<s>
In	O
the	O
branch	O
of	O
mathematics	O
called	O
functional	B-Application
analysis	I-Application
,	O
a	O
complemented	B-Algorithm
subspace	I-Algorithm
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
a	O
vector	O
subspace	O
for	O
which	O
there	O
exists	O
some	O
other	O
vector	O
subspace	O
of	O
called	O
its	O
(	O
topological	O
)	O
complement	O
in	O
,	O
such	O
that	O
is	O
the	O
direct	O
sum	O
in	O
the	O
category	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
Formally	O
,	O
topological	O
direct	O
sums	O
strengthen	O
the	O
algebraic	O
direct	O
sum	O
by	O
requiring	O
certain	O
maps	O
be	O
continuous	B-Algorithm
;	O
the	O
result	O
retains	O
many	O
nice	O
properties	O
from	O
the	O
operation	O
of	O
direct	O
sum	O
in	O
finite-dimensional	O
vector	O
spaces	O
.	O
</s>
<s>
In	O
general	O
,	O
classifying	O
all	O
complemented	B-Algorithm
subspaces	I-Algorithm
is	O
a	O
difficult	O
problem	O
,	O
which	O
has	O
been	O
solved	O
only	O
for	O
some	O
well-known	O
Banach	O
spaces	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
complemented	B-Algorithm
subspace	I-Algorithm
is	O
analogous	O
to	O
,	O
but	O
distinct	O
from	O
,	O
that	O
of	O
a	O
set	O
complement	O
.	O
</s>
<s>
The	O
map	O
is	O
a	O
morphism	O
in	O
the	O
category	O
of	O
vector	O
spaces	O
—	O
that	O
is	O
to	O
say	O
,	O
linear	B-Architecture
.	O
</s>
<s>
The	O
addition	O
map	O
is	O
bijective	B-Algorithm
.	O
</s>
<s>
In	O
the	O
category	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
that	O
algebraic	O
decomposition	O
becomes	O
less	O
useful	O
.	O
</s>
<s>
The	O
definition	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
requires	O
the	O
addition	O
map	O
to	O
be	O
continuous	B-Algorithm
;	O
its	O
inverse	O
may	O
not	O
be	O
.	O
</s>
<s>
The	O
categorical	O
definition	O
of	O
direct	O
sum	O
,	O
however	O
,	O
requires	O
and	O
to	O
be	O
morphisms	O
—	O
that	O
is	O
,	O
continuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
The	O
addition	O
map	O
is	O
a	O
TVS-isomorphism	O
(	O
that	O
is	O
,	O
a	O
surjective	B-Algorithm
linear	B-Architecture
homeomorphism	O
)	O
.	O
</s>
<s>
is	O
the	O
direct	O
sum	O
of	O
and	O
in	O
the	O
category	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
The	O
map	O
is	O
bijective	B-Algorithm
and	O
open	O
.	O
</s>
<s>
Even	O
if	O
both	O
and	O
are	O
closed	O
in	O
,	O
may	O
still	O
fail	O
to	O
be	O
continuous	B-Algorithm
.	O
</s>
<s>
Condition	O
1(d )	O
above	O
implies	O
that	O
any	O
topological	O
complement	O
of	O
is	O
isomorphic	O
,	O
as	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
,	O
to	O
the	O
quotient	O
vector	O
space	O
.	O
</s>
<s>
Because	O
a	O
linear	B-Architecture
map	I-Architecture
between	O
two	O
normed	O
(	O
or	O
Banach	O
)	O
spaces	O
is	O
bounded	O
if	O
and	O
only	O
if	O
it	O
is	O
continuous	B-Algorithm
,	O
the	O
definition	O
in	O
the	O
categories	O
of	O
normed	O
(	O
resp	O
.	O
</s>
<s>
Banach	O
)	O
spaces	O
is	O
the	O
same	O
as	O
in	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
There	O
exists	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
with	O
image	O
such	O
that	O
;	O
</s>
<s>
There	O
exists	O
a	O
continuous	B-Algorithm
linear	B-Architecture
projection	O
with	O
image	O
such	O
that	O
.	O
</s>
<s>
For	O
every	O
TVS	O
the	O
restriction	O
map	O
is	O
surjective	B-Algorithm
.	O
</s>
<s>
For	O
any	O
two	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
and	O
,	O
the	O
subspaces	O
and	O
are	O
topological	O
complements	O
in	O
.	O
</s>
<s>
This	O
is	O
because	O
has	O
the	O
indiscrete	O
topology	O
,	O
and	O
so	O
the	O
algebraic	O
projection	O
is	O
continuous	B-Algorithm
.	O
</s>
<s>
If	O
and	O
is	O
surjective	B-Algorithm
,	O
then	O
.	O
</s>
<s>
Suppose	O
is	O
Hausdorff	O
and	O
locally	B-Algorithm
convex	I-Algorithm
and	O
a	O
free	O
topological	O
vector	O
subspace	O
:	O
for	O
some	O
set	O
,	O
we	O
have	O
(	O
as	O
a	O
t.v.s.	O
)	O
.	O
</s>
<s>
In	O
arbitrary	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
a	O
finite-dimensional	O
vector	O
subspace	O
is	O
topologically	B-Algorithm
complemented	I-Algorithm
if	O
and	O
only	O
if	O
for	O
every	O
non-zero	O
,	O
there	O
exists	O
a	O
continuous	B-Algorithm
linear	B-Architecture
functional	O
on	O
that	O
separates	O
from	O
.	O
</s>
<s>
In	O
a	O
Hilbert	O
space	O
,	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
any	O
closed	O
vector	O
subspace	O
is	O
always	O
a	O
topological	O
complement	O
of	O
.	O
</s>
<s>
Let	O
be	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
over	O
the	O
field	O
.	O
</s>
<s>
From	O
the	O
existence	O
of	O
Hamel	O
bases	O
,	O
every	O
infinite-dimensional	O
Banach	O
space	O
contains	O
unclosed	O
linear	B-Architecture
subspaces	O
.	O
</s>
<s>
Since	O
any	O
complemented	B-Algorithm
subspace	I-Algorithm
is	O
closed	O
,	O
none	O
of	O
those	O
subspaces	O
is	O
complemented	O
.	O
</s>
<s>
If	O
is	O
a	O
continuous	B-Algorithm
linear	B-Architecture
surjection	B-Algorithm
,	O
then	O
the	O
following	O
conditions	O
are	O
equivalent	O
:	O
</s>
<s>
There	O
exists	O
a	O
"	O
right	O
inverse	O
"	O
:	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
such	O
that	O
,	O
where	O
is	O
the	O
identity	O
map	O
.	O
</s>
<s>
Topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
admit	O
the	O
following	O
Cantor-Schröder-Bernstein	O
–	O
type	O
theorem	O
:	O
</s>
<s>
Understanding	O
the	O
complemented	B-Algorithm
subspaces	I-Algorithm
of	O
an	O
arbitrary	O
Banach	O
space	O
up	O
to	O
isomorphism	O
is	O
a	O
classical	O
problem	O
that	O
has	O
motivated	O
much	O
work	O
in	O
basis	O
theory	O
,	O
particularly	O
the	O
development	O
of	O
absolutely	O
summing	O
operators	O
.	O
</s>
<s>
Most	O
famously	O
,	O
if	O
then	O
the	O
only	O
complemented	B-Algorithm
subspaces	I-Algorithm
of	O
are	O
isomorphic	O
to	O
and	O
the	O
same	O
goes	O
for	O
Such	O
spaces	O
are	O
called	O
(	O
when	O
their	O
only	O
infinite-dimensional	O
complemented	B-Algorithm
subspaces	I-Algorithm
are	O
isomorphic	O
to	O
the	O
original	O
)	O
.	O
</s>
<s>
The	O
spaces	O
are	O
not	O
prime	O
whenever	O
in	O
fact	O
,	O
they	O
admit	O
uncountably	O
many	O
non-isomorphic	O
complemented	B-Algorithm
subspaces	I-Algorithm
.	O
</s>
<s>
No	O
other	O
complemented	B-Algorithm
subspaces	I-Algorithm
of	O
are	O
currently	O
known	O
.	O
</s>
<s>
An	O
infinite-dimensional	O
Banach	O
space	O
is	O
called	O
indecomposable	O
whenever	O
its	O
only	O
complemented	B-Algorithm
subspaces	I-Algorithm
are	O
either	O
finite-dimensional	O
or	O
-codimensional	O
.	O
</s>
