<s>
In	O
the	O
mathematical	O
fields	O
of	O
linear	B-Language
algebra	I-Language
and	O
functional	B-Application
analysis	I-Application
,	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
a	O
subspace	O
W	O
of	O
a	O
vector	O
space	O
V	O
equipped	O
with	O
a	O
bilinear	O
form	O
B	O
is	O
the	O
set	O
W⊥	O
of	O
all	O
vectors	O
in	O
V	O
that	O
are	O
orthogonal	O
to	O
every	O
vector	O
in	O
W	O
.	O
Informally	O
,	O
it	O
is	O
called	O
the	O
perp	O
,	O
short	O
for	O
perpendicular	O
complement	O
.	O
</s>
<s>
Finally	O
,	O
the	O
fact	O
that	O
these	O
spaces	O
are	O
orthogonal	B-Algorithm
complements	I-Algorithm
follows	O
from	O
the	O
dimension	O
relationships	O
given	O
below	O
.	O
</s>
<s>
There	O
is	O
a	O
corresponding	O
definition	O
of	O
right	O
orthogonal	B-Algorithm
complement	I-Algorithm
.	O
</s>
<s>
The	O
definition	O
extends	O
to	O
a	O
bilinear	O
form	O
on	O
a	O
free	O
module	O
over	O
a	O
commutative	O
ring	O
,	O
and	O
to	O
a	O
sesquilinear	B-Algorithm
form	I-Algorithm
extended	O
to	O
include	O
any	O
free	O
module	O
over	O
a	O
commutative	O
ring	O
with	O
conjugation	O
.	O
</s>
<s>
An	O
orthogonal	B-Algorithm
complement	I-Algorithm
is	O
a	O
subspace	O
of	O
;	O
</s>
<s>
The	O
radical	O
of	O
is	O
a	O
subspace	O
of	O
every	O
orthogonal	B-Algorithm
complement	I-Algorithm
;	O
</s>
<s>
The	O
orthogonal	B-Algorithm
complement	I-Algorithm
is	O
always	O
closed	O
in	O
the	O
metric	O
topology	O
.	O
</s>
<s>
In	O
infinite-dimensional	O
Hilbert	O
spaces	O
,	O
some	O
subspaces	O
are	O
not	O
closed	O
,	O
but	O
all	O
orthogonal	B-Algorithm
complements	I-Algorithm
are	O
closed	O
.	O
</s>
<s>
If	O
is	O
a	O
vector	O
subspace	O
of	O
an	O
inner	O
product	O
space	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
is	O
the	O
closure	O
of	O
that	O
is	O
,	O
</s>
<s>
The	O
orthogonal	B-Algorithm
complement	I-Algorithm
generalizes	O
to	O
the	O
annihilator	O
,	O
and	O
gives	O
a	O
Galois	O
connection	O
on	O
subsets	O
of	O
the	O
inner	O
product	O
space	O
,	O
with	O
associated	O
closure	O
operator	O
the	O
topological	O
closure	O
of	O
the	O
span	O
.	O
</s>
<s>
For	O
a	O
finite-dimensional	O
inner	O
product	O
space	O
of	O
dimension	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
a	O
-dimensional	O
subspace	O
is	O
an	O
-dimensional	O
subspace	O
,	O
and	O
the	O
double	O
orthogonal	B-Algorithm
complement	I-Algorithm
is	O
the	O
original	O
subspace	O
:	O
</s>
<s>
In	O
special	O
relativity	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
is	O
used	O
to	O
determine	O
the	O
simultaneous	O
hyperplane	O
at	O
a	O
point	O
of	O
a	O
world	O
line	O
.	O
</s>
