<s>
In	O
mathematics	O
,	O
specifically	O
in	O
functional	B-Application
analysis	I-Application
and	O
Hilbert	O
space	O
theory	O
,	O
the	O
fundamental	B-Algorithm
theorem	I-Algorithm
of	I-Algorithm
Hilbert	I-Algorithm
spaces	I-Algorithm
gives	O
a	O
necessarily	O
and	O
sufficient	O
condition	O
for	O
a	O
Hausdorff	O
pre-Hilbert	O
space	O
to	O
be	O
a	O
Hilbert	O
space	O
in	O
terms	O
of	O
the	O
canonical	O
isometry	O
of	O
a	O
pre-Hilbert	O
space	O
into	O
its	O
anti-dual	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
.	O
</s>
<s>
A	O
function	O
is	O
called	O
semilinear	B-Algorithm
or	O
antilinear	B-Algorithm
if	O
for	O
all	O
and	O
all	O
scalars	O
,	O
</s>
<s>
The	O
vector	O
space	O
of	O
all	O
continuous	O
antilinear	B-Algorithm
functions	O
on	O
is	O
called	O
the	O
anti-dual	O
space	O
or	O
complex	O
conjugate	O
dual	O
space	O
of	O
and	O
is	O
denoted	O
by	O
(	O
in	O
contrast	O
,	O
the	O
continuous	O
dual	O
space	O
of	O
is	O
denoted	O
by	O
)	O
,	O
which	O
we	O
make	O
into	O
a	O
normed	O
space	O
by	O
endowing	O
it	O
with	O
the	O
canonical	O
norm	O
(	O
defined	O
in	O
the	O
same	O
way	O
as	O
the	O
canonical	O
norm	O
on	O
the	O
continuous	O
dual	O
space	O
of	O
)	O
.	O
</s>
<s>
A	O
sesquilinear	B-Algorithm
form	I-Algorithm
is	O
a	O
map	O
such	O
that	O
for	O
all	O
,	O
the	O
map	O
defined	O
by	O
is	O
linear	B-Architecture
,	O
and	O
for	O
all	O
,	O
the	O
map	O
defined	O
by	O
is	O
antilinear	B-Algorithm
.	O
</s>
<s>
Note	O
that	O
in	O
Physics	O
,	O
the	O
convention	O
is	O
that	O
a	O
sesquilinear	B-Algorithm
form	I-Algorithm
is	O
linear	B-Architecture
in	O
its	O
second	O
coordinate	O
and	O
antilinear	B-Algorithm
in	O
its	O
first	O
coordinate	O
.	O
</s>
<s>
A	O
sesquilinear	B-Algorithm
form	I-Algorithm
on	O
is	O
called	O
positive	O
definite	O
if	O
for	O
all	O
non-0	O
;	O
it	O
is	O
called	O
non-negative	O
if	O
for	O
all	O
.	O
</s>
<s>
A	O
sesquilinear	B-Algorithm
form	I-Algorithm
on	O
is	O
called	O
a	O
Hermitian	O
form	O
if	O
in	O
addition	O
it	O
has	O
the	O
property	O
that	O
for	O
all	O
.	O
</s>
<s>
A	O
pre-Hilbert	O
space	O
is	O
a	O
pair	O
consisting	O
of	O
a	O
vector	O
space	O
and	O
a	O
non-negative	O
sesquilinear	B-Algorithm
form	I-Algorithm
on	O
;	O
</s>
<s>
if	O
in	O
addition	O
this	O
sesquilinear	B-Algorithm
form	I-Algorithm
is	O
positive	O
definite	O
then	O
is	O
called	O
a	O
Hausdorff	O
pre-Hilbert	O
space	O
.	O
</s>
<s>
The	O
sesquilinear	B-Algorithm
form	I-Algorithm
is	O
separately	O
uniformly	O
continuous	O
in	O
each	O
of	O
its	O
two	O
arguments	O
and	O
hence	O
can	O
be	O
extended	O
to	O
a	O
separately	O
continuous	O
sesquilinear	B-Algorithm
form	I-Algorithm
on	O
the	O
completion	B-Algorithm
of	O
;	O
if	O
is	O
Hausdorff	O
then	O
this	O
completion	B-Algorithm
is	O
a	O
Hilbert	O
space	O
.	O
</s>
<s>
If	O
is	O
a	O
pre-Hilbert	O
space	O
then	O
this	O
canonical	O
map	O
is	O
linear	B-Architecture
and	O
continuous	O
;	O
</s>
<s>
There	O
is	O
of	O
course	O
a	O
canonical	O
antilinear	B-Algorithm
surjective	B-Algorithm
isometry	O
that	O
sends	O
a	O
continuous	O
linear	B-Architecture
functional	O
on	O
to	O
the	O
continuous	O
antilinear	B-Algorithm
functional	O
denoted	O
by	O
and	O
defined	O
by	O
.	O
</s>
<s>
Fundamental	B-Algorithm
theorem	I-Algorithm
of	I-Algorithm
Hilbert	I-Algorithm
spaces	I-Algorithm
:	O
Suppose	O
that	O
is	O
a	O
Hausdorff	O
pre-Hilbert	O
space	O
where	O
is	O
a	O
sesquilinear	B-Algorithm
form	I-Algorithm
that	O
is	O
linear	B-Architecture
in	O
its	O
first	O
coordinate	O
and	O
antilinear	B-Algorithm
in	O
its	O
second	O
coordinate	O
.	O
</s>
<s>
Then	O
the	O
canonical	O
linear	B-Architecture
mapping	I-Architecture
from	O
into	O
the	O
anti-dual	O
space	O
of	O
is	O
surjective	B-Algorithm
if	O
and	O
only	O
if	O
is	O
a	O
Hilbert	O
space	O
,	O
in	O
which	O
case	O
the	O
canonical	O
map	O
is	O
a	O
surjective	B-Algorithm
isometry	O
of	O
onto	O
its	O
anti-dual	O
.	O
</s>
