<s>
In	O
mathematics	O
,	O
especially	O
functional	B-Application
analysis	I-Application
,	O
a	O
bornology	B-Algorithm
on	O
a	O
set	O
X	O
is	O
a	O
collection	O
of	O
subsets	O
of	O
X	O
satisfying	O
axioms	O
that	O
generalize	O
the	O
notion	O
of	O
boundedness	B-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
key	O
motivations	O
behind	O
bornologies	B-Algorithm
and	O
bornological	B-Algorithm
analysis	O
is	O
the	O
fact	O
that	O
bornological	B-Algorithm
spaces	I-Algorithm
provide	O
a	O
convenient	O
setting	O
for	O
homological	O
algebra	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
This	O
is	O
becausepg	O
9	O
the	O
category	O
of	O
bornological	B-Algorithm
spaces	I-Algorithm
is	O
additive	O
,	O
complete	O
,	O
cocomplete	O
,	O
and	O
has	O
a	O
tensor	O
product	O
adjoint	O
to	O
an	O
internal	O
hom	O
,	O
all	O
necessary	O
components	O
for	O
homological	O
algebra	O
.	O
</s>
<s>
Bornology	B-Algorithm
originates	O
from	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
There	O
are	O
two	O
natural	O
ways	O
of	O
studying	O
the	O
problems	O
of	O
functional	B-Application
analysis	I-Application
:	O
one	O
way	O
is	O
to	O
study	O
notions	O
related	O
to	O
topologies	O
(	O
vector	O
topologies	O
,	O
continuous	B-Algorithm
operators	I-Algorithm
,	O
open/compact	O
subsets	O
,	O
etc	O
.	O
)	O
</s>
<s>
and	O
the	O
other	O
is	O
to	O
study	O
notions	O
related	O
to	O
boundedness	B-Algorithm
(	O
vector	B-Algorithm
bornologies	I-Algorithm
,	O
bounded	B-Algorithm
operators	O
,	O
bounded	B-Algorithm
subsets	I-Algorithm
,	O
etc	O
.	O
</s>
<s>
For	O
normed	O
spaces	O
,	O
from	O
which	O
functional	B-Application
analysis	I-Application
arose	O
,	O
topological	O
and	O
bornological	B-Algorithm
notions	O
are	O
distinct	O
but	O
complementary	O
and	O
closely	O
related	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
unit	O
ball	O
centered	O
at	O
the	O
origin	O
is	O
both	O
a	O
neighborhood	O
of	O
the	O
origin	O
and	O
a	O
bounded	B-Algorithm
subset	O
.	O
</s>
<s>
Furthermore	O
,	O
a	O
subset	O
of	O
a	O
normed	O
space	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
(	O
respectively	O
,	O
is	O
a	O
bounded	B-Algorithm
set	I-Algorithm
)	O
exactly	O
when	O
it	O
contains	O
(	O
respectively	O
,	O
it	O
is	O
contained	O
in	O
)	O
a	O
non-zero	O
scalar	O
multiple	O
of	O
this	O
ball	O
;	O
so	O
this	O
is	O
one	O
instance	O
where	O
the	O
topological	O
and	O
bornological	B-Algorithm
notions	O
are	O
distinct	O
but	O
complementary	O
(	O
in	O
the	O
sense	O
that	O
their	O
definitions	O
differ	O
only	O
by	O
which	O
of	O
and	O
is	O
used	O
)	O
.	O
</s>
<s>
Other	O
times	O
,	O
the	O
distinction	O
between	O
topological	O
and	O
bornological	B-Algorithm
notions	O
may	O
even	O
be	O
unnecessary	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
linear	O
maps	O
between	O
normed	O
spaces	O
,	O
being	O
continuous	B-Algorithm
(	O
a	O
topological	O
notion	O
)	O
is	O
equivalent	O
to	O
being	O
bounded	B-Algorithm
(	O
a	O
bornological	B-Algorithm
notion	O
)	O
.	O
</s>
<s>
Although	O
the	O
distinction	O
between	O
topology	O
and	O
bornology	B-Algorithm
is	O
often	O
blurred	O
or	O
unnecessary	O
for	O
normed	O
space	O
,	O
it	O
becomes	O
more	O
important	O
when	O
studying	O
generalizations	O
of	O
normed	O
spaces	O
.	O
</s>
<s>
Nevertheless	O
,	O
bornology	B-Algorithm
and	O
topology	O
can	O
still	O
be	O
thought	O
of	O
as	O
two	O
necessary	O
,	O
distinct	O
,	O
and	O
complementary	O
aspects	O
of	O
one	O
and	O
the	O
same	O
reality	O
.	O
</s>
<s>
The	O
general	O
theory	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
arose	O
first	O
from	O
the	O
theory	O
normed	O
spaces	O
and	O
then	O
bornology	B-Algorithm
emerged	O
this	O
general	O
theory	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
although	O
bornology	B-Algorithm
has	O
since	O
become	O
recognized	O
as	O
a	O
fundamental	O
notion	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
Born	O
from	O
the	O
work	O
of	O
George	O
Mackey	O
(	O
after	O
whom	O
Mackey	B-Algorithm
spaces	I-Algorithm
are	O
named	O
)	O
,	O
the	O
importance	O
of	O
bounded	B-Algorithm
subsets	I-Algorithm
first	O
became	O
apparent	O
in	O
duality	B-Algorithm
theory	I-Algorithm
,	O
especially	O
because	O
of	O
the	O
Mackey	B-Algorithm
–	I-Algorithm
Arens	I-Algorithm
theorem	I-Algorithm
and	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
Starting	O
around	O
the	O
1950s	O
,	O
it	O
became	O
apparent	O
that	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
were	O
inadequate	O
for	O
the	O
study	O
of	O
certain	O
major	O
problems	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
multiplication	O
operation	O
of	O
some	O
important	O
topological	O
algebras	O
was	O
not	O
continuous	B-Algorithm
,	O
although	O
it	O
was	O
often	O
bounded	B-Algorithm
.	O
</s>
<s>
Other	O
major	O
problems	O
for	O
which	O
TVSs	O
were	O
found	O
to	O
be	O
inadequate	O
was	O
in	O
developing	O
a	O
more	O
general	O
theory	O
of	O
differential	O
calculus	O
,	O
generalizing	O
distributions	O
from	O
(	O
the	O
usual	O
)	O
scalar-valued	O
distributions	O
to	O
vector	O
or	O
operator-valued	O
distributions	O
,	O
and	O
extending	O
the	O
holomorphic	O
functional	B-Algorithm
calculus	I-Algorithm
of	O
Gelfand	O
(	O
which	O
is	O
primarily	O
concerted	O
with	O
Banach	O
algebras	O
or	O
locally	B-Algorithm
convex	I-Algorithm
algebras	O
)	O
to	O
a	O
broader	O
class	O
of	O
operators	O
,	O
including	O
those	O
whose	O
spectra	O
is	O
not	O
compact	O
.	O
</s>
<s>
Bornology	B-Algorithm
has	O
been	O
found	O
to	O
be	O
a	O
useful	O
tool	O
for	O
investigating	O
these	O
problems	O
and	O
others	O
,	O
including	O
problems	O
in	O
algebraic	O
geometry	O
and	O
general	O
topology	O
.	O
</s>
<s>
Elements	O
of	O
a	O
bornology	B-Algorithm
are	O
called	O
.	O
</s>
<s>
Stated	O
in	O
plain	O
English	O
,	O
this	O
says	O
that	O
subsets	O
of	O
bounded	B-Algorithm
sets	I-Algorithm
are	O
bounded	B-Algorithm
.	O
</s>
<s>
Assuming	O
(	O
1	O
)	O
,	O
this	O
condition	O
may	O
be	O
replaced	O
with	O
:	O
For	O
every	O
In	O
plain	O
English	O
,	O
this	O
says	O
that	O
every	O
point	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
In	O
plain	O
English	O
,	O
this	O
says	O
that	O
the	O
union	O
of	O
two	O
bounded	B-Algorithm
sets	I-Algorithm
is	O
a	O
bounded	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
Thus	O
a	O
bornology	B-Algorithm
can	O
equivalently	O
be	O
defined	O
as	O
a	O
downward	O
closed	O
cover	O
that	O
is	O
closed	O
under	O
binary	O
unions	O
.	O
</s>
<s>
Properties	O
(	O
1	O
)	O
and	O
(	O
2	O
)	O
imply	O
that	O
every	O
singleton	O
subset	O
of	O
is	O
an	O
element	O
of	O
every	O
bornology	B-Algorithm
on	O
property	O
(	O
3	O
)	O
,	O
in	O
turn	O
,	O
guarantees	O
that	O
the	O
same	O
is	O
true	O
of	O
every	O
finite	O
subset	O
of	O
In	O
other	O
words	O
,	O
points	O
and	O
finite	O
subsets	O
are	O
always	O
bounded	B-Algorithm
in	O
every	O
bornology	B-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
empty	O
set	O
is	O
always	O
bounded	B-Algorithm
.	O
</s>
<s>
Every	O
base	O
for	O
a	O
bornology	B-Algorithm
is	O
also	O
a	O
subbase	O
for	O
it	O
.	O
</s>
<s>
Such	O
an	O
intersection	O
of	O
bornologies	B-Algorithm
will	O
cover	O
because	O
every	O
bornology	B-Algorithm
on	O
contains	O
every	O
finite	O
subset	O
of	O
(	O
that	O
is	O
,	O
if	O
is	O
a	O
bornology	B-Algorithm
on	O
and	O
is	O
finite	O
then	O
)	O
.	O
</s>
<s>
Given	O
a	O
collection	O
of	O
subsets	O
of	O
the	O
smallest	O
bornology	B-Algorithm
on	O
containing	O
is	O
called	O
the	O
.	O
</s>
<s>
It	O
is	O
equal	O
to	O
the	O
intersection	O
of	O
all	O
bornologies	B-Algorithm
on	O
that	O
contain	O
as	O
a	O
subset	O
.	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
bounded	B-Algorithm
structures	O
.	O
</s>
<s>
Since	O
the	O
composition	O
of	O
two	O
locally	O
bounded	B-Algorithm
map	O
is	O
again	O
locally	O
bounded	B-Algorithm
,	O
it	O
is	O
clear	O
that	O
the	O
class	O
of	O
all	O
bounded	B-Algorithm
structures	O
forms	O
a	O
category	O
whose	O
morphisms	O
are	O
bounded	B-Algorithm
maps	O
.	O
</s>
<s>
An	O
isomorphism	O
in	O
this	O
category	O
is	O
called	O
a	O
and	O
it	O
is	O
a	O
bijective	B-Algorithm
locally	O
bounded	B-Algorithm
map	O
whose	O
inverse	O
is	O
also	O
locally	O
bounded	B-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
and	O
is	O
a	O
linear	O
map	O
.	O
</s>
<s>
is	O
a	O
(	O
locally	O
)	O
bounded	B-Algorithm
map	O
;	O
</s>
<s>
For	O
every	O
bornivorous	B-Algorithm
(	O
that	O
is	O
,	O
bounded	B-Algorithm
in	O
the	O
bornological	B-Algorithm
sense	O
)	O
disk	O
in	O
is	O
also	O
bornivorous	B-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
then	O
this	O
list	O
may	O
be	O
extended	O
to	O
include	O
:	O
</s>
<s>
takes	O
bounded	B-Algorithm
disks	O
to	O
bounded	B-Algorithm
disks	O
;	O
</s>
<s>
If	O
is	O
a	O
seminormed	O
space	O
and	O
is	O
locally	B-Algorithm
convex	I-Algorithm
then	O
this	O
list	O
may	O
be	O
extended	O
to	O
include	O
:	O
</s>
<s>
If	O
is	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
between	O
two	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
they	O
need	O
not	O
even	O
be	O
Hausdorff	O
)	O
,	O
then	O
it	O
is	O
a	O
bounded	B-Algorithm
linear	O
operator	O
(	O
when	O
and	O
have	O
their	O
von-Neumann	B-Algorithm
bornologies	I-Algorithm
)	O
.	O
</s>
<s>
A	O
sequentially	O
continuous	B-Algorithm
map	O
between	O
two	O
TVSs	O
is	O
necessarily	O
locally	O
bounded	B-Algorithm
.	O
</s>
<s>
For	O
any	O
set	O
the	O
power	O
set	O
of	O
is	O
a	O
bornology	B-Algorithm
on	O
called	O
the	O
.	O
</s>
<s>
For	O
any	O
set	O
the	O
set	O
of	O
all	O
finite	O
subsets	O
of	O
is	O
a	O
bornology	B-Algorithm
on	O
called	O
the	O
.	O
</s>
<s>
The	O
on	O
determined	O
by	O
these	O
maps	O
is	O
the	O
strongest	O
bornology	B-Algorithm
on	O
making	O
each	O
locally	O
bounded	B-Algorithm
.	O
</s>
<s>
The	O
on	O
determined	O
by	O
these	O
maps	O
is	O
the	O
weakest	O
bornology	B-Algorithm
on	O
making	O
each	O
locally	O
bounded	B-Algorithm
.	O
</s>
<s>
If	O
for	O
each	O
denotes	O
the	O
bornology	B-Algorithm
generated	O
by	O
then	O
this	O
bornology	B-Algorithm
is	O
equal	O
to	O
the	O
collection	O
of	O
all	O
subsets	O
of	O
of	O
the	O
form	O
where	O
each	O
and	O
all	O
but	O
finitely	O
many	O
are	O
empty	O
.	O
</s>
<s>
The	O
on	O
is	O
the	O
finest	O
bornology	B-Algorithm
on	O
making	O
the	O
inclusion	B-Algorithm
map	I-Algorithm
of	O
into	O
(	O
defined	O
by	O
)	O
locally	O
bounded	B-Algorithm
.	O
</s>
<s>
Let	O
be	O
an	O
-indexed	O
family	O
of	O
bounded	B-Algorithm
structures	O
,	O
let	O
and	O
for	O
each	O
let	O
denote	O
the	O
canonical	O
projection	O
.	O
</s>
<s>
That	O
is	O
,	O
it	O
is	O
the	O
strongest	O
bornology	B-Algorithm
on	O
making	O
each	O
of	O
the	O
canonical	O
projections	O
locally	O
bounded	B-Algorithm
.	O
</s>
<s>
Every	O
continuous	B-Algorithm
map	O
between	O
T1	O
spaces	O
is	O
bounded	B-Algorithm
with	O
respect	O
to	O
their	O
compact	O
bornologies	B-Algorithm
.	O
</s>
<s>
The	O
bornology	B-Algorithm
is	O
called	O
if	O
it	O
satisfies	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
:	O
</s>
<s>
The	O
bornology	B-Algorithm
is	O
called	O
if	O
is	O
both	O
open	O
and	O
closed	O
.	O
</s>
<s>
Every	O
compact	O
subset	O
of	O
a	O
locally	O
bounded	B-Algorithm
topological	O
space	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
then	O
the	O
set	O
of	O
all	O
bounded	B-Algorithm
subsets	I-Algorithm
of	O
form	O
a	O
bornology	B-Algorithm
(	O
indeed	O
,	O
even	O
a	O
vector	B-Algorithm
bornology	I-Algorithm
)	O
on	O
called	O
the	O
,	O
the	O
,	O
or	O
simply	O
of	O
and	O
is	O
referred	O
to	O
as	O
.	O
</s>
<s>
A	O
linear	O
map	O
between	O
two	O
bornological	B-Algorithm
spaces	I-Algorithm
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
bounded	B-Algorithm
(	O
with	O
respect	O
to	O
the	O
usual	O
bornologies	B-Algorithm
)	O
.	O
</s>
