<s>
In	O
mathematics	O
,	O
specifically	O
in	O
functional	B-Application
analysis	I-Application
and	O
Hilbert	O
space	O
theory	O
,	O
vector-valued	B-Algorithm
Hahn	I-Algorithm
–	I-Algorithm
Banach	I-Algorithm
theorems	I-Algorithm
are	O
generalizations	O
of	O
the	O
Hahn	O
–	O
Banach	O
theorems	O
from	O
linear	O
functionals	O
(	O
which	O
are	O
always	O
valued	O
in	O
the	O
real	O
numbers	O
or	O
the	O
complex	O
numbers	O
)	O
to	O
linear	O
operators	O
valued	O
in	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
.	O
</s>
<s>
Throughout	O
and	O
will	O
be	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
over	O
the	O
field	O
and	O
will	O
denote	O
the	O
vector	O
space	O
of	O
all	O
continuous	O
linear	O
maps	O
from	O
to	O
,	O
where	O
if	O
and	O
are	O
normed	O
spaces	O
then	O
we	O
endow	O
with	O
its	O
canonical	O
operator	O
norm	O
.	O
</s>
<s>
A	O
TVS	O
has	O
the	O
extension	O
property	O
if	O
for	O
every	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
and	O
every	O
vector	O
subspace	O
of	O
,	O
has	O
the	O
extension	O
property	O
from	O
to	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
is	O
injective	O
if	O
for	O
every	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
containing	O
as	O
a	O
topological	O
vector	O
subspace	O
,	O
there	O
exists	O
a	O
continuous	O
projection	B-Algorithm
from	O
onto	O
.	O
</s>
<s>
the	O
norm	O
of	O
is	O
identical	O
to	O
the	O
usual	O
restriction	O
to	O
of	O
'	O
s	O
norm	O
)	O
,	O
there	O
exists	O
a	O
continuous	O
projection	B-Algorithm
from	O
onto	O
having	O
norm	O
1	O
.	O
</s>
<s>
In	O
order	O
for	O
a	O
TVS	O
to	O
have	O
the	O
extension	O
property	O
,	O
it	O
must	O
be	O
complete	B-Algorithm
(	O
since	O
it	O
must	O
be	O
possible	O
to	O
extend	O
the	O
identity	O
map	O
from	O
to	O
the	O
completion	O
of	O
;	O
that	O
is	O
,	O
to	O
the	O
map	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
continuous	O
linear	O
map	O
from	O
a	O
vector	O
subspace	O
of	O
into	O
a	O
complete	B-Algorithm
Hausdorff	O
space	O
then	O
there	O
always	O
exists	O
a	O
unique	O
continuous	O
linear	O
extension	O
of	O
from	O
to	O
the	O
closure	O
of	O
in	O
.	O
</s>
<s>
Consequently	O
,	O
it	O
suffices	O
to	O
only	O
consider	O
maps	O
from	O
closed	O
vector	O
subspaces	O
into	O
complete	B-Algorithm
Hausdorff	O
spaces	O
.	O
</s>
<s>
Any	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
having	O
the	O
extension	O
property	O
is	O
injective	O
.	O
</s>
<s>
is	O
linearly	O
isometric	O
to	O
a	O
complete	B-Algorithm
Archimedean	B-Algorithm
ordered	I-Algorithm
vector	B-Algorithm
lattice	I-Algorithm
with	O
order	O
unit	O
and	O
endowed	O
with	O
the	O
order	O
unit	O
norm	O
.	O
</s>
<s>
Give	O
its	O
usual	O
product	O
topology	O
,	O
which	O
makes	O
it	O
into	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
.	O
</s>
