<s>
In	O
linear	B-Language
algebra	I-Language
,	O
two	O
vectors	O
in	O
an	O
inner	O
product	O
space	O
are	O
orthonormal	B-Algorithm
if	O
they	O
are	O
orthogonal	B-Application
(	O
or	O
perpendicular	O
along	O
a	O
line	O
)	O
unit	O
vectors	O
.	O
</s>
<s>
A	O
set	O
of	O
vectors	O
form	O
an	O
orthonormal	B-Algorithm
set	I-Algorithm
if	O
all	O
vectors	O
in	O
the	O
set	O
are	O
mutually	O
orthogonal	B-Application
and	O
all	O
of	O
unit	O
length	O
.	O
</s>
<s>
An	O
orthonormal	B-Algorithm
set	I-Algorithm
which	O
forms	O
a	O
basis	O
is	O
called	O
an	O
orthonormal	B-Algorithm
basis	O
.	O
</s>
<s>
The	O
construction	O
of	O
orthogonality	B-Application
of	O
vectors	O
is	O
motivated	O
by	O
a	O
desire	O
to	O
extend	O
the	O
intuitive	O
notion	O
of	O
perpendicular	O
vectors	O
to	O
higher-dimensional	O
spaces	O
.	O
</s>
<s>
This	O
definition	O
can	O
be	O
formalized	O
in	O
Cartesian	O
space	O
by	O
defining	O
the	O
dot	O
product	O
and	O
specifying	O
that	O
two	O
vectors	O
in	O
the	O
plane	O
are	O
orthogonal	B-Application
if	O
their	O
dot	O
product	O
is	O
zero	O
.	O
</s>
<s>
Many	O
important	O
results	O
in	O
linear	B-Language
algebra	I-Language
deal	O
with	O
collections	O
of	O
two	O
or	O
more	O
orthogonal	B-Application
vectors	O
.	O
</s>
<s>
The	O
notion	O
of	O
restricting	O
orthogonal	B-Application
pairs	O
of	O
vectors	O
to	O
only	O
those	O
of	O
unit	O
length	O
is	O
important	O
enough	O
to	O
be	O
given	O
a	O
special	O
name	O
.	O
</s>
<s>
Two	O
vectors	O
which	O
are	O
orthogonal	B-Application
and	O
of	O
length	O
1	O
are	O
said	O
to	O
be	O
orthonormal	B-Algorithm
.	O
</s>
<s>
What	O
does	O
a	O
pair	O
of	O
orthonormal	B-Algorithm
vectors	I-Algorithm
in	O
2-D	O
Euclidean	O
space	O
look	O
like	O
?	O
</s>
<s>
Consider	O
the	O
restrictions	O
on	O
x1	O
,	O
x2	O
,	O
y1	O
,	O
y2	O
required	O
to	O
make	O
u	O
and	O
v	O
form	O
an	O
orthonormal	B-Algorithm
pair	O
.	O
</s>
<s>
From	O
the	O
orthogonality	B-Application
restriction	O
,	O
u	O
•	O
v	O
=	O
0	O
.	O
</s>
<s>
It	O
is	O
clear	O
that	O
in	O
the	O
plane	O
,	O
orthonormal	B-Algorithm
vectors	I-Algorithm
are	O
simply	O
radii	O
of	O
the	O
unit	O
circle	O
whose	O
difference	O
in	O
angles	O
equals	O
90°	O
.	O
</s>
<s>
Orthonormal	B-Algorithm
sets	I-Algorithm
are	O
not	O
especially	O
significant	O
on	O
their	O
own	O
.	O
</s>
<s>
However	O
,	O
they	O
display	O
certain	O
features	O
that	O
make	O
them	O
fundamental	O
in	O
exploring	O
the	O
notion	O
of	O
diagonalizability	B-Algorithm
of	O
certain	O
operators	B-Architecture
on	O
vector	O
spaces	O
.	O
</s>
<s>
Orthonormal	B-Algorithm
sets	I-Algorithm
have	O
certain	O
very	O
appealing	O
properties	O
,	O
which	O
make	O
them	O
particularly	O
easy	O
to	O
work	O
with	O
.	O
</s>
<s>
Every	O
orthonormal	B-Algorithm
list	O
of	O
vectors	O
is	O
linearly	O
independent	O
.	O
</s>
<s>
Gram-Schmidt	B-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
If	O
{	O
v1	O
,	O
v2	O
,...,	O
vn}	O
is	O
a	O
linearly	O
independent	O
list	O
of	O
vectors	O
in	O
an	O
inner-product	O
space	O
,	O
then	O
there	O
exists	O
an	O
orthonormal	B-Algorithm
list	O
{	O
e1	O
,	O
e2	O
,...,	O
en}	O
of	O
vectors	O
in	O
such	O
that	O
span( e1	O
,	O
e2	O
,...,	O
en	O
)	O
=	O
span( v1	O
,	O
v2	O
,...,	O
vn	O
)	O
.	O
</s>
<s>
Proof	O
of	O
the	O
Gram-Schmidt	B-Algorithm
theorem	I-Algorithm
is	O
constructive	O
,	O
and	O
discussed	B-Algorithm
at	I-Algorithm
length	I-Algorithm
elsewhere	O
.	O
</s>
<s>
The	O
Gram-Schmidt	B-Algorithm
theorem	I-Algorithm
,	O
together	O
with	O
the	O
axiom	O
of	O
choice	O
,	O
guarantees	O
that	O
every	O
vector	O
space	O
admits	O
an	O
orthonormal	B-Algorithm
basis	O
.	O
</s>
<s>
This	O
is	O
possibly	O
the	O
most	O
significant	O
use	O
of	O
orthonormality	B-Algorithm
,	O
as	O
this	O
fact	O
permits	O
operators	B-Architecture
on	O
inner-product	O
spaces	O
to	O
be	O
discussed	O
in	O
terms	O
of	O
their	O
action	O
on	O
the	O
space	O
's	O
orthonormal	B-Algorithm
basis	O
vectors	O
.	O
</s>
<s>
What	O
results	O
is	O
a	O
deep	O
relationship	O
between	O
the	O
diagonalizability	B-Algorithm
of	O
an	O
operator	O
and	O
how	O
it	O
acts	O
on	O
the	O
orthonormal	B-Algorithm
basis	O
vectors	O
.	O
</s>
<s>
Any	O
two	O
vectors	O
ei	O
,	O
ej	O
where	O
i≠j	O
are	O
orthogonal	B-Application
,	O
and	O
all	O
vectors	O
are	O
clearly	O
of	O
unit	O
length	O
.	O
</s>
<s>
So	O
{	O
e1	O
,	O
e2	O
,...,	O
en}	O
forms	O
an	O
orthonormal	B-Algorithm
basis	O
.	O
</s>
<s>
forms	O
an	O
orthonormal	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
But	O
,	O
removing	O
the	O
restriction	O
that	O
n	O
be	O
finite	O
makes	O
the	O
set	O
dense	O
in	O
C[−π,π]	O
and	O
therefore	O
an	O
orthonormal	B-Algorithm
basis	O
of	O
C[−π,π]	O
.	O
</s>
