<s>
In	O
mathematics	O
,	O
a	O
linear	B-Algorithm
form	I-Algorithm
(	O
also	O
known	O
as	O
a	O
linear	B-Algorithm
functional	I-Algorithm
,	O
a	O
one-form	B-Algorithm
,	O
or	O
a	O
covector	B-Algorithm
)	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
a	O
vector	O
space	O
to	O
its	O
field	O
of	O
scalars	O
(	O
often	O
,	O
the	O
real	O
numbers	O
or	O
the	O
complex	O
numbers	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
vector	O
space	O
over	O
a	O
field	O
,	O
the	O
set	O
of	O
all	O
linear	B-Algorithm
functionals	I-Algorithm
from	O
to	O
is	O
itself	O
a	O
vector	O
space	O
over	O
with	O
addition	O
and	O
scalar	O
multiplication	O
defined	O
pointwise	O
.	O
</s>
<s>
It	O
is	O
often	O
denoted	O
,	O
or	O
,	O
when	O
the	O
field	O
is	O
understood	O
,	O
;	O
other	O
notations	O
are	O
also	O
used	O
,	O
such	O
as	O
,	O
or	O
When	O
vectors	O
are	O
represented	O
by	O
column	O
vectors	O
(	O
as	O
is	O
common	O
when	O
a	O
basis	O
is	O
fixed	O
)	O
,	O
then	O
linear	B-Algorithm
functionals	I-Algorithm
are	O
represented	O
as	O
row	O
vectors	O
,	O
and	O
their	O
values	O
on	O
specific	O
vectors	O
are	O
given	O
by	O
matrix	O
products	O
(	O
with	O
the	O
row	O
vector	O
on	O
the	O
left	O
)	O
.	O
</s>
<s>
The	O
constant	O
zero	O
function	O
,	O
mapping	O
every	O
vector	O
to	O
zero	O
,	O
is	O
trivially	O
a	O
linear	B-Algorithm
functional	I-Algorithm
.	O
</s>
<s>
Every	O
other	O
linear	B-Algorithm
functional	I-Algorithm
(	O
such	O
as	O
the	O
ones	O
below	O
)	O
is	O
surjective	B-Algorithm
(	O
that	O
is	O
,	O
its	O
range	O
is	O
all	O
of	O
)	O
.	O
</s>
<s>
Mean	O
:	O
The	O
mean	O
element	O
of	O
an	O
-vector	O
is	O
given	O
by	O
the	O
one-form	B-Algorithm
That	O
is	O
,	O
</s>
<s>
Sampling	B-Algorithm
:	O
Sampling	B-Algorithm
with	O
a	O
kernel	B-Algorithm
can	O
be	O
considered	O
a	O
one-form	B-Algorithm
,	O
where	O
the	O
one-form	B-Algorithm
is	O
the	O
kernel	B-Algorithm
shifted	O
to	O
the	O
appropriate	O
location	O
.	O
</s>
<s>
Net	O
present	O
value	O
of	O
a	O
net	O
cash	O
flow	O
,	O
is	O
given	O
by	O
the	O
one-form	B-Algorithm
where	O
is	O
the	O
discount	O
rate	O
.	O
</s>
<s>
and	O
each	O
linear	B-Algorithm
functional	I-Algorithm
can	O
be	O
expressed	O
in	O
this	O
form	O
.	O
</s>
<s>
The	O
trace	O
of	O
a	O
square	O
matrix	O
is	O
the	O
sum	O
of	O
all	O
elements	O
on	O
its	O
main	B-Algorithm
diagonal	I-Algorithm
.	O
</s>
<s>
Linear	B-Algorithm
functionals	I-Algorithm
first	O
appeared	O
in	O
functional	B-Application
analysis	I-Application
,	O
the	O
study	O
of	O
vector	B-Algorithm
spaces	I-Algorithm
of	I-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
is	O
a	O
linear	B-Algorithm
functional	I-Algorithm
from	O
the	O
vector	O
space	O
of	O
continuous	O
functions	O
on	O
the	O
interval	O
to	O
the	O
real	O
numbers	O
.	O
</s>
<s>
A	O
function	O
having	O
the	O
equation	O
of	O
a	O
line	O
with	O
(	O
for	O
example	O
,	O
)	O
is	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
,	O
since	O
it	O
is	O
not	O
linear	O
.	O
</s>
<s>
It	O
is	O
,	O
however	O
,	O
affine-linear	B-Algorithm
.	O
</s>
<s>
In	O
finite	O
dimensions	O
,	O
a	O
linear	B-Algorithm
functional	I-Algorithm
can	O
be	O
visualized	O
in	O
terms	O
of	O
its	O
level	B-Algorithm
sets	I-Algorithm
,	O
the	O
sets	O
of	O
vectors	O
which	O
map	O
to	O
a	O
given	O
value	O
.	O
</s>
<s>
In	O
three	O
dimensions	O
,	O
the	O
level	B-Algorithm
sets	I-Algorithm
of	O
a	O
linear	B-Algorithm
functional	I-Algorithm
are	O
a	O
family	O
of	O
mutually	O
parallel	O
planes	O
;	O
in	O
higher	O
dimensions	O
,	O
they	O
are	O
parallel	O
hyperplanes	O
.	O
</s>
<s>
This	O
method	O
of	O
visualizing	O
linear	B-Algorithm
functionals	I-Algorithm
is	O
sometimes	O
introduced	O
in	O
general	O
relativity	O
texts	O
,	O
such	O
as	O
Gravitation	O
by	O
.	O
</s>
<s>
If	O
are	O
distinct	O
points	O
in	O
,	O
then	O
the	O
linear	B-Algorithm
functionals	I-Algorithm
defined	O
above	O
form	O
a	O
basis	O
of	O
the	O
dual	O
space	O
of	O
,	O
the	O
space	O
of	O
polynomials	O
of	O
degree	O
The	O
integration	O
functional	O
is	O
also	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
,	O
and	O
so	O
can	O
be	O
expressed	O
as	O
a	O
linear	O
combination	O
of	O
these	O
basis	O
elements	O
.	O
</s>
<s>
for	O
all	O
This	O
forms	O
the	O
foundation	O
of	O
the	O
theory	O
of	O
numerical	B-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
Linear	B-Algorithm
functionals	I-Algorithm
are	O
particularly	O
important	O
in	O
quantum	O
mechanics	O
.	O
</s>
<s>
Quantum	O
mechanical	O
systems	O
are	O
represented	O
by	O
Hilbert	O
spaces	O
,	O
which	O
are	O
anti	O
–	O
isomorphic	B-Architecture
to	O
their	O
own	O
dual	O
spaces	O
.	O
</s>
<s>
A	O
state	O
of	O
a	O
quantum	O
mechanical	O
system	O
can	O
be	O
identified	O
with	O
a	O
linear	B-Algorithm
functional	I-Algorithm
.	O
</s>
<s>
In	O
the	O
theory	O
of	O
generalized	O
functions	O
,	O
certain	O
kinds	O
of	O
generalized	O
functions	O
called	O
distributions	O
can	O
be	O
realized	O
as	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
spaces	O
of	O
test	O
functions	O
.	O
</s>
<s>
A	O
linear	B-Algorithm
functional	I-Algorithm
belonging	O
to	O
the	O
dual	O
space	O
can	O
be	O
expressed	O
as	O
a	O
linear	O
combination	O
of	O
basis	O
functionals	O
,	O
with	O
coefficients	O
(	O
"	O
components	O
"	O
)	O
,	O
</s>
<s>
So	O
each	O
component	O
of	O
a	O
linear	B-Algorithm
functional	I-Algorithm
can	O
be	O
extracted	O
by	O
applying	O
the	O
functional	O
to	O
the	O
corresponding	O
basis	O
vector	O
.	O
</s>
<s>
Given	O
a	O
module	O
over	O
a	O
ring	O
,	O
a	O
linear	B-Algorithm
form	I-Algorithm
on	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
to	O
,	O
where	O
the	O
latter	O
is	O
considered	O
as	O
a	O
module	O
over	O
itself	O
.	O
</s>
<s>
The	O
space	O
of	O
linear	B-Algorithm
forms	I-Algorithm
is	O
always	O
denoted	O
,	O
whether	O
is	O
a	O
field	O
or	O
not	O
.	O
</s>
<s>
The	O
existence	O
of	O
"	O
enough	O
"	O
linear	B-Algorithm
forms	I-Algorithm
on	O
a	O
module	O
is	O
equivalent	O
to	O
projectivity	O
.	O
</s>
<s>
Every	O
linear	B-Algorithm
functional	I-Algorithm
on	O
is	O
complex-valued	O
while	O
every	O
linear	B-Algorithm
functional	I-Algorithm
on	O
is	O
real-valued	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
only	O
function	O
on	O
that	O
is	O
both	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
and	O
a	O
linear	O
function	O
on	O
is	O
the	O
trivial	O
functional	O
;	O
in	O
other	O
words	O
,	O
where	O
denotes	O
the	O
space	O
's	O
algebraic	O
dual	O
space	O
.	O
</s>
<s>
Assume	O
that	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
.	O
</s>
<s>
This	O
conclusion	O
extends	O
to	O
the	O
analogous	O
statement	O
for	O
polars	B-Algorithm
of	O
balanced	O
sets	O
in	O
general	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
,	O
the	O
space	O
of	O
continuous	O
linear	B-Algorithm
functionals	I-Algorithm
—	O
the	O
—	O
is	O
often	O
simply	O
called	O
the	O
dual	O
space	O
.	O
</s>
<s>
In	O
finite	O
dimensions	O
,	O
every	O
linear	B-Algorithm
functional	I-Algorithm
is	O
continuous	O
,	O
so	O
the	O
continuous	O
dual	O
is	O
the	O
same	O
as	O
the	O
algebraic	O
dual	O
,	O
but	O
in	O
infinite	O
dimensions	O
the	O
continuous	O
dual	O
is	O
a	O
proper	O
subspace	O
of	O
the	O
algebraic	O
dual	O
.	O
</s>
<s>
Continuous	O
linear	B-Algorithm
functionals	I-Algorithm
have	O
nice	O
properties	O
for	O
analysis	O
:	O
a	O
linear	B-Algorithm
functional	I-Algorithm
is	O
continuous	O
if	O
and	O
only	O
if	O
its	O
kernel	B-Algorithm
is	O
closed	O
,	O
and	O
a	O
non-trivial	O
continuous	O
linear	B-Algorithm
functional	I-Algorithm
is	O
an	O
open	O
map	O
,	O
even	O
if	O
the	O
(	O
topological	O
)	O
vector	O
space	O
is	O
not	O
complete	O
.	O
</s>
<s>
A	O
vector	O
subspace	O
of	O
is	O
called	O
maximal	O
if	O
(	O
meaning	O
and	O
)	O
and	O
does	O
not	O
exist	O
a	O
vector	O
subspace	O
of	O
such	O
that	O
A	O
vector	O
subspace	O
of	O
is	O
maximal	O
if	O
and	O
only	O
if	O
it	O
is	O
the	O
kernel	B-Algorithm
of	O
some	O
non-trivial	O
linear	B-Algorithm
functional	I-Algorithm
on	O
(	O
that	O
is	O
,	O
for	O
some	O
linear	B-Algorithm
functional	I-Algorithm
on	O
that	O
is	O
not	O
identically	O
)	O
.	O
</s>
<s>
Any	O
two	O
linear	B-Algorithm
functionals	I-Algorithm
with	O
the	O
same	O
kernel	B-Algorithm
are	O
proportional	O
(	O
i.e.	O
</s>
<s>
Any	O
(	O
algebraic	O
)	O
linear	B-Algorithm
functional	I-Algorithm
on	O
a	O
vector	O
subspace	O
can	O
be	O
extended	O
to	O
the	O
whole	O
space	O
;	O
for	O
example	O
,	O
the	O
evaluation	O
functionals	O
described	O
above	O
can	O
be	O
extended	O
to	O
the	O
vector	O
space	O
of	O
polynomials	O
on	O
all	O
of	O
However	O
,	O
this	O
extension	O
cannot	O
always	O
be	O
done	O
while	O
keeping	O
the	O
linear	B-Algorithm
functional	I-Algorithm
continuous	O
.	O
</s>
<s>
is	O
contained	O
in	O
the	O
polar	B-Algorithm
of	O
some	O
neighborhood	O
of	O
in	O
;	O
</s>
<s>
the	O
(	O
pre	O
)	O
polar	B-Algorithm
of	O
is	O
a	O
neighborhood	O
of	O
in	O
;	O
</s>
<s>
Moreover	O
,	O
Alaoglu	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
implies	O
that	O
the	O
weak-*	O
closure	O
of	O
an	O
equicontinuous	O
subset	O
of	O
is	O
weak-*	O
compact	O
(	O
and	O
thus	O
that	O
every	O
equicontinuous	O
subset	O
weak-*	O
relatively	O
compact	O
)	O
.	O
</s>
