<s>
In	O
mathematics	O
,	O
linear	B-Architecture
maps	I-Architecture
form	O
an	O
important	O
class	O
of	O
"	O
simple	O
"	O
functions	O
which	O
preserve	O
the	O
algebraic	O
structure	O
of	O
linear	O
spaces	O
and	O
are	O
often	O
used	O
as	O
approximations	O
to	O
more	O
general	O
functions	O
(	O
see	O
linear	B-Algorithm
approximation	I-Algorithm
)	O
.	O
</s>
<s>
If	O
the	O
spaces	O
involved	O
are	O
also	O
topological	O
spaces	O
(	O
that	O
is	O
,	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
)	O
,	O
then	O
it	O
makes	O
sense	O
to	O
ask	O
whether	O
all	O
linear	B-Architecture
maps	I-Architecture
are	O
continuous	O
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
for	O
maps	O
defined	O
on	O
infinite-dimensional	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
e.g.	O
,	O
infinite-dimensional	O
normed	O
spaces	O
)	O
,	O
the	O
answer	O
is	O
generally	O
no	O
:	O
there	O
exist	O
discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
Let	O
X	O
and	O
Y	O
be	O
two	O
normed	O
spaces	O
and	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
X	O
to	O
Y	O
.	O
</s>
<s>
Thus	O
,	O
is	O
a	O
bounded	O
linear	B-Architecture
operator	I-Architecture
and	O
so	O
is	O
continuous	O
.	O
</s>
<s>
Examples	O
of	O
discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
are	O
easy	O
to	O
construct	O
in	O
spaces	O
that	O
are	O
not	O
complete	O
;	O
on	O
any	O
Cauchy	O
sequence	O
of	O
linearly	O
independent	O
vectors	O
which	O
does	O
not	O
have	O
a	O
limit	O
,	O
there	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
such	O
that	O
the	O
quantities	O
grow	O
without	O
bound	O
.	O
</s>
<s>
In	O
a	O
sense	O
,	O
the	O
linear	B-Architecture
operators	I-Architecture
are	O
not	O
continuous	O
because	O
the	O
space	O
has	O
"	O
holes	O
"	O
.	O
</s>
<s>
Note	O
that	O
T	O
is	O
real-valued	O
,	O
and	O
so	O
is	O
actually	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
X	O
(	O
an	O
element	O
of	O
the	O
algebraic	O
dual	O
space	O
X*	O
)	O
.	O
</s>
<s>
The	O
linear	B-Architecture
map	I-Architecture
X	O
→	O
X	O
which	O
assigns	O
to	O
each	O
function	O
its	O
derivative	B-Algorithm
is	O
similarly	O
discontinuous	O
.	O
</s>
<s>
Note	O
that	O
although	O
the	O
derivative	B-Algorithm
operator	O
is	O
not	O
continuous	O
,	O
it	O
is	O
closed	O
.	O
</s>
<s>
This	O
example	O
can	O
be	O
extended	O
into	O
a	O
general	O
theorem	O
about	O
the	O
existence	O
of	O
discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
on	O
any	O
infinite-dimensional	O
normed	O
space	O
(	O
as	O
long	O
as	O
the	O
codomain	O
is	O
not	O
trivial	O
)	O
.	O
</s>
<s>
Discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
can	O
be	O
proven	O
to	O
exist	O
more	O
generally	O
,	O
even	O
if	O
the	O
space	O
is	O
complete	O
.	O
</s>
<s>
We	O
will	O
find	O
a	O
discontinuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
f	O
from	O
X	O
to	O
K	O
,	O
which	O
will	O
imply	O
the	O
existence	O
of	O
a	O
discontinuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
g	O
from	O
X	O
to	O
Y	O
given	O
by	O
the	O
formula	O
where	O
is	O
an	O
arbitrary	O
nonzero	O
vector	O
in	O
Y	O
.	O
</s>
<s>
If	O
X	O
is	O
infinite-dimensional	O
,	O
to	O
show	O
the	O
existence	O
of	O
a	B-Algorithm
linear	I-Algorithm
functional	I-Algorithm
which	I-Algorithm
is	I-Algorithm
not	I-Algorithm
continuous	I-Algorithm
then	O
amounts	O
to	O
constructing	O
f	O
which	O
is	O
not	O
bounded	O
.	O
</s>
<s>
T	O
so	O
defined	O
will	O
extend	O
uniquely	O
to	O
a	O
linear	B-Architecture
map	I-Architecture
on	O
X	O
,	O
and	O
since	O
it	O
is	O
clearly	O
not	O
bounded	O
,	O
it	O
is	O
not	O
continuous	O
.	O
</s>
<s>
As	O
noted	O
above	O
,	O
the	O
axiom	O
of	O
choice	O
(	O
AC	O
)	O
is	O
used	O
in	O
the	O
general	O
existence	O
theorem	O
of	O
discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
there	O
are	O
no	O
constructive	O
examples	O
of	O
discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
with	O
complete	O
domain	O
(	O
for	O
example	O
,	O
Banach	O
spaces	O
)	O
.	O
</s>
<s>
In	O
analysis	O
as	O
it	O
is	O
usually	O
practiced	O
by	O
working	O
mathematicians	O
,	O
the	O
axiom	O
of	O
choice	O
is	O
always	O
employed	O
(	O
it	O
is	O
an	O
axiom	O
of	O
ZFC	O
set	O
theory	O
)	O
;	O
thus	O
,	O
to	O
the	O
analyst	O
,	O
all	O
infinite-dimensional	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
admit	O
discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
Solovay	O
's	O
result	O
shows	O
that	O
it	O
is	O
not	O
necessary	O
to	O
assume	O
that	O
all	O
infinite-dimensional	O
vector	O
spaces	O
admit	O
discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
,	O
and	O
there	O
are	O
schools	O
of	O
analysis	O
which	O
adopt	O
a	O
more	O
constructivist	O
viewpoint	O
.	O
</s>
<s>
For	O
example	O
,	O
H	O
.	O
G	O
.	O
Garnir	O
,	O
in	O
searching	O
for	O
so-called	O
"	O
dream	O
spaces	O
"	O
(	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
on	O
which	O
every	O
linear	B-Architecture
map	I-Architecture
into	O
a	O
normed	O
space	O
is	O
continuous	O
)	O
,	O
was	O
led	O
to	O
adopt	O
ZF	O
+	O
DC	O
+	O
BP	O
(	O
dependent	O
choice	O
is	O
a	O
weakened	O
form	O
and	O
the	O
Baire	O
property	O
is	O
a	O
negation	O
of	O
strong	O
AC	O
)	O
as	O
his	O
axioms	O
to	O
prove	O
the	O
Garnir	O
–	O
Wright	O
closed	O
graph	O
theorem	O
which	O
states	O
,	O
among	O
other	O
things	O
,	O
that	O
any	O
linear	B-Architecture
map	I-Architecture
from	O
an	O
F-space	B-Algorithm
to	O
a	O
TVS	O
is	O
continuous	O
.	O
</s>
<s>
The	O
upshot	O
is	O
that	O
the	O
existence	O
of	O
discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
depends	O
on	O
AC	O
;	O
it	O
is	O
consistent	O
with	O
set	O
theory	O
without	O
AC	O
that	O
there	O
are	O
no	O
discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
on	O
complete	O
spaces	O
.	O
</s>
<s>
In	O
particular	O
,	O
no	O
concrete	O
construction	O
such	O
as	O
the	O
derivative	B-Algorithm
can	O
succeed	O
in	O
defining	O
a	O
discontinuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
everywhere	O
on	O
a	O
complete	O
space	O
.	O
</s>
<s>
It	O
makes	O
sense	O
to	O
ask	O
which	O
linear	B-Architecture
operators	I-Architecture
on	O
a	O
given	O
space	O
are	O
closed	O
.	O
</s>
<s>
To	O
be	O
more	O
concrete	O
,	O
let	O
be	O
a	O
map	O
from	O
to	O
with	O
domain	O
written	O
We	O
do	O
n't	O
lose	O
much	O
if	O
we	O
replace	O
X	O
by	O
the	O
closure	O
of	O
That	O
is	O
,	O
in	O
studying	O
operators	O
that	O
are	O
not	O
everywhere-defined	O
,	O
one	O
may	O
restrict	O
one	O
's	O
attention	O
to	O
densely	B-Algorithm
defined	I-Algorithm
operators	I-Algorithm
without	O
loss	O
of	O
generality	O
.	O
</s>
<s>
So	O
the	O
natural	O
question	O
to	O
ask	O
about	O
linear	B-Architecture
operators	I-Architecture
that	O
are	O
not	O
everywhere-defined	O
is	O
whether	O
they	O
are	O
closable	O
.	O
</s>
<s>
The	O
answer	O
is	O
,	O
"	O
not	O
necessarily	O
"	O
;	O
indeed	O
,	O
every	O
infinite-dimensional	O
normed	O
space	O
admits	O
linear	B-Architecture
operators	I-Architecture
that	O
are	O
not	O
closable	O
.	O
</s>
<s>
In	O
fact	O
,	O
there	O
is	O
even	O
an	O
example	O
of	O
a	O
linear	B-Architecture
operator	I-Architecture
whose	O
graph	O
has	O
closure	O
all	O
of	O
Such	O
an	O
operator	O
is	O
not	O
closable	O
.	O
</s>
<s>
As	O
a	O
consequence	O
of	O
the	O
Stone	O
–	O
Weierstrass	O
theorem	O
,	O
the	O
graph	O
of	O
this	O
operator	O
is	O
dense	O
in	O
so	O
this	O
provides	O
a	O
sort	O
of	O
maximally	O
discontinuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
(	O
confer	O
nowhere	O
continuous	O
function	O
)	O
.	O
</s>
<s>
The	O
dual	O
space	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
the	O
collection	O
of	O
continuous	O
linear	B-Architecture
maps	I-Architecture
from	O
the	O
space	O
into	O
the	O
underlying	O
field	O
.	O
</s>
<s>
Thus	O
the	O
failure	O
of	O
some	O
linear	B-Architecture
maps	I-Architecture
to	O
be	O
continuous	O
for	O
infinite-dimensional	O
normed	O
spaces	O
implies	O
that	O
for	O
these	O
spaces	O
,	O
one	O
needs	O
to	O
distinguish	O
the	O
algebraic	O
dual	O
space	O
from	O
the	O
continuous	O
dual	O
space	O
which	O
is	O
then	O
a	O
proper	O
subset	O
.	O
</s>
<s>
The	O
argument	O
for	O
the	O
existence	O
of	O
discontinuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
on	O
normed	O
spaces	O
can	O
be	O
generalized	O
to	O
all	O
metrizable	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
especially	O
to	O
all	O
Fréchet	O
spaces	O
,	O
but	O
there	O
exist	O
infinite-dimensional	O
locally	O
convex	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
such	O
that	O
every	O
functional	O
is	O
continuous	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
,	O
which	O
applies	O
to	O
all	O
locally	O
convex	O
spaces	O
,	O
guarantees	O
the	O
existence	O
of	O
many	O
continuous	O
linear	B-Algorithm
functionals	I-Algorithm
,	O
and	O
so	O
a	O
large	O
dual	O
space	O
.	O
</s>
<s>
In	O
fact	O
,	O
to	O
every	O
convex	O
set	O
,	O
the	O
Minkowski	O
gauge	O
associates	O
a	O
continuous	O
linear	B-Algorithm
functional	I-Algorithm
.	O
</s>
