<s>
In	O
mathematics	O
,	O
a	O
function	B-Algorithm
space	I-Algorithm
is	O
a	O
set	O
of	O
functions	O
between	O
two	O
fixed	O
sets	O
.	O
</s>
<s>
Often	O
,	O
the	O
domain	B-Algorithm
and/or	O
codomain	B-Algorithm
will	O
have	O
additional	O
structure	O
which	O
is	O
inherited	O
by	O
the	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
In	O
other	O
scenarios	O
,	O
the	O
function	B-Algorithm
space	I-Algorithm
might	O
inherit	O
a	O
topological	O
or	O
metric	O
structure	O
,	O
hence	O
the	O
name	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
When	O
the	O
domain	B-Algorithm
has	O
additional	O
structure	O
,	O
one	O
might	O
consider	O
instead	O
the	O
subset	O
(	O
or	O
subspace	O
)	O
of	O
all	O
such	O
functions	O
which	O
respect	O
that	O
structure	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
is	O
also	O
a	O
vector	O
space	O
over	O
,	O
the	O
set	O
of	O
linear	B-Architecture
maps	I-Architecture
→	O
form	O
a	O
vector	O
space	O
over	O
with	O
pointwise	O
operations	O
(	O
often	O
denoted	O
Hom( , )	O
)	O
.	O
</s>
<s>
One	O
such	O
space	O
is	O
the	O
dual	O
space	O
of	O
:	O
the	O
set	O
of	O
linear	B-Algorithm
functionals	I-Algorithm
→	O
with	O
addition	O
and	O
scalar	O
multiplication	O
defined	O
pointwise	O
.	O
</s>
<s>
Function	B-Algorithm
spaces	I-Algorithm
appear	O
in	O
various	O
areas	O
of	O
mathematics	O
:	O
</s>
<s>
The	O
set	O
of	O
bijections	B-Algorithm
from	O
X	O
to	O
Y	O
is	O
denoted	O
.	O
</s>
<s>
In	O
functional	B-Application
analysis	I-Application
,	O
the	O
same	O
is	O
seen	O
for	O
continuous	O
linear	B-Architecture
transformations	I-Architecture
,	O
including	O
topologies	B-Architecture
on	I-Architecture
the	I-Architecture
vector	I-Architecture
spaces	I-Architecture
in	O
the	O
above	O
,	O
and	O
many	O
of	O
the	O
major	O
examples	O
are	O
function	B-Algorithm
spaces	I-Algorithm
carrying	O
a	O
topology	B-Architecture
;	O
the	O
best	O
known	O
examples	O
include	O
Hilbert	O
spaces	O
and	O
Banach	O
spaces	O
.	O
</s>
<s>
In	O
functional	B-Application
analysis	I-Application
,	O
the	O
set	O
of	O
all	O
functions	O
from	O
the	O
natural	O
numbers	O
to	O
some	O
set	O
X	O
is	O
called	O
a	O
sequence	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
In	O
topology	B-Architecture
,	O
one	O
may	O
attempt	O
to	O
put	O
a	O
topology	B-Architecture
on	O
the	O
space	O
of	O
continuous	O
functions	O
from	O
a	O
topological	O
space	O
X	O
to	O
another	O
one	O
Y	O
,	O
with	O
utility	O
depending	O
on	O
the	O
nature	O
of	O
the	O
spaces	O
.	O
</s>
<s>
A	O
commonly	O
used	O
example	O
is	O
the	O
compact-open	B-Algorithm
topology	I-Algorithm
,	O
e.g.	O
</s>
<s>
Also	O
available	O
is	O
the	O
product	O
topology	B-Architecture
on	O
the	O
space	O
of	O
set	O
theoretic	O
functions	O
(	O
i.e.	O
</s>
<s>
In	O
this	O
context	O
,	O
this	O
topology	B-Architecture
is	O
also	O
referred	O
to	O
as	O
the	O
topology	B-Architecture
of	O
pointwise	O
convergence	O
.	O
</s>
<s>
In	O
algebraic	O
topology	B-Architecture
,	O
the	O
study	O
of	O
homotopy	O
theory	O
is	O
essentially	O
that	O
of	O
discrete	O
invariants	O
of	O
function	B-Algorithm
spaces	I-Algorithm
;	O
</s>
<s>
In	O
the	O
theory	O
of	O
stochastic	O
processes	O
,	O
the	O
basic	O
technical	O
problem	O
is	O
how	O
to	O
construct	O
a	O
probability	O
measure	O
on	O
a	O
function	B-Algorithm
space	I-Algorithm
of	O
paths	O
of	O
the	O
process	O
(	O
functions	O
of	O
time	O
)	O
;	O
</s>
<s>
In	O
category	O
theory	O
,	O
the	O
function	B-Algorithm
space	I-Algorithm
is	O
called	O
an	O
exponential	O
object	O
or	O
map	O
object	O
.	O
</s>
<s>
In	O
functional	B-Language
programming	I-Language
and	O
lambda	B-Language
calculus	I-Language
,	O
function	O
types	O
are	O
used	O
to	O
express	O
the	O
idea	O
of	O
higher-order	B-Language
functions	I-Language
.	O
</s>
<s>
In	O
domain	B-Algorithm
theory	O
,	O
the	O
basic	O
idea	O
is	O
to	O
find	O
constructions	O
from	O
partial	O
orders	O
that	O
can	O
model	O
lambda	B-Language
calculus	I-Language
,	O
by	O
creating	O
a	O
well-behaved	O
Cartesian	B-Application
closed	I-Application
category	I-Application
.	O
</s>
<s>
In	O
the	O
representation	O
theory	O
of	O
finite	O
groups	O
,	O
given	O
two	O
finite-dimensional	O
representations	O
and	O
of	O
a	O
group	O
,	O
one	O
can	O
form	O
a	O
representation	O
of	O
over	O
the	O
vector	O
space	O
of	O
linear	B-Architecture
maps	I-Architecture
Hom( , )	O
called	O
the	O
Hom	O
representation	O
.	O
</s>
<s>
Functional	B-Application
analysis	I-Application
is	O
organized	O
around	O
adequate	O
techniques	O
to	O
bring	O
function	B-Algorithm
spaces	I-Algorithm
as	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
within	O
reach	O
of	O
the	O
ideas	O
that	O
would	O
apply	O
to	O
normed	O
spaces	O
of	O
finite	O
dimension	O
.	O
</s>
<s>
If	O
is	O
an	O
element	O
of	O
the	O
function	B-Algorithm
space	I-Algorithm
of	O
all	O
continuous	O
functions	O
that	O
are	O
defined	O
on	O
a	O
closed	O
interval	O
,	O
the	O
norm	O
defined	O
on	O
is	O
the	O
maximum	O
absolute	O
value	O
of	O
for	O
,	O
</s>
