{\displaystyle C(T,\mathbb {K} ).}





See also


Bornivorous set
Bornological space
Bornology
Space of linear maps
Ultrabornological space




Citations






Bibliography

Kata Kunci Pencarian:

• Vector bornology
• Bornological space
• Bornology
• Bornivorous set
• Bounded set (topological vector space)
• Ultrabornological space
• Montel space
• Absorbing set
• Nuclear space
• Convenient vector space