<s>
In	O
linear	B-Language
algebra	I-Language
,	O
the	O
transpose	B-Algorithm
of	I-Algorithm
a	I-Algorithm
linear	I-Algorithm
map	I-Algorithm
between	O
two	O
vector	O
spaces	O
,	O
defined	O
over	O
the	O
same	O
field	O
,	O
is	O
an	O
induced	O
map	O
between	O
the	O
dual	O
spaces	O
of	O
the	O
two	O
vector	O
spaces	O
.	O
</s>
<s>
The	O
transpose	O
or	O
algebraic	B-Algorithm
adjoint	I-Algorithm
of	O
a	O
linear	B-Architecture
map	I-Architecture
is	O
often	O
used	O
to	O
study	O
the	O
original	O
linear	B-Architecture
map	I-Architecture
.	O
</s>
<s>
In	O
the	O
language	O
of	O
category	O
theory	O
,	O
taking	O
the	O
dual	O
of	O
vector	O
spaces	O
and	O
the	O
transpose	O
of	O
linear	B-Architecture
maps	I-Architecture
is	O
therefore	O
a	O
contravariant	O
functor	O
from	O
the	O
category	O
of	O
vector	O
spaces	O
over	O
to	O
itself	O
.	O
</s>
<s>
One	O
can	O
identify	O
with	O
using	O
the	O
natural	B-Algorithm
injection	I-Algorithm
into	O
the	O
double	O
dual	O
.	O
</s>
<s>
Suppose	O
now	O
that	O
is	O
a	O
weakly	O
continuous	O
linear	B-Architecture
operator	I-Architecture
between	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
and	O
with	O
continuous	O
dual	O
spaces	O
and	O
respectively	O
.	O
</s>
<s>
Suppose	O
and	O
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
and	O
is	O
a	O
weakly	O
continuous	O
linear	B-Architecture
operator	I-Architecture
(	O
so	O
)	O
.	O
</s>
<s>
The	O
kernel	B-Algorithm
of	O
is	O
the	O
subspace	O
of	O
orthogonal	O
to	O
the	O
image	O
of	O
:	O
</s>
<s>
The	O
linear	B-Architecture
map	I-Architecture
is	O
injective	O
if	O
and	O
only	O
if	O
its	O
image	O
is	O
a	O
weakly	O
dense	O
subset	O
of	O
(	O
that	O
is	O
,	O
the	O
image	O
of	O
is	O
dense	O
in	O
when	O
is	O
given	O
the	O
weak	O
topology	O
induced	O
by	O
)	O
.	O
</s>
<s>
both	O
endowed	O
with	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
,	O
both	O
endowed	O
with	O
the	O
topology	O
of	O
uniform	O
convergence	O
on	O
compact	O
convex	O
subsets	O
,	O
both	O
endowed	O
with	O
the	O
topology	O
of	O
uniform	O
convergence	O
on	O
compact	O
subsets	O
)	O
.	O
</s>
<s>
(	O
Surjection	B-Algorithm
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
)	O
:	O
If	O
and	O
are	O
Fréchet	B-Algorithm
spaces	I-Algorithm
then	O
the	O
continuous	O
linear	B-Architecture
operator	I-Architecture
is	O
surjective	B-Algorithm
if	O
and	O
only	O
if	O
(	O
1	O
)	O
the	O
transpose	O
is	O
injective	O
,	O
and	O
(	O
2	O
)	O
the	O
image	O
of	O
the	O
transpose	O
of	O
is	O
a	O
weakly	O
closed	O
(	O
i.e.	O
</s>
<s>
whose	O
kernel	B-Algorithm
is	O
the	O
annihilator	O
and	O
which	O
is	O
surjective	B-Algorithm
by	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
.	O
</s>
<s>
If	O
the	O
linear	B-Architecture
map	I-Architecture
is	O
represented	O
by	O
the	O
matrix	B-Architecture
with	O
respect	O
to	O
two	O
bases	O
of	O
and	O
then	O
is	O
represented	O
by	O
the	O
transpose	O
matrix	B-Architecture
with	O
respect	O
to	O
the	O
dual	O
bases	O
of	O
and	O
hence	O
the	O
name	O
.	O
</s>
<s>
Alternatively	O
,	O
as	O
is	O
represented	O
by	O
acting	O
to	O
the	O
right	O
on	O
column	O
vectors	O
,	O
is	O
represented	O
by	O
the	O
same	O
matrix	B-Architecture
acting	O
to	O
the	O
left	O
on	O
row	O
vectors	O
.	O
</s>
<s>
The	O
transpose	O
is	O
a	O
map	O
and	O
is	O
defined	O
for	O
linear	B-Architecture
maps	I-Architecture
between	O
any	O
vector	O
spaces	O
and	O
without	O
requiring	O
any	O
additional	O
structure	O
.	O
</s>
<s>
The	O
Hermitian	O
adjoint	O
maps	O
and	O
is	O
only	O
defined	O
for	O
linear	B-Architecture
maps	I-Architecture
between	O
Hilbert	O
spaces	O
,	O
as	O
it	O
is	O
defined	O
in	O
terms	O
of	O
the	O
inner	O
product	O
on	O
the	O
Hilbert	O
space	O
.	O
</s>
<s>
More	O
precisely	O
:	O
if	O
and	O
are	O
Hilbert	O
spaces	O
and	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
then	O
the	O
transpose	O
of	O
and	O
the	O
Hermitian	O
adjoint	O
of	O
which	O
we	O
will	O
denote	O
respectively	O
by	O
and	O
are	O
related	O
.	O
</s>
