<s>
In	O
mathematics	O
,	O
a	O
Hilbert	B-Algorithm
manifold	I-Algorithm
is	O
a	O
manifold	B-Architecture
modeled	O
on	O
Hilbert	O
spaces	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
Hilbert	B-Algorithm
manifold	I-Algorithm
provides	O
a	O
possibility	O
of	O
extending	O
the	O
theory	O
of	O
manifolds	B-Architecture
to	O
infinite-dimensional	O
setting	O
.	O
</s>
<s>
Analogously	O
to	O
the	O
finite-dimensional	O
situation	O
,	O
one	O
can	O
define	O
a	O
differentiable	O
Hilbert	B-Algorithm
manifold	I-Algorithm
by	O
considering	O
a	O
maximal	O
atlas	O
in	O
which	O
the	O
transition	O
maps	O
are	O
differentiable	O
.	O
</s>
<s>
Many	O
basic	O
constructions	O
of	O
the	O
manifold	B-Architecture
theory	I-Architecture
,	O
such	O
as	O
the	O
tangent	O
space	O
of	O
a	O
manifold	B-Architecture
and	O
a	O
tubular	O
neighbourhood	O
of	O
a	O
submanifold	O
(	O
of	O
finite	O
codimension	O
)	O
carry	O
over	O
from	O
the	O
finite	O
dimensional	O
situation	O
to	O
the	O
Hilbert	O
setting	O
with	O
little	O
change	O
.	O
</s>
<s>
However	O
,	O
in	O
statements	O
involving	O
maps	O
between	O
manifolds	B-Architecture
,	O
one	O
often	O
has	O
to	O
restrict	O
consideration	O
to	O
Fredholm	O
maps	O
,	O
that	O
is	O
,	O
maps	O
whose	O
differential	O
at	O
every	O
point	O
is	O
Fredholm	O
.	O
</s>
<s>
Notwithstanding	O
this	O
difference	O
,	O
Hilbert	B-Algorithm
manifolds	I-Algorithm
have	O
several	O
very	O
nice	O
properties	O
.	O
</s>
<s>
In	O
particular	O
,	O
every	O
Hilbert	B-Algorithm
manifold	I-Algorithm
is	O
parallelizable	O
.	O
</s>
<s>
Every	O
smooth	O
Hilbert	B-Algorithm
manifold	I-Algorithm
can	O
be	O
smoothly	O
embedded	O
onto	O
an	O
open	O
subset	O
of	O
the	O
model	O
Hilbert	O
space	O
.	O
</s>
<s>
Every	O
homotopy	O
equivalence	O
between	O
two	O
Hilbert	B-Algorithm
manifolds	I-Algorithm
is	O
homotopic	O
to	O
a	O
diffeomorphism	O
.	O
</s>
<s>
In	O
particular	O
every	O
two	O
homotopy	O
equivalent	O
Hilbert	B-Algorithm
manifolds	I-Algorithm
are	O
already	O
diffeomorphic	O
.	O
</s>
<s>
This	O
stands	O
in	O
contrast	O
to	O
lens	O
spaces	O
and	O
exotic	O
spheres	O
,	O
which	O
demonstrate	O
that	O
in	O
the	O
finite-dimensional	O
situation	O
,	O
homotopy	O
equivalence	O
,	O
homeomorphism	O
,	O
and	O
diffeomorphism	O
of	O
manifolds	B-Architecture
are	O
distinct	O
properties	O
.	O
</s>
<s>
Although	O
Sard	O
's	O
Theorem	O
does	O
not	O
hold	O
in	O
general	O
,	O
every	O
continuous	O
map	O
from	O
a	O
Hilbert	B-Algorithm
manifold	I-Algorithm
can	O
be	O
arbitrary	O
closely	O
approximated	O
by	O
a	O
smooth	O
map	O
which	O
has	O
no	O
critical	O
points	O
.	O
</s>
<s>
Similarly	O
,	O
any	O
open	O
subset	O
of	O
a	O
Hilbert	O
space	O
is	O
a	O
Hilbert	B-Algorithm
manifold	I-Algorithm
and	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
under	O
the	O
same	O
construction	O
as	O
for	O
the	O
whole	O
space	O
.	O
</s>
<s>
There	O
are	O
several	O
mapping	B-Algorithm
spaces	I-Algorithm
between	O
manifolds	B-Architecture
which	O
can	O
be	O
viewed	O
as	O
Hilbert	O
spaces	O
by	O
only	O
considering	O
maps	O
of	O
suitable	O
Sobolev	O
class	O
.	O
</s>
<s>
For	O
example	O
we	O
can	O
consider	O
the	O
space	O
of	O
all	O
maps	O
from	O
the	O
unit	O
circle	O
into	O
a	O
manifold	B-Architecture
This	O
can	O
be	O
topologized	O
via	O
the	O
compact	B-Algorithm
open	I-Algorithm
topology	I-Algorithm
as	O
a	O
subspace	O
of	O
the	O
space	O
of	O
all	O
continuous	O
mappings	O
from	O
the	O
circle	O
to	O
that	O
is	O
,	O
the	O
free	O
loop	O
space	O
of	O
The	O
Sobolev	O
kind	O
mapping	O
space	O
described	O
above	O
is	O
homotopy	O
equivalent	O
to	O
the	O
free	O
loop	O
space	O
.	O
</s>
