<s>
In	O
mathematics	O
,	O
particularly	O
in	O
functional	B-Application
analysis	I-Application
and	O
topology	B-Architecture
,	O
closed	B-Algorithm
graph	I-Algorithm
is	O
a	O
property	O
of	O
functions	O
.	O
</s>
<s>
A	O
function	O
between	O
topological	O
spaces	O
has	O
a	O
closed	B-Algorithm
graph	I-Algorithm
if	O
its	O
graph	B-Application
is	O
a	O
closed	O
subset	O
of	O
the	O
product	O
space	O
.	O
</s>
<s>
A	O
related	O
property	O
is	O
open	B-Algorithm
graph	I-Algorithm
.	O
</s>
<s>
This	O
property	O
is	O
studied	O
because	O
there	O
are	O
many	O
theorems	O
,	O
known	O
as	O
closed	B-Algorithm
graph	I-Algorithm
theorems	O
,	O
giving	O
conditions	O
under	O
which	O
a	O
function	O
with	O
a	O
closed	B-Algorithm
graph	I-Algorithm
is	O
necessarily	O
continuous	O
.	O
</s>
<s>
One	O
particularly	O
well-known	O
class	O
of	O
closed	B-Algorithm
graph	I-Algorithm
theorems	O
are	O
the	O
closed	B-Algorithm
graph	I-Algorithm
theorems	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
Definition	O
:	O
If	O
and	O
are	O
sets	O
,	O
a	O
set-valued	O
function	O
in	O
on	O
(	O
also	O
called	O
a	O
-valued	O
multifunction	O
on	O
)	O
is	O
a	O
function	O
with	O
domain	B-Algorithm
that	O
is	O
valued	O
in	O
.	O
</s>
<s>
We	O
give	O
the	O
more	O
general	O
definition	O
of	O
when	O
a	O
-valued	O
function	O
or	O
set-valued	O
function	O
defined	O
on	O
a	O
subset	O
of	O
has	O
a	O
closed	B-Algorithm
graph	I-Algorithm
since	O
this	O
generality	O
is	O
needed	O
in	O
the	O
study	O
of	O
closed	O
linear	B-Architecture
operators	I-Architecture
that	O
are	O
defined	O
on	O
a	O
dense	O
subspace	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
and	O
not	O
necessarily	O
defined	O
on	O
all	O
of	O
)	O
.	O
</s>
<s>
This	O
particular	O
case	O
is	O
one	O
of	O
the	O
main	O
reasons	O
why	O
functions	O
with	O
closed	B-Algorithm
graphs	I-Algorithm
are	O
studied	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
will	O
always	O
be	O
endowed	O
with	O
the	O
product	O
topology	B-Architecture
.	O
</s>
<s>
Definition	O
:	O
We	O
say	O
that	O
has	O
a	O
closed	B-Algorithm
graph	I-Algorithm
(	O
resp	O
.	O
</s>
<s>
open	B-Algorithm
graph	I-Algorithm
,	O
sequentially	O
closed	B-Algorithm
graph	I-Algorithm
,	O
sequentially	O
open	B-Algorithm
graph	I-Algorithm
)	O
in	O
if	O
the	O
graph	B-Application
of	O
,	O
,	O
is	O
a	O
closed	O
(	O
resp	O
.	O
</s>
<s>
open	O
,	O
sequentially	O
closed	O
,	O
sequentially	O
open	O
)	O
subset	O
of	O
when	O
is	O
endowed	O
with	O
the	O
product	O
topology	B-Architecture
.	O
</s>
<s>
sequentially	O
closed	O
,	O
open	O
,	O
sequentially	O
open	O
)	O
graph	B-Application
in	O
if	O
and	O
only	O
if	O
the	O
same	O
is	O
true	O
of	O
.	O
</s>
<s>
set-valued	O
function	O
)	O
whose	O
graph	B-Application
is	O
equal	O
to	O
the	O
closure	O
of	O
the	O
set	O
in	O
.	O
</s>
<s>
Additional	O
assumptions	O
for	O
linear	B-Architecture
maps	I-Architecture
:	O
If	O
in	O
addition	O
,	O
,	O
,	O
and	O
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
and	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
then	O
to	O
call	O
closable	O
we	O
also	O
require	O
that	O
the	O
set	O
be	O
a	O
vector	O
subspace	O
of	O
and	O
the	O
closure	O
of	O
be	O
a	O
linear	B-Architecture
map	I-Architecture
.	O
</s>
<s>
Definition	O
:	O
If	O
is	O
closable	O
on	O
then	O
a	O
core	O
or	O
essential	O
domain	B-Algorithm
of	O
is	O
a	O
subset	O
such	O
that	O
the	O
closure	O
in	O
of	O
the	O
graph	B-Application
of	O
the	O
restriction	O
of	O
to	O
is	O
equal	O
to	O
the	O
closure	O
of	O
the	O
graph	B-Application
of	O
in	O
(	O
i.e.	O
</s>
<s>
Definition	O
and	O
notation	O
:	O
When	O
we	O
write	O
then	O
we	O
mean	O
that	O
is	O
a	O
-valued	O
function	O
with	O
domain	B-Algorithm
where	O
.	O
</s>
<s>
sequentially	O
closed	O
)	O
or	O
has	O
a	O
closed	B-Algorithm
graph	I-Algorithm
(	O
resp	O
.	O
</s>
<s>
has	O
a	O
sequentially	O
closed	B-Algorithm
graph	I-Algorithm
)	O
then	O
we	O
mean	O
that	O
the	O
graph	B-Application
of	O
is	O
closed	O
(	O
resp	O
.	O
</s>
<s>
When	O
reading	O
literature	O
in	O
functional	B-Application
analysis	I-Application
,	O
if	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
between	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
(	O
e.g.	O
</s>
<s>
Definition	O
:	O
A	O
map	O
is	O
called	O
closed	O
if	O
its	O
graph	B-Application
is	O
closed	O
in	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
term	O
"	O
closed	O
linear	B-Architecture
operator	I-Architecture
"	O
will	O
almost	O
certainly	O
refer	O
to	O
a	O
linear	B-Architecture
map	I-Architecture
whose	O
graph	B-Application
is	O
closed	O
.	O
</s>
<s>
Otherwise	O
,	O
especially	O
in	O
literature	O
about	O
point-set	O
topology	B-Architecture
,	O
"	O
is	O
closed	O
"	O
may	O
instead	O
mean	O
the	O
following	O
:	O
</s>
<s>
has	O
a	O
closed	B-Algorithm
graph	I-Algorithm
(	O
in	O
)	O
;	O
</s>
<s>
(	O
definition	O
)	O
the	O
graph	B-Application
of	O
,	O
,	O
is	O
a	O
closed	O
subset	O
of	O
;	O
</s>
<s>
Thus	O
to	O
show	O
that	O
the	O
function	O
has	O
a	O
closed	B-Algorithm
graph	I-Algorithm
we	O
may	O
assume	O
that	O
converges	O
in	O
to	O
some	O
(	O
and	O
then	O
show	O
that	O
)	O
while	O
to	O
show	O
that	O
is	O
continuous	O
we	O
may	O
not	O
assume	O
that	O
converges	O
in	O
to	O
some	O
and	O
we	O
must	O
instead	O
prove	O
that	O
this	O
is	O
true	O
(	O
and	O
moreover	O
,	O
we	O
must	O
more	O
specifically	O
prove	O
that	O
converges	O
to	O
in	O
)	O
.	O
</s>
<s>
has	O
a	O
sequentially	O
closed	B-Algorithm
graph	I-Algorithm
(	O
in	O
)	O
;	O
</s>
<s>
has	O
a	O
sequentially	O
closed	B-Algorithm
graph	I-Algorithm
(	O
in	O
)	O
;	O
</s>
<s>
(	O
definition	O
)	O
the	O
graph	B-Application
of	O
is	O
a	O
sequentially	O
closed	O
subset	O
of	O
;	O
</s>
<s>
has	O
a	O
closed	B-Algorithm
graph	I-Algorithm
(	O
in	O
)	O
;	O
</s>
<s>
(	O
definition	O
)	O
the	O
graph	B-Application
of	O
is	O
a	O
closed	O
subset	O
of	O
;	O
</s>
<s>
If	O
is	O
a	O
continuous	O
function	O
between	O
topological	O
spaces	O
and	O
if	O
is	O
Hausdorff	O
then	O
has	O
a	O
closed	B-Algorithm
graph	I-Algorithm
in	O
.	O
</s>
<s>
Note	O
that	O
if	O
is	O
a	O
function	O
between	O
Hausdorff	O
topological	O
spaces	O
then	O
it	O
is	O
possible	O
for	O
to	O
have	O
a	O
closed	B-Algorithm
graph	I-Algorithm
in	O
but	O
not	O
be	O
continuous	O
.	O
</s>
<s>
Conditions	O
that	O
guarantee	O
that	O
a	O
function	O
with	O
a	O
closed	B-Algorithm
graph	I-Algorithm
is	O
necessarily	O
continuous	O
are	O
called	O
closed	B-Algorithm
graph	I-Algorithm
theorems	O
.	O
</s>
<s>
Closed	B-Algorithm
graph	I-Algorithm
theorems	O
are	O
of	O
particular	O
interest	O
in	O
functional	B-Application
analysis	I-Application
where	O
there	O
are	O
many	O
theorems	O
giving	O
conditions	O
under	O
which	O
a	O
linear	B-Architecture
map	I-Architecture
with	O
a	O
closed	B-Algorithm
graph	I-Algorithm
is	O
necessarily	O
continuous	O
.	O
</s>
<s>
If	O
is	O
a	O
function	O
between	O
topological	O
spaces	O
whose	O
graph	B-Application
is	O
closed	O
in	O
and	O
if	O
is	O
a	O
compact	O
space	O
then	O
is	O
continuous	O
.	O
</s>
<s>
Let	O
denote	O
the	O
real	O
numbers	O
with	O
the	O
usual	O
Euclidean	O
topology	B-Architecture
and	O
let	O
denote	O
with	O
the	O
indiscrete	O
topology	B-Architecture
(	O
where	O
note	O
that	O
is	O
not	O
Hausdorff	O
and	O
that	O
every	O
function	O
valued	O
in	O
is	O
continuous	O
)	O
.	O
</s>
<s>
Then	O
is	O
continuous	O
but	O
its	O
graph	B-Application
is	O
not	O
closed	O
in	O
.	O
</s>
<s>
If	O
is	O
any	O
space	O
then	O
the	O
identity	O
map	O
is	O
continuous	O
but	O
its	O
graph	B-Application
,	O
which	O
is	O
the	O
diagonal	O
,	O
is	O
closed	O
in	O
if	O
and	O
only	O
if	O
is	O
Hausdorff	O
.	O
</s>
<s>
If	O
is	O
a	O
continuous	O
map	O
whose	O
graph	B-Application
is	O
not	O
closed	O
then	O
is	O
not	O
a	O
Hausdorff	O
space	O
.	O
</s>
<s>
Let	O
and	O
both	O
denote	O
the	O
real	O
numbers	O
with	O
the	O
usual	O
Euclidean	O
topology	B-Architecture
.	O
</s>
<s>
Then	O
has	O
a	O
closed	B-Algorithm
graph	I-Algorithm
(	O
and	O
a	O
sequentially	O
closed	B-Algorithm
graph	I-Algorithm
)	O
in	O
but	O
it	O
is	O
not	O
continuous	O
(	O
since	O
it	O
has	O
a	O
discontinuity	O
at	O
)	O
.	O
</s>
<s>
Let	O
denote	O
the	O
real	O
numbers	O
with	O
the	O
usual	O
Euclidean	O
topology	B-Architecture
,	O
let	O
denote	O
with	O
the	O
discrete	O
topology	B-Architecture
,	O
and	O
let	O
be	O
the	O
identity	O
map	O
(	O
i.e.	O
</s>
<s>
Then	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
whose	O
graph	B-Application
is	O
closed	O
in	O
but	O
it	O
is	O
clearly	O
not	O
continuous	O
(	O
since	O
singleton	O
sets	O
are	O
open	O
in	O
but	O
not	O
in	O
)	O
.	O
</s>
<s>
Let	O
be	O
a	O
Hausdorff	O
TVS	O
and	O
let	O
be	O
a	O
vector	O
topology	B-Architecture
on	O
that	O
is	O
strictly	O
finer	O
than	O
.	O
</s>
<s>
Then	O
the	O
identity	O
map	O
a	O
closed	O
discontinuous	O
linear	B-Architecture
operator	I-Architecture
.	O
</s>
<s>
Every	O
continuous	O
linear	B-Architecture
operator	I-Architecture
valued	O
in	O
a	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
has	O
a	O
closed	B-Algorithm
graph	I-Algorithm
and	O
recall	O
that	O
a	O
linear	B-Architecture
operator	I-Architecture
between	O
two	O
normed	O
spaces	O
is	O
continuous	O
if	O
and	O
only	O
if	O
it	O
is	O
bounded	O
.	O
</s>
<s>
Definition	O
:	O
If	O
and	O
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
then	O
we	O
call	O
a	O
linear	B-Architecture
map	I-Architecture
a	O
closed	O
linear	B-Architecture
operator	I-Architecture
if	O
its	O
graph	B-Application
is	O
closed	O
in	O
.	O
</s>
<s>
The	O
closed	B-Algorithm
graph	I-Algorithm
theorem	O
states	O
that	O
any	O
closed	O
linear	B-Architecture
operator	I-Architecture
between	O
two	O
F-spaces	B-Algorithm
(	O
such	O
as	O
Banach	O
spaces	O
)	O
is	O
continuous	O
,	O
where	O
recall	O
that	O
if	O
and	O
are	O
Banach	O
spaces	O
then	O
being	O
continuous	O
is	O
equivalent	O
to	O
being	O
bounded	O
.	O
</s>
<s>
The	O
following	O
properties	O
are	O
easily	O
checked	O
for	O
a	O
linear	B-Architecture
operator	I-Architecture
between	O
Banach	O
spaces	O
:	O
</s>
<s>
If	O
is	O
closed	O
,	O
then	O
its	O
kernel	B-Algorithm
(	O
or	O
nullspace	B-Algorithm
)	O
is	O
a	O
closed	O
vector	O
subspace	O
of	O
;	O
</s>
<s>
A	O
linear	B-Architecture
operator	I-Architecture
admits	O
a	O
closure	O
if	O
and	O
only	O
if	O
for	O
every	O
and	O
every	O
pair	O
of	O
sequences	O
and	O
in	O
both	O
converging	O
to	O
in	O
,	O
such	O
that	O
both	O
and	O
converge	O
in	O
,	O
one	O
has	O
.	O
</s>
<s>
Consider	O
the	O
derivative	B-Algorithm
operator	O
where	O
is	O
the	O
Banach	O
space	O
of	O
all	O
continuous	O
functions	O
on	O
an	O
interval	O
.	O
</s>
<s>
If	O
one	O
takes	O
its	O
domain	B-Algorithm
to	O
be	O
,	O
then	O
is	O
a	O
closed	O
operator	O
,	O
which	O
is	O
not	O
bounded	O
.	O
</s>
