<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
or	O
continuous	B-Algorithm
linear	I-Algorithm
mapping	I-Algorithm
is	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
transformation	I-Algorithm
between	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
An	O
operator	O
between	O
two	O
normed	O
spaces	O
is	O
a	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
if	O
and	O
only	O
if	O
it	O
is	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
between	O
two	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
.	O
</s>
<s>
is	O
continuous	B-Algorithm
.	O
</s>
<s>
If	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
and	O
are	O
both	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	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
sequential	O
space	O
(	O
such	O
as	O
a	O
pseudometrizable	B-Algorithm
space	I-Algorithm
)	O
then	O
this	O
list	O
may	O
be	O
extended	O
to	O
include	O
:	O
</s>
<s>
is	O
sequentially	O
continuous	B-Algorithm
at	O
some	O
(	O
or	O
equivalently	O
,	O
at	O
every	O
)	O
point	O
of	O
its	O
domain	O
.	O
</s>
<s>
If	O
is	O
pseudometrizable	B-Algorithm
or	O
metrizable	O
(	O
such	O
as	O
a	O
normed	O
or	O
Banach	O
space	O
)	O
then	O
we	O
may	O
add	O
to	O
this	O
list	O
:	O
</s>
<s>
is	O
a	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
(	O
that	O
is	O
,	O
it	O
maps	O
bounded	B-Algorithm
subsets	O
of	O
to	O
bounded	B-Algorithm
subsets	O
of	O
)	O
.	O
</s>
<s>
If	O
and	O
are	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
with	O
finite-dimensional	O
then	O
this	O
list	O
may	O
be	O
extended	O
to	O
include	O
:	O
</s>
<s>
Throughout	O
,	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
</s>
<s>
The	O
notion	O
of	O
"	O
bounded	B-Algorithm
set	I-Algorithm
"	O
for	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
that	O
of	O
being	O
a	O
von	B-Algorithm
Neumann	I-Algorithm
bounded	I-Algorithm
set	I-Algorithm
.	O
</s>
<s>
A	O
linear	B-Architecture
map	I-Architecture
is	O
bounded	B-Algorithm
on	O
a	O
set	O
if	O
and	O
only	O
if	O
it	O
is	O
bounded	B-Algorithm
on	O
for	O
every	O
(	O
because	O
and	O
any	O
translation	O
of	O
a	O
bounded	B-Algorithm
set	I-Algorithm
is	O
again	O
bounded	B-Algorithm
)	O
.	O
</s>
<s>
By	O
definition	O
,	O
a	O
linear	B-Architecture
map	I-Architecture
between	O
TVSs	O
is	O
said	O
to	O
be	O
and	O
is	O
called	O
a	O
if	O
for	O
every	O
(	O
von	O
Neumann	O
)	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
its	O
domain	O
,	O
is	O
a	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
it	O
codomain	O
;	O
or	O
said	O
more	O
briefly	O
,	O
if	O
it	O
is	O
bounded	B-Algorithm
on	O
every	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
its	O
domain	O
.	O
</s>
<s>
Explicitly	O
,	O
if	O
denotes	O
this	O
ball	O
then	O
is	O
a	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
if	O
and	O
only	O
if	O
is	O
a	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
if	O
is	O
also	O
a	O
(	O
semi	O
)	O
normed	O
space	O
then	O
this	O
happens	O
if	O
and	O
only	O
if	O
the	O
operator	O
norm	O
is	O
finite	O
.	O
</s>
<s>
Every	O
sequentially	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
is	O
bounded	B-Algorithm
.	O
</s>
<s>
It	O
is	O
""	O
(	O
of	O
some	O
point	O
)	O
if	O
there	O
exists	O
point	O
in	O
its	O
domain	O
at	O
which	O
it	O
is	O
locally	O
bounded	B-Algorithm
,	O
in	O
which	O
case	O
this	O
linear	B-Architecture
map	I-Architecture
is	O
necessarily	O
locally	O
bounded	B-Algorithm
at	O
point	O
of	O
its	O
domain	O
.	O
</s>
<s>
The	O
term	O
""	O
is	O
sometimes	O
used	O
to	O
refer	O
to	O
a	O
map	O
that	O
is	O
locally	O
bounded	B-Algorithm
at	O
every	O
point	O
of	O
its	O
domain	O
,	O
but	O
some	O
functional	B-Application
analysis	I-Application
authors	O
define	O
"	O
locally	O
bounded	B-Algorithm
"	O
to	O
instead	O
be	O
a	O
synonym	O
of	O
"	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
"	O
,	O
which	O
are	O
related	O
but	O
equivalent	O
concepts	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
this	O
article	O
will	O
avoid	O
the	O
term	O
"	O
locally	O
bounded	B-Algorithm
"	O
and	O
instead	O
say	O
"	O
locally	O
bounded	B-Algorithm
at	O
every	O
point	O
"	O
(	O
there	O
is	O
no	O
disagreement	O
about	O
the	O
definition	O
of	O
"	O
locally	O
bounded	B-Algorithm
"	O
)	O
.	O
</s>
<s>
A	O
linear	B-Architecture
map	I-Architecture
is	O
"	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
"	O
(	O
of	O
some	O
point	O
)	O
if	O
and	O
only	O
if	O
it	O
is	O
locally	O
bounded	B-Algorithm
at	O
every	O
point	O
of	O
its	O
domain	O
,	O
in	O
which	O
case	O
it	O
is	O
necessarily	O
continuous	B-Algorithm
(	O
even	O
if	O
its	O
domain	O
is	O
not	O
a	O
normed	O
space	O
)	O
and	O
thus	O
also	O
bounded	B-Algorithm
(	O
because	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
is	O
always	O
a	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
)	O
.	O
</s>
<s>
For	O
any	O
linear	B-Architecture
map	I-Architecture
,	O
if	O
it	O
is	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
then	O
it	O
is	O
continuous	B-Algorithm
,	O
and	O
if	O
it	O
is	O
continuous	B-Algorithm
then	O
it	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
The	O
converse	O
statements	O
are	O
not	O
true	O
in	O
general	O
but	O
they	O
are	O
both	O
true	O
when	O
the	O
linear	B-Architecture
map	I-Architecture
's	O
domain	O
is	O
a	O
normed	O
space	O
.	O
</s>
<s>
The	O
next	O
example	O
shows	O
that	O
it	O
is	O
possible	O
for	O
a	O
linear	B-Architecture
map	I-Architecture
to	O
be	O
continuous	B-Algorithm
(	O
and	O
thus	O
also	O
bounded	B-Algorithm
)	O
but	O
not	O
bounded	B-Algorithm
on	O
any	O
neighborhood	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
demonstrates	O
that	O
being	O
"	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
"	O
is	O
always	O
synonymous	O
with	O
being	O
"	O
bounded	B-Algorithm
"	O
.	O
</s>
<s>
:	O
If	O
is	O
the	O
identity	O
map	O
on	O
some	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
then	O
this	O
linear	B-Architecture
map	I-Architecture
is	O
always	O
continuous	B-Algorithm
(	O
indeed	O
,	O
even	O
a	O
TVS-isomorphism	O
)	O
and	O
bounded	B-Algorithm
,	O
but	O
is	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
if	O
and	O
only	O
if	O
there	O
exists	O
a	O
bounded	B-Algorithm
neighborhood	O
of	O
the	O
origin	O
in	O
which	O
is	O
equivalent	O
to	O
being	O
a	O
seminormable	O
space	O
(	O
which	O
if	O
is	O
Hausdorff	O
,	O
is	O
the	O
same	O
as	O
being	O
a	O
normable	O
space	O
)	O
.	O
</s>
<s>
This	O
shows	O
that	O
it	O
is	O
possible	O
for	O
a	O
linear	B-Architecture
map	I-Architecture
to	O
be	O
continuous	B-Algorithm
but	O
bounded	B-Algorithm
on	O
any	O
neighborhood	O
.	O
</s>
<s>
Indeed	O
,	O
this	O
example	O
shows	O
that	O
every	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
that	O
is	O
not	O
seminormable	O
has	O
a	O
linear	O
TVS-automorphism	O
that	O
is	O
not	O
bounded	B-Algorithm
on	O
any	O
neighborhood	O
of	O
any	O
point	O
.	O
</s>
<s>
Thus	O
although	O
every	O
linear	B-Architecture
map	I-Architecture
that	O
is	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
is	O
necessarily	O
continuous	B-Algorithm
,	O
the	O
converse	O
is	O
not	O
guaranteed	O
in	O
general	O
.	O
</s>
<s>
To	O
summarize	O
the	O
discussion	O
below	O
,	O
for	O
a	O
linear	B-Architecture
map	I-Architecture
on	O
a	O
normed	O
(	O
or	O
seminormed	O
)	O
space	O
,	O
being	O
continuous	B-Algorithm
,	O
being	O
bounded	B-Algorithm
,	O
and	O
being	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
are	O
all	O
equivalent	O
.	O
</s>
<s>
A	O
linear	B-Architecture
map	I-Architecture
whose	O
domain	O
codomain	O
is	O
normable	O
(	O
or	O
seminormable	O
)	O
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
.	O
</s>
<s>
And	O
a	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
valued	O
in	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
will	O
be	O
continuous	B-Algorithm
if	O
its	O
domain	O
is	O
(	O
pseudo	O
)	O
metrizable	O
or	O
bornological	B-Algorithm
.	O
</s>
<s>
A	O
TVS	B-Architecture
is	O
said	O
to	O
be	O
if	O
there	O
exists	O
a	O
neighborhood	O
that	O
is	O
also	O
a	O
bounded	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
every	O
normed	O
or	O
seminormed	O
space	O
is	O
a	O
locally	O
bounded	B-Algorithm
TVS	B-Architecture
since	O
the	O
unit	O
ball	O
centered	O
at	O
the	O
origin	O
is	O
a	O
bounded	B-Algorithm
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
If	O
is	O
a	O
bounded	B-Algorithm
neighborhood	O
of	O
the	O
origin	O
in	O
a	O
(	O
locally	O
bounded	B-Algorithm
)	O
TVS	B-Architecture
then	O
its	O
image	O
under	O
any	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
will	O
be	O
a	O
bounded	B-Algorithm
set	I-Algorithm
(	O
so	O
this	O
map	O
is	O
thus	O
bounded	B-Algorithm
on	O
this	O
neighborhood	O
)	O
.	O
</s>
<s>
Consequently	O
,	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
a	O
locally	O
bounded	B-Algorithm
TVS	B-Architecture
into	O
any	O
other	O
TVS	B-Architecture
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
.	O
</s>
<s>
Moreover	O
,	O
any	O
TVS	B-Architecture
with	O
this	O
property	O
must	O
be	O
a	O
locally	O
bounded	B-Algorithm
TVS	B-Architecture
.	O
</s>
<s>
Explicitly	O
,	O
if	O
is	O
a	O
TVS	B-Architecture
such	O
that	O
every	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
(	O
into	O
any	O
TVS	B-Architecture
)	O
whose	O
domain	O
is	O
is	O
necessarily	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
,	O
then	O
must	O
be	O
a	O
locally	O
bounded	B-Algorithm
TVS	B-Architecture
(	O
because	O
the	O
identity	O
function	O
is	O
always	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
)	O
.	O
</s>
<s>
Any	O
linear	B-Architecture
map	I-Architecture
from	O
a	O
TVS	B-Architecture
into	O
a	O
locally	O
bounded	B-Algorithm
TVS	B-Architecture
(	O
such	O
as	O
any	O
linear	B-Algorithm
functional	I-Algorithm
)	O
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
is	O
a	O
TVS	B-Architecture
such	O
that	O
every	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
(	O
from	O
any	O
TVS	B-Architecture
)	O
with	O
codomain	O
is	O
necessarily	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
,	O
then	O
must	O
be	O
a	O
locally	O
bounded	B-Algorithm
TVS	B-Architecture
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
a	O
arbitrary	O
TVS	B-Architecture
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
.	O
</s>
<s>
Thus	O
when	O
the	O
domain	O
the	O
codomain	O
of	O
a	O
linear	B-Architecture
map	I-Architecture
is	O
normable	O
or	O
seminormable	O
,	O
then	O
continuity	O
will	O
be	O
equivalent	O
to	O
being	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
.	O
</s>
<s>
A	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
is	O
always	O
a	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
.	O
</s>
<s>
But	O
importantly	O
,	O
in	O
the	O
most	O
general	O
setting	O
of	O
a	O
linear	B-Architecture
operator	I-Architecture
between	O
arbitrary	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
it	O
is	O
possible	O
for	O
a	O
linear	B-Architecture
operator	I-Architecture
to	O
be	O
bounded	B-Algorithm
but	O
to	O
be	O
continuous	B-Algorithm
.	O
</s>
<s>
A	O
linear	B-Architecture
map	I-Architecture
whose	O
domain	O
is	O
pseudometrizable	B-Algorithm
(	O
such	O
as	O
any	O
normed	O
space	O
)	O
is	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
The	O
same	O
is	O
true	O
of	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
a	O
bornological	B-Algorithm
space	I-Algorithm
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
without	O
additional	O
information	O
about	O
either	O
the	O
linear	B-Architecture
map	I-Architecture
or	O
its	O
domain	O
or	O
codomain	O
,	O
the	O
map	O
being	O
"	O
bounded	B-Algorithm
"	O
is	O
not	O
equivalent	O
to	O
it	O
being	O
"	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
"	O
.	O
</s>
<s>
If	O
is	O
a	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
from	O
a	O
normed	O
space	O
into	O
some	O
TVS	B-Architecture
then	O
is	O
necessarily	O
continuous	B-Algorithm
;	O
this	O
is	O
because	O
any	O
open	O
ball	O
centered	O
at	O
the	O
origin	O
in	O
is	O
both	O
a	O
bounded	B-Algorithm
subset	I-Algorithm
(	O
which	O
implies	O
that	O
is	O
bounded	B-Algorithm
since	O
is	O
a	O
bounded	B-Algorithm
linear	B-Architecture
map	I-Architecture
)	O
and	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
so	O
that	O
is	O
thus	O
bounded	B-Algorithm
on	O
this	O
neighborhood	O
of	O
the	O
origin	O
,	O
which	O
(	O
as	O
mentioned	O
above	O
)	O
guarantees	O
continuity	O
.	O
</s>
<s>
Every	O
linear	B-Algorithm
functional	I-Algorithm
on	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	B-Architecture
)	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
so	O
all	O
of	O
the	O
properties	O
described	O
above	O
for	O
continuous	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
apply	O
to	O
them	O
.	O
</s>
<s>
However	O
,	O
because	O
of	O
their	O
specialized	O
nature	O
,	O
we	O
can	O
say	O
even	O
more	O
about	O
continuous	B-Algorithm
linear	B-Algorithm
functionals	I-Algorithm
than	O
we	O
can	O
about	O
more	O
general	O
continuous	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
is	O
continuous	B-Algorithm
.	O
</s>
<s>
is	O
continuous	B-Algorithm
at	O
the	O
origin	O
.	O
</s>
<s>
This	O
is	O
one	O
of	O
several	O
reasons	O
why	O
many	O
definitions	O
involving	O
linear	B-Algorithm
functionals	I-Algorithm
,	O
such	O
as	O
polar	B-Algorithm
sets	I-Algorithm
for	O
example	O
,	O
involve	O
closed	O
(	O
rather	O
than	O
open	O
)	O
neighborhoods	O
and	O
non-strict	O
(	O
rather	O
than	O
strict	O
)	O
inequalities	O
.	O
</s>
<s>
is	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
(	O
of	O
some	O
point	O
)	O
.	O
</s>
<s>
Said	O
differently	O
,	O
is	O
a	O
locally	O
bounded	B-Algorithm
at	O
some	O
point	O
of	O
its	O
domain	O
.	O
</s>
<s>
Importantly	O
,	O
a	O
linear	B-Algorithm
functional	I-Algorithm
being	O
"	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
"	O
is	O
in	O
general	O
equivalent	O
to	O
being	O
a	O
"	O
bounded	B-Algorithm
linear	B-Algorithm
functional	I-Algorithm
"	O
because	O
(	O
as	O
described	O
above	O
)	O
it	O
is	O
possible	O
for	O
a	O
linear	B-Architecture
map	I-Architecture
to	O
be	O
bounded	B-Algorithm
but	O
continuous	B-Algorithm
.	O
</s>
<s>
However	O
,	O
continuity	O
and	O
boundedness	B-Algorithm
are	O
equivalent	O
if	O
the	O
domain	O
is	O
a	O
normed	O
or	O
seminormed	O
space	O
;	O
that	O
is	O
,	O
for	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
a	O
normed	O
space	O
,	O
being	O
"	O
bounded	B-Algorithm
"	O
is	O
equivalent	O
to	O
being	O
"	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
"	O
.	O
</s>
<s>
is	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Said	O
differently	O
,	O
is	O
a	O
locally	O
bounded	B-Algorithm
at	O
the	O
origin	O
.	O
</s>
<s>
By	O
definition	O
of	O
the	O
set	O
which	O
is	O
called	O
the	O
(	O
absolute	O
)	O
polar	O
of	O
the	O
inequality	O
holds	O
if	O
and	O
only	O
if	O
Polar	B-Algorithm
sets	I-Algorithm
,	O
and	O
so	O
also	O
this	O
particular	O
inequality	O
,	O
play	O
important	O
roles	O
in	O
duality	B-Algorithm
theory	I-Algorithm
.	O
</s>
<s>
is	O
a	O
locally	O
bounded	B-Algorithm
at	O
every	O
point	O
of	O
its	O
domain	O
.	O
</s>
<s>
In	O
particular	O
,	O
is	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
the	O
seminorm	O
is	O
a	O
continuous	B-Algorithm
.	O
</s>
<s>
The	O
imaginary	O
part	O
of	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
is	O
sequentially	O
continuous	B-Algorithm
at	O
some	O
(	O
or	O
equivalently	O
,	O
at	O
every	O
)	O
point	O
of	O
its	O
domain	O
.	O
</s>
<s>
If	O
the	O
domain	O
is	O
metrizable	B-Algorithm
or	I-Algorithm
pseudometrizable	I-Algorithm
(	O
for	O
example	O
,	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
or	O
a	O
normed	O
space	O
)	O
then	O
this	O
list	O
may	O
be	O
extended	O
to	O
include	O
:	O
</s>
<s>
is	O
a	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
(	O
that	O
is	O
,	O
it	O
maps	O
bounded	B-Algorithm
subsets	O
of	O
its	O
domain	O
to	O
bounded	B-Algorithm
subsets	O
of	O
its	O
codomain	O
)	O
.	O
</s>
<s>
If	O
the	O
domain	O
is	O
a	O
bornological	B-Algorithm
space	I-Algorithm
(	O
for	O
example	O
,	O
a	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
)	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>
is	O
a	O
bounded	B-Algorithm
linear	B-Architecture
operator	I-Architecture
.	O
</s>
<s>
is	O
sequentially	O
continuous	B-Algorithm
at	O
some	O
(	O
or	O
equivalently	O
,	O
at	O
every	O
)	O
point	O
of	O
its	O
domain	O
.	O
</s>
<s>
is	O
sequentially	O
continuous	B-Algorithm
at	O
the	O
origin	O
.	O
</s>
<s>
If	O
is	O
complex	O
then	O
either	O
all	O
three	O
of	O
and	O
are	O
continuous	B-Algorithm
(	O
respectively	O
,	O
bounded	B-Algorithm
)	O
,	O
or	O
else	O
all	O
three	O
are	O
discontinuous	B-Algorithm
(	O
respectively	O
,	O
unbounded	O
)	O
.	O
</s>
<s>
Every	O
linear	B-Architecture
map	I-Architecture
whose	O
domain	O
is	O
a	O
finite-dimensional	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	B-Architecture
)	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
This	O
is	O
not	O
true	O
if	O
the	O
finite-dimensional	O
TVS	B-Architecture
is	O
not	O
Hausdorff	O
.	O
</s>
<s>
Every	O
(	O
constant	O
)	O
map	O
between	O
TVS	B-Architecture
that	O
is	O
identically	O
equal	O
to	O
zero	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
that	O
is	O
continuous	B-Algorithm
,	O
bounded	B-Algorithm
,	O
and	O
bounded	B-Algorithm
on	O
the	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
In	O
particular	O
,	O
every	O
TVS	B-Architecture
has	O
a	O
non-empty	O
continuous	B-Algorithm
dual	O
space	O
(	O
although	O
it	O
is	O
possible	O
for	O
the	O
constant	O
zero	O
map	O
to	O
be	O
its	O
only	O
continuous	B-Algorithm
linear	B-Algorithm
functional	I-Algorithm
)	O
.	O
</s>
<s>
Suppose	O
is	O
any	O
Hausdorff	O
TVS	B-Architecture
.	O
</s>
<s>
Then	O
linear	B-Algorithm
functional	I-Algorithm
on	O
is	O
necessarily	O
continuous	B-Algorithm
if	O
and	O
only	O
if	O
every	O
vector	O
subspace	O
of	O
is	O
closed	O
.	O
</s>
<s>
Every	O
linear	B-Algorithm
functional	I-Algorithm
on	O
is	O
necessarily	O
a	O
bounded	B-Algorithm
linear	B-Algorithm
functional	I-Algorithm
if	O
and	O
only	O
if	O
every	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
is	O
contained	O
in	O
a	O
finite-dimensional	O
vector	O
subspace	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
is	O
normable	O
if	O
and	O
only	O
if	O
every	O
bounded	B-Algorithm
linear	B-Algorithm
functional	I-Algorithm
on	O
it	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
A	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
maps	O
bounded	B-Algorithm
sets	I-Algorithm
into	O
bounded	B-Algorithm
sets	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
complex	O
normed	O
space	O
and	O
is	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
then	O
(	O
where	O
in	O
particular	O
,	O
one	O
side	O
is	O
infinite	O
if	O
and	O
only	O
if	O
the	O
other	O
side	O
is	O
infinite	O
)	O
.	O
</s>
<s>
Every	O
non-trivial	O
continuous	B-Algorithm
linear	B-Algorithm
functional	I-Algorithm
on	O
a	O
TVS	B-Architecture
is	O
an	O
open	O
map	O
.	O
</s>
