<s>
Functional	B-Application
analysis	I-Application
is	O
a	O
branch	O
of	O
mathematical	O
analysis	O
,	O
the	O
core	O
of	O
which	O
is	O
formed	O
by	O
the	O
study	O
of	O
vector	O
spaces	O
endowed	O
with	O
some	O
kind	O
of	O
limit-related	O
structure	O
(	O
for	O
example	O
,	O
inner	O
product	O
,	O
norm	O
,	O
or	O
topology	O
)	O
and	O
the	O
linear	O
functions	O
defined	O
on	O
these	O
spaces	O
and	O
respecting	O
these	O
structures	O
in	O
a	O
suitable	O
sense	O
.	O
</s>
<s>
The	O
historical	O
roots	O
of	O
functional	B-Application
analysis	I-Application
lie	O
in	O
the	O
study	O
of	O
spaces	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
and	O
the	O
formulation	O
of	O
properties	O
of	O
transformations	O
of	O
functions	O
such	O
as	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
as	O
transformations	O
defining	O
,	O
for	O
example	O
,	O
continuous	O
or	O
unitary	B-Algorithm
operators	I-Algorithm
between	O
function	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
This	O
point	O
of	O
view	O
turned	O
out	O
to	O
be	O
particularly	O
useful	O
for	O
the	O
study	O
of	O
differential	O
and	O
integral	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
The	O
usage	O
of	O
the	O
word	O
functional	O
as	O
a	O
noun	O
goes	O
back	O
to	O
the	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
,	O
implying	O
a	O
function	B-Language
whose	I-Language
argument	I-Language
is	I-Language
a	I-Language
function	I-Language
.	O
</s>
<s>
Hadamard	O
also	O
founded	O
the	O
modern	O
school	O
of	O
linear	B-Algorithm
functional	I-Algorithm
analysis	O
further	O
developed	O
by	O
Riesz	O
and	O
the	O
group	O
of	O
Polish	O
mathematicians	O
around	O
Stefan	O
Banach	O
.	O
</s>
<s>
In	O
modern	O
introductory	O
texts	O
on	O
functional	B-Application
analysis	I-Application
,	O
the	O
subject	O
is	O
seen	O
as	O
the	O
study	O
of	O
vector	O
spaces	O
endowed	O
with	O
a	O
topology	O
,	O
in	O
particular	O
infinite-dimensional	O
spaces	O
.	O
</s>
<s>
In	O
contrast	O
,	O
linear	B-Language
algebra	I-Language
deals	O
mostly	O
with	O
finite-dimensional	O
spaces	O
,	O
and	O
does	O
not	O
use	O
topology	O
.	O
</s>
<s>
An	O
important	O
part	O
of	O
functional	B-Application
analysis	I-Application
is	O
the	O
extension	O
of	O
the	O
theories	O
of	O
measure	O
,	O
integration	O
,	O
and	O
probability	O
to	O
infinite	O
dimensional	O
spaces	O
,	O
also	O
known	O
as	O
infinite	B-Application
dimensional	I-Application
analysis	I-Application
.	O
</s>
<s>
The	O
basic	O
and	O
historically	O
first	O
class	O
of	O
spaces	O
studied	O
in	O
functional	B-Application
analysis	I-Application
are	O
complete	O
normed	O
vector	O
spaces	O
over	O
the	O
real	O
or	O
complex	O
numbers	O
.	O
</s>
<s>
More	O
generally	O
,	O
functional	B-Application
analysis	I-Application
includes	O
the	O
study	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
and	O
other	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
not	O
endowed	O
with	O
a	O
norm	O
.	O
</s>
<s>
An	O
important	O
object	O
of	O
study	O
in	O
functional	B-Application
analysis	I-Application
are	O
the	O
continuous	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
defined	O
on	O
Banach	O
and	O
Hilbert	O
spaces	O
.	O
</s>
<s>
These	O
lead	O
naturally	O
to	O
the	O
definition	O
of	O
C*	B-Algorithm
-algebras	I-Algorithm
and	O
other	O
operator	B-Algorithm
algebras	I-Algorithm
.	O
</s>
<s>
Finite-dimensional	O
Hilbert	O
spaces	O
are	O
fully	O
understood	O
in	O
linear	B-Language
algebra	I-Language
,	O
and	O
infinite-dimensional	O
separable	O
Hilbert	O
spaces	O
are	O
isomorphic	O
to	O
.	O
</s>
<s>
Separability	O
being	O
important	O
for	O
applications	O
,	O
functional	B-Application
analysis	I-Application
of	O
Hilbert	O
spaces	O
consequently	O
mostly	O
deals	O
with	O
this	O
space	O
.	O
</s>
<s>
One	O
of	O
the	O
open	O
problems	O
in	O
functional	B-Application
analysis	I-Application
is	O
to	O
prove	O
that	O
every	O
bounded	O
linear	B-Architecture
operator	I-Architecture
on	O
a	O
Hilbert	O
space	O
has	O
a	O
proper	O
invariant	O
subspace	O
.	O
</s>
<s>
In	O
Banach	O
spaces	O
,	O
a	O
large	O
part	O
of	O
the	O
study	O
involves	O
the	O
dual	O
space	O
:	O
the	O
space	O
of	O
all	O
continuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
from	O
the	O
space	O
into	O
its	O
underlying	O
field	O
,	O
so-called	O
functionals	O
.	O
</s>
<s>
Also	O
,	O
the	O
notion	O
of	O
derivative	B-Algorithm
can	O
be	O
extended	O
to	O
arbitrary	O
functions	O
between	O
Banach	O
spaces	O
.	O
</s>
<s>
See	O
,	O
for	O
instance	O
,	O
the	O
Fréchet	O
derivative	B-Algorithm
article	O
.	O
</s>
<s>
There	O
are	O
four	O
major	O
theorems	O
which	O
are	O
sometimes	O
called	O
the	O
four	O
pillars	O
of	O
functional	B-Application
analysis	I-Application
:	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
,	O
the	O
open	O
mapping	O
theorem	O
,	O
the	O
closed	O
graph	O
theorem	O
and	O
the	O
uniform	O
boundedness	O
principle	O
,	O
also	O
known	O
as	O
the	O
Banach	O
–	O
Steinhaus	O
theorem	O
.	O
</s>
<s>
Important	O
results	O
of	O
functional	B-Application
analysis	I-Application
include	O
:	O
</s>
<s>
The	O
uniform	O
boundedness	O
principle	O
or	O
Banach	O
–	O
Steinhaus	O
theorem	O
is	O
one	O
of	O
the	O
fundamental	O
results	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
In	O
its	O
basic	O
form	O
,	O
it	O
asserts	O
that	O
for	O
a	O
family	O
of	O
continuous	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
(	O
and	O
thus	O
bounded	O
operators	O
)	O
whose	O
domain	O
is	O
a	O
Banach	O
space	O
,	O
pointwise	O
boundedness	O
is	O
equivalent	O
to	O
uniform	O
boundedness	O
in	O
operator	O
norm	O
.	O
</s>
<s>
There	O
are	O
many	O
theorems	O
known	O
as	O
the	O
spectral	O
theorem	O
,	O
but	O
one	O
in	O
particular	O
has	O
many	O
applications	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
This	O
is	O
the	O
beginning	O
of	O
the	O
vast	O
research	O
area	O
of	O
functional	B-Application
analysis	I-Application
called	O
operator	O
theory	O
;	O
see	O
also	O
the	O
spectral	O
measure	O
.	O
</s>
<s>
The	O
Hahn	O
–	O
Banach	O
theorem	O
is	O
a	O
central	O
tool	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
It	O
allows	O
the	O
extension	O
of	O
bounded	O
linear	B-Algorithm
functionals	I-Algorithm
defined	O
on	O
a	O
subspace	O
of	O
some	O
vector	O
space	O
to	O
the	O
whole	O
space	O
,	O
and	O
it	O
also	O
shows	O
that	O
there	O
are	O
"	O
enough	O
"	O
continuous	O
linear	B-Algorithm
functionals	I-Algorithm
defined	O
on	O
every	O
normed	O
vector	O
space	O
to	O
make	O
the	O
study	O
of	O
the	O
dual	O
space	O
"	O
interesting	O
"	O
.	O
</s>
<s>
The	O
open	O
mapping	O
theorem	O
,	O
also	O
known	O
as	O
the	O
Banach	O
–	O
Schauder	O
theorem	O
(	O
named	O
after	O
Stefan	O
Banach	O
and	O
Juliusz	O
Schauder	O
)	O
,	O
is	O
a	O
fundamental	O
result	O
which	O
states	O
that	O
if	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
between	O
Banach	O
spaces	O
is	O
surjective	B-Algorithm
then	O
it	O
is	O
an	O
open	O
map	O
.	O
</s>
<s>
The	O
statement	O
of	O
the	O
theorem	O
is	O
no	O
longer	O
true	O
if	O
either	O
space	O
is	O
just	O
assumed	O
to	O
be	O
a	O
normed	O
space	O
,	O
but	O
is	O
true	O
if	O
and	O
are	O
taken	O
to	O
be	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
topological	O
space	O
and	O
is	O
a	O
compact	O
Hausdorff	O
space	O
,	O
then	O
the	O
graph	O
of	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
to	O
is	O
closed	O
if	O
and	O
only	O
if	O
is	O
continuous	O
.	O
</s>
<s>
Most	O
spaces	O
considered	O
in	O
functional	B-Application
analysis	I-Application
have	O
infinite	O
dimension	O
.	O
</s>
<s>
However	O
,	O
a	O
somewhat	O
different	O
concept	O
,	O
the	O
Schauder	O
basis	O
,	O
is	O
usually	O
more	O
relevant	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
Functional	B-Application
analysis	I-Application
in	O
its	O
includes	O
the	O
following	O
tendencies	O
:	O
</s>
<s>
An	O
approach	O
to	O
analysis	O
based	O
on	O
topological	O
groups	O
,	O
topological	O
rings	O
,	O
and	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
