<s>
In	O
linear	B-Language
algebra	I-Language
and	O
functional	B-Application
analysis	I-Application
,	O
a	O
projection	B-Algorithm
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
from	O
a	O
vector	O
space	O
to	O
itself	O
(	O
an	O
endomorphism	O
)	O
such	O
that	O
.	O
</s>
<s>
This	O
definition	O
of	O
"	O
projection	B-Algorithm
"	O
formalizes	O
and	O
generalizes	O
the	O
idea	O
of	O
graphical	B-Algorithm
projection	I-Algorithm
.	O
</s>
<s>
One	O
can	O
also	O
consider	O
the	O
effect	O
of	O
a	O
projection	B-Algorithm
on	O
a	O
geometrical	O
object	O
by	O
examining	O
the	O
effect	O
of	O
the	O
projection	B-Algorithm
on	O
points	O
in	O
the	O
object	O
.	O
</s>
<s>
A	O
projection	B-Algorithm
on	O
a	O
vector	O
space	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
such	O
that	O
.	O
</s>
<s>
when	O
is	O
a	O
Hilbert	O
space	O
)	O
the	O
concept	O
of	O
orthogonality	B-Application
can	O
be	O
used	O
.	O
</s>
<s>
A	O
projection	B-Algorithm
on	O
a	O
Hilbert	O
space	O
is	O
called	O
an	O
orthogonal	O
projection	B-Algorithm
if	O
it	O
satisfies	O
for	O
all	O
.	O
</s>
<s>
A	O
projection	B-Algorithm
on	O
a	O
Hilbert	O
space	O
that	O
is	O
not	O
orthogonal	O
is	O
called	O
an	O
oblique	O
projection	B-Algorithm
.	O
</s>
<s>
In	O
the	O
finite-dimensional	O
case	O
,	O
a	O
square	B-Algorithm
matrix	I-Algorithm
is	O
called	O
a	O
projection	B-Algorithm
matrix	B-Architecture
if	O
it	O
is	O
equal	O
to	O
its	O
square	O
,	O
i.e.	O
</s>
<s>
A	O
square	B-Algorithm
matrix	I-Algorithm
is	O
called	O
an	O
orthogonal	O
projection	B-Algorithm
matrix	B-Architecture
if	O
for	O
a	O
real	O
matrix	B-Architecture
,	O
and	O
respectively	O
for	O
a	O
complex	O
matrix	B-Architecture
,	O
where	O
denotes	O
the	O
transpose	O
of	O
and	O
denotes	O
the	O
adjoint	O
or	O
Hermitian	B-Algorithm
transpose	I-Algorithm
of	O
.	O
</s>
<s>
A	O
projection	B-Algorithm
matrix	B-Architecture
that	O
is	O
not	O
an	O
orthogonal	O
projection	B-Algorithm
matrix	B-Architecture
is	O
called	O
an	O
oblique	O
projection	B-Algorithm
matrix	B-Architecture
.	O
</s>
<s>
The	O
eigenvalues	O
of	O
a	O
projection	B-Algorithm
matrix	B-Architecture
must	O
be	O
0	O
or	O
1	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
function	O
which	O
maps	O
the	O
point	O
in	O
three-dimensional	O
space	O
to	O
the	O
point	O
is	O
an	O
orthogonal	O
projection	B-Algorithm
onto	B-Algorithm
the	O
xy-plane	O
.	O
</s>
<s>
Observing	O
that	O
shows	O
that	O
the	O
projection	B-Algorithm
is	O
an	O
orthogonal	O
projection	B-Algorithm
.	O
</s>
<s>
showing	O
that	O
is	O
indeed	O
a	O
projection	B-Algorithm
.	O
</s>
<s>
By	O
definition	O
,	O
a	O
projection	B-Algorithm
is	O
idempotent	O
(	O
i.e.	O
</s>
<s>
Every	O
projection	B-Algorithm
is	O
an	O
open	O
map	O
,	O
meaning	O
that	O
it	O
maps	O
each	O
open	O
set	O
in	O
the	O
domain	B-Algorithm
to	O
an	O
open	O
set	O
in	O
the	O
subspace	O
topology	O
of	O
the	O
image	O
.	O
</s>
<s>
Let	O
be	O
a	O
finite-dimensional	O
vector	O
space	O
and	O
be	O
a	O
projection	B-Algorithm
on	O
.	O
</s>
<s>
Suppose	O
the	O
subspaces	O
and	O
are	O
the	O
image	O
and	O
kernel	B-Algorithm
of	O
respectively	O
.	O
</s>
<s>
The	O
image	O
and	O
kernel	B-Algorithm
of	O
a	O
projection	B-Algorithm
are	O
complementary	O
,	O
as	O
are	O
and	O
.	O
</s>
<s>
The	O
operator	O
is	O
also	O
a	O
projection	B-Algorithm
as	O
the	O
image	O
and	O
kernel	B-Algorithm
of	O
become	O
the	O
kernel	B-Algorithm
and	O
image	O
of	O
and	O
vice	O
versa	O
.	O
</s>
<s>
We	O
say	O
is	O
a	O
projection	B-Algorithm
along	O
onto	B-Algorithm
(	O
kernel/image	O
)	O
and	O
is	O
a	O
projection	B-Algorithm
along	O
onto	B-Algorithm
.	O
</s>
<s>
Only	O
0	O
or	O
1	O
can	O
be	O
an	O
eigenvalue	O
of	O
a	O
projection	B-Algorithm
.	O
</s>
<s>
This	O
implies	O
that	O
an	O
orthogonal	O
projection	B-Algorithm
is	O
always	O
a	O
positive	B-Algorithm
semi-definite	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
the	O
corresponding	O
eigenspaces	O
are	O
(	O
respectively	O
)	O
the	O
kernel	B-Algorithm
and	O
range	O
of	O
the	O
projection	B-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
given	O
a	O
subspace	O
,	O
there	O
may	O
be	O
many	O
projections	B-Algorithm
whose	O
range	O
(	O
or	O
kernel	B-Algorithm
)	O
is	O
.	O
</s>
<s>
If	O
a	O
projection	B-Algorithm
is	O
nontrivial	O
it	O
has	O
minimal	O
polynomial	O
,	O
which	O
factors	O
into	O
distinct	O
linear	O
factors	O
,	O
and	O
thus	O
is	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
The	O
product	O
of	O
projections	B-Algorithm
is	O
not	O
in	O
general	O
a	O
projection	B-Algorithm
,	O
even	O
if	O
they	O
are	O
orthogonal	O
.	O
</s>
<s>
If	O
two	O
projections	B-Algorithm
commute	O
then	O
their	O
product	O
is	O
a	O
projection	B-Algorithm
,	O
but	O
the	O
converse	O
is	O
false	O
:	O
the	O
product	O
of	O
two	O
non-commuting	O
projections	B-Algorithm
may	O
be	O
a	O
projection	B-Algorithm
.	O
</s>
<s>
If	O
two	O
orthogonal	O
projections	B-Algorithm
commute	O
then	O
their	O
product	O
is	O
an	O
orthogonal	O
projection	B-Algorithm
.	O
</s>
<s>
If	O
the	O
product	O
of	O
two	O
orthogonal	O
projections	B-Algorithm
is	O
an	O
orthogonal	O
projection	B-Algorithm
,	O
then	O
the	O
two	O
orthogonal	O
projections	B-Algorithm
commute	O
(	O
more	O
generally	O
:	O
two	O
self-adjoint	O
endomorphisms	O
commute	O
if	O
and	O
only	O
if	O
their	O
product	O
is	O
self-adjoint	O
)	O
.	O
</s>
<s>
When	O
the	O
vector	O
space	O
has	O
an	O
inner	O
product	O
and	O
is	O
complete	O
(	O
is	O
a	O
Hilbert	O
space	O
)	O
the	O
concept	O
of	O
orthogonality	B-Application
can	O
be	O
used	O
.	O
</s>
<s>
An	O
orthogonal	O
projection	B-Algorithm
is	O
a	O
projection	B-Algorithm
for	O
which	O
the	O
range	O
and	O
the	O
null	B-Algorithm
space	I-Algorithm
are	O
orthogonal	B-Application
subspaces	I-Application
.	O
</s>
<s>
A	O
projection	B-Algorithm
is	O
orthogonal	O
if	O
and	O
only	O
if	O
it	O
is	O
self-adjoint	O
.	O
</s>
<s>
Therefore	O
,	O
and	O
are	O
orthogonal	O
projections	B-Algorithm
.	O
</s>
<s>
An	O
orthogonal	O
projection	B-Algorithm
is	O
a	O
bounded	O
operator	O
.	O
</s>
<s>
A	O
simple	O
case	O
occurs	O
when	O
the	O
orthogonal	O
projection	B-Algorithm
is	O
onto	B-Algorithm
a	O
line	O
.	O
</s>
<s>
(	O
If	O
is	O
complex-valued	O
,	O
the	O
transpose	O
in	O
the	O
above	O
equation	O
is	O
replaced	O
by	O
a	O
Hermitian	B-Algorithm
transpose	I-Algorithm
)	O
.	O
</s>
<s>
This	O
operator	O
leaves	O
u	O
invariant	O
,	O
and	O
it	O
annihilates	O
all	O
vectors	O
orthogonal	O
to	O
,	O
proving	O
that	O
it	O
is	O
indeed	O
the	O
orthogonal	O
projection	B-Algorithm
onto	B-Algorithm
the	O
line	O
containing	O
u	O
.	O
</s>
<s>
This	O
formula	O
can	O
be	O
generalized	O
to	O
orthogonal	O
projections	B-Algorithm
on	O
a	O
subspace	O
of	O
arbitrary	O
dimension	O
.	O
</s>
<s>
Let	O
be	O
an	O
orthonormal	O
basis	O
of	O
the	O
subspace	O
,	O
with	O
the	O
assumption	O
that	O
the	O
integer	O
,	O
and	O
let	O
denote	O
the	O
matrix	B-Architecture
whose	O
columns	O
are	O
,	O
i.e.	O
,	O
.	O
</s>
<s>
Then	O
the	O
projection	B-Algorithm
is	O
given	O
by	O
:	O
</s>
<s>
The	O
matrix	B-Architecture
is	O
the	O
partial	O
isometry	O
that	O
vanishes	O
on	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
and	O
is	O
the	O
isometry	O
that	O
embeds	O
into	O
the	O
underlying	O
vector	O
space	O
.	O
</s>
<s>
If	O
is	O
a	O
(	O
not	O
necessarily	O
orthonormal	O
)	O
basis	O
with	O
,	O
and	O
is	O
the	O
matrix	B-Architecture
with	O
these	O
vectors	O
as	O
columns	O
,	O
then	O
the	O
projection	B-Algorithm
is	O
:	O
</s>
<s>
The	O
matrix	B-Architecture
still	O
embeds	O
into	O
the	O
underlying	O
vector	O
space	O
but	O
is	O
no	O
longer	O
an	O
isometry	O
in	O
general	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
is	O
a	O
"	O
normalizing	O
factor	O
"	O
that	O
recovers	O
the	O
norm	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
rank-1	O
operator	O
is	O
not	O
a	O
projection	B-Algorithm
if	O
After	O
dividing	O
by	O
we	O
obtain	O
the	O
projection	B-Algorithm
onto	B-Algorithm
the	O
subspace	O
spanned	O
by	O
.	O
</s>
<s>
In	O
the	O
general	O
case	O
,	O
we	O
can	O
have	O
an	O
arbitrary	O
positive	B-Algorithm
definite	I-Algorithm
matrix	I-Algorithm
defining	O
an	O
inner	O
product	O
,	O
and	O
the	O
projection	B-Algorithm
is	O
given	O
by	O
.	O
</s>
<s>
When	O
the	O
range	O
space	O
of	O
the	O
projection	B-Algorithm
is	O
generated	O
by	O
a	O
frame	O
(	O
i.e.	O
</s>
<s>
the	O
number	O
of	O
generators	O
is	O
greater	O
than	O
its	O
dimension	O
)	O
,	O
the	O
formula	O
for	O
the	O
projection	B-Algorithm
takes	O
the	O
form	O
:	O
.	O
</s>
<s>
This	O
is	O
just	O
one	O
of	O
many	O
ways	O
to	O
construct	O
the	O
projection	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
non-singular	O
matrix	B-Architecture
and	O
(	O
i.e.	O
,	O
is	O
the	O
null	B-Algorithm
space	I-Algorithm
matrix	B-Architecture
of	O
)	O
,	O
the	O
following	O
holds	O
:	O
</s>
<s>
All	O
these	O
formulas	O
also	O
hold	O
for	O
complex	O
inner	O
product	O
spaces	O
,	O
provided	O
that	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
is	O
used	O
instead	O
of	O
the	O
transpose	O
.	O
</s>
<s>
Further	O
details	O
on	O
sums	O
of	O
projectors	B-Algorithm
can	O
be	O
found	O
in	O
Banerjee	O
and	O
Roy	O
(	O
2014	O
)	O
.	O
</s>
<s>
Also	O
see	O
Banerjee	O
(	O
2004	O
)	O
for	O
application	O
of	O
sums	O
of	O
projectors	B-Algorithm
in	O
basic	O
spherical	O
trigonometry	O
.	O
</s>
<s>
The	O
term	O
oblique	O
projections	B-Algorithm
is	O
sometimes	O
used	O
to	O
refer	O
to	O
non-orthogonal	O
projections	B-Algorithm
.	O
</s>
<s>
These	O
projections	B-Algorithm
are	O
also	O
used	O
to	O
represent	O
spatial	O
figures	O
in	O
two-dimensional	O
drawings	O
(	O
see	O
oblique	O
projection	B-Algorithm
)	O
,	O
though	O
not	O
as	O
frequently	O
as	O
orthogonal	O
projections	B-Algorithm
.	O
</s>
<s>
Whereas	O
calculating	O
the	O
fitted	O
value	O
of	O
an	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
regression	I-General_Concept
requires	O
an	O
orthogonal	O
projection	B-Algorithm
,	O
calculating	O
the	O
fitted	O
value	O
of	O
an	O
instrumental	O
variables	O
regression	O
requires	O
an	O
oblique	O
projection	B-Algorithm
.	O
</s>
<s>
Projections	B-Algorithm
are	O
defined	O
by	O
their	O
null	B-Algorithm
space	I-Algorithm
and	O
the	O
basis	O
vectors	O
used	O
to	O
characterize	O
their	O
range	O
(	O
which	O
is	O
the	O
complement	O
of	O
the	O
null	B-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
When	O
these	O
basis	O
vectors	O
are	O
orthogonal	O
to	O
the	O
null	B-Algorithm
space	I-Algorithm
,	O
then	O
the	O
projection	B-Algorithm
is	O
an	O
orthogonal	O
projection	B-Algorithm
.	O
</s>
<s>
When	O
these	O
basis	O
vectors	O
are	O
not	O
orthogonal	O
to	O
the	O
null	B-Algorithm
space	I-Algorithm
,	O
the	O
projection	B-Algorithm
is	O
an	O
oblique	O
projection	B-Algorithm
,	O
or	O
just	O
a	O
general	O
projection	B-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
linear	B-Architecture
operator	I-Architecture
such	O
that	O
and	O
assume	O
that	O
is	O
not	O
the	O
zero	O
operator	O
.	O
</s>
<s>
Let	O
the	O
vectors	O
form	O
a	O
basis	O
for	O
the	O
range	O
of	O
the	O
projection	B-Algorithm
,	O
and	O
assemble	O
these	O
vectors	O
in	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
range	O
and	O
the	O
null	B-Algorithm
space	I-Algorithm
are	O
complementary	O
spaces	O
,	O
so	O
the	O
null	B-Algorithm
space	I-Algorithm
has	O
dimension	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
the	O
null	B-Algorithm
space	I-Algorithm
has	O
dimension	O
.	O
</s>
<s>
Let	O
form	O
a	O
basis	O
for	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
the	O
null	B-Algorithm
space	I-Algorithm
of	O
the	O
projection	B-Algorithm
,	O
and	O
assemble	O
these	O
vectors	O
in	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
This	O
expression	O
generalizes	O
the	O
formula	O
for	O
orthogonal	O
projections	B-Algorithm
given	O
above	O
.	O
</s>
<s>
So	O
,	O
and	O
then	O
is	O
in	O
the	O
null	B-Algorithm
space	I-Algorithm
of	O
.	O
</s>
<s>
Since	O
matrices	O
and	O
are	O
of	O
full	O
rank	O
by	O
their	O
construction	O
,	O
the	O
-matrix	O
is	O
invertible	O
.	O
</s>
<s>
In	O
the	O
case	O
that	O
is	O
an	O
orthogonal	O
projection	B-Algorithm
,	O
we	O
can	O
take	O
,	O
and	O
it	O
follows	O
that	O
.	O
</s>
<s>
In	O
general	O
,	O
if	O
the	O
vector	O
space	O
is	O
over	O
complex	O
number	O
field	O
,	O
one	O
then	O
uses	O
the	O
Hermitian	B-Algorithm
transpose	I-Algorithm
and	O
has	O
the	O
formula	O
.	O
</s>
<s>
Recall	O
that	O
one	O
can	O
define	O
the	O
Moore	O
–	O
Penrose	O
inverse	O
of	O
the	O
matrix	B-Architecture
by	O
since	O
has	O
full	O
column	O
rank	O
,	O
so	O
.	O
</s>
<s>
Note	O
that	O
is	O
also	O
an	O
oblique	O
projection	B-Algorithm
.	O
</s>
<s>
be	O
an	O
orthonormal	O
basis	O
of	O
and	O
let	O
be	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
.	O
</s>
<s>
This	O
implies	O
that	O
the	O
largest	O
singular	O
values	O
of	O
and	O
are	O
equal	O
,	O
and	O
thus	O
that	O
the	O
matrix	B-Architecture
norm	O
of	O
the	O
oblique	O
projections	B-Algorithm
are	O
the	O
same	O
.	O
</s>
<s>
However	O
,	O
the	O
condition	B-Algorithm
number	I-Algorithm
satisfies	O
the	O
relation	O
,	O
and	O
is	O
therefore	O
not	O
necessarily	O
equal	O
.	O
</s>
<s>
There	O
is	O
a	O
theorem	O
in	O
linear	B-Language
algebra	I-Language
that	O
states	O
that	O
this	O
is	O
the	O
smallest	O
distance	O
(	O
the	O
orthogonal	O
distance	O
)	O
from	O
to	O
and	O
is	O
commonly	O
used	O
in	O
areas	O
such	O
as	O
machine	O
learning	O
.	O
</s>
<s>
Any	O
projection	B-Algorithm
on	O
a	O
vector	O
space	O
of	O
dimension	O
over	O
a	O
field	O
is	O
a	O
diagonalizable	B-Algorithm
matrix	I-Algorithm
,	O
since	O
its	O
minimal	O
polynomial	O
divides	O
,	O
which	O
splits	O
into	O
distinct	O
linear	O
factors	O
.	O
</s>
<s>
Here	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
of	O
size	O
,	O
is	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
of	O
size	O
,	O
and	O
is	O
the	O
direct	O
sum	O
operator	O
.	O
</s>
<s>
The	O
factor	O
corresponds	O
to	O
the	O
maximal	O
invariant	O
subspace	O
on	O
which	O
acts	O
as	O
an	O
orthogonal	O
projection	B-Algorithm
(	O
so	O
that	O
P	O
itself	O
is	O
orthogonal	O
if	O
and	O
only	O
if	O
)	O
and	O
the	O
-blocks	O
correspond	O
to	O
the	O
oblique	O
components	O
.	O
</s>
<s>
A	O
given	O
direct	O
sum	O
decomposition	O
of	O
into	O
complementary	O
subspaces	O
still	O
specifies	O
a	O
projection	B-Algorithm
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
If	O
is	O
the	O
direct	O
sum	O
,	O
then	O
the	O
operator	O
defined	O
by	O
is	O
still	O
a	O
projection	B-Algorithm
with	O
range	O
and	O
kernel	B-Algorithm
.	O
</s>
<s>
Conversely	O
,	O
if	O
is	O
projection	B-Algorithm
on	O
,	O
i.e.	O
</s>
<s>
In	O
other	O
words	O
,	O
is	O
also	O
a	O
projection	B-Algorithm
.	O
</s>
<s>
However	O
,	O
in	O
contrast	O
to	O
the	O
finite-dimensional	O
case	O
,	O
projections	B-Algorithm
need	O
not	O
be	O
continuous	O
in	O
general	O
.	O
</s>
<s>
If	O
a	O
subspace	O
of	O
is	O
not	O
closed	O
in	O
the	O
norm	O
topology	O
,	O
then	O
the	O
projection	B-Algorithm
onto	B-Algorithm
is	O
not	O
continuous	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
range	O
of	O
a	O
continuous	O
projection	B-Algorithm
must	O
be	O
a	O
closed	O
subspace	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
kernel	B-Algorithm
of	O
a	O
continuous	O
projection	B-Algorithm
(	O
in	O
fact	O
,	O
a	O
continuous	O
linear	B-Architecture
operator	I-Architecture
in	O
general	O
)	O
is	O
closed	O
.	O
</s>
<s>
Thus	O
a	O
continuous	O
projection	B-Algorithm
gives	O
a	O
decomposition	O
of	O
into	O
two	O
complementary	O
closed	O
subspaces	O
:	O
.	O
</s>
<s>
If	O
there	O
exists	O
a	O
closed	O
subspace	O
such	O
that	O
,	O
then	O
the	O
projection	B-Algorithm
with	O
range	O
and	O
kernel	B-Algorithm
is	O
continuous	O
.	O
</s>
<s>
In	O
general	O
,	O
given	O
a	O
closed	O
subspace	O
,	O
there	O
need	O
not	O
exist	O
a	O
complementary	O
closed	O
subspace	O
,	O
although	O
for	O
Hilbert	O
spaces	O
this	O
can	O
always	O
be	O
done	O
by	O
taking	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
.	O
</s>
<s>
By	O
Hahn	O
–	O
Banach	O
,	O
there	O
exists	O
a	O
bounded	O
linear	B-Algorithm
functional	I-Algorithm
such	O
that	O
.	O
</s>
<s>
it	O
is	O
a	O
projection	B-Algorithm
.	O
</s>
<s>
Projections	B-Algorithm
(	O
orthogonal	O
and	O
otherwise	O
)	O
play	O
a	O
major	O
role	O
in	O
algorithms	O
for	O
certain	O
linear	B-Language
algebra	I-Language
problems	O
:	O
</s>
<s>
QR	O
decomposition	O
(	O
see	O
Householder	B-Algorithm
transformation	I-Algorithm
and	O
Gram	B-Algorithm
–	I-Algorithm
Schmidt	I-Algorithm
decomposition	I-Algorithm
)	O
;	O
</s>
<s>
As	O
stated	O
above	O
,	O
projections	B-Algorithm
are	O
a	O
special	O
case	O
of	O
idempotents	O
.	O
</s>
<s>
Analytically	O
,	O
orthogonal	O
projections	B-Algorithm
are	O
non-commutative	O
generalizations	O
of	O
characteristic	O
functions	O
.	O
</s>
<s>
Therefore	O
,	O
as	O
one	O
can	O
imagine	O
,	O
projections	B-Algorithm
are	O
very	O
often	O
encountered	O
in	O
the	O
context	O
of	O
operator	B-Algorithm
algebras	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
von	O
Neumann	O
algebra	O
is	O
generated	O
by	O
its	O
complete	O
lattice	O
of	O
projections	B-Algorithm
.	O
</s>
<s>
More	O
generally	O
,	O
given	O
a	O
map	O
between	O
normed	O
vector	O
spaces	O
one	O
can	O
analogously	O
ask	O
for	O
this	O
map	O
to	O
be	O
an	O
isometry	O
on	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
the	O
kernel	B-Algorithm
:	O
that	O
be	O
an	O
isometry	O
(	O
compare	O
Partial	O
isometry	O
)	O
;	O
in	O
particular	O
it	O
must	O
be	O
onto	B-Algorithm
.	O
</s>
<s>
The	O
case	O
of	O
an	O
orthogonal	O
projection	B-Algorithm
is	O
when	O
W	O
is	O
a	O
subspace	O
of	O
V	O
.	O
In	O
Riemannian	O
geometry	O
,	O
this	O
is	O
used	O
in	O
the	O
definition	O
of	O
a	O
Riemannian	O
submersion	O
.	O
</s>
