<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
branches	O
of	O
mathematics	O
,	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
(	O
also	O
known	O
as	O
Alaoglu	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
)	O
states	O
that	O
the	O
closed	O
unit	O
ball	O
of	O
the	O
dual	O
space	O
of	O
a	O
normed	O
vector	O
space	O
is	O
compact	O
in	O
the	O
weak*	O
topology	O
.	O
</s>
<s>
According	O
to	O
Lawrence	O
Narici	O
and	O
Edward	O
Beckenstein	O
,	O
the	O
Alaoglu	B-Algorithm
theorem	I-Algorithm
is	O
a	O
"	O
very	O
important	O
result	O
-	O
maybe	O
most	O
important	O
fact	O
about	O
the	O
weak-*	O
topology	O
-	O
 [ that ] 	O
echos	O
throughout	O
functional	O
analysis.	O
"	O
</s>
<s>
The	O
Bourbaki	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
is	O
a	O
generalization	O
of	O
the	O
original	O
theorem	O
by	O
Bourbaki	O
to	O
dual	B-Algorithm
topologies	I-Algorithm
on	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
This	O
theorem	O
is	O
also	O
called	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
or	O
the	O
weak-*	O
compactness	O
theorem	O
and	O
it	O
is	O
commonly	O
called	O
simply	O
the	O
Alaoglu	B-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
where	O
the	O
triple	O
forms	O
a	O
dual	B-Algorithm
system	I-Algorithm
called	O
the	O
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
then	O
its	O
continuous	O
dual	O
space	O
will	O
be	O
denoted	O
by	O
where	O
always	O
holds	O
.	O
</s>
<s>
If	O
is	O
a	O
normed	O
vector	O
space	O
,	O
then	O
the	O
polar	B-Algorithm
of	O
a	O
neighborhood	O
is	O
closed	O
and	O
norm-bounded	O
in	O
the	O
dual	O
space	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
is	O
the	O
open	O
(	O
or	O
closed	O
)	O
unit	O
ball	O
in	O
then	O
the	O
polar	B-Algorithm
of	O
is	O
the	O
closed	O
unit	O
ball	O
in	O
the	O
continuous	O
dual	O
space	O
of	O
(	O
with	O
the	O
usual	O
dual	O
norm	O
)	O
.	O
</s>
<s>
F	B-Algorithm
.	I-Algorithm
Riesz	I-Algorithm
theorem	I-Algorithm
)	O
.	O
</s>
<s>
It	O
should	O
be	O
cautioned	O
that	O
despite	O
appearances	O
,	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
does	O
imply	O
that	O
the	O
weak-*	O
topology	O
is	O
locally	O
compact	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
closed	O
unit	O
ball	O
is	O
only	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
the	O
strong	B-Algorithm
topology	I-Algorithm
,	O
but	O
is	O
usually	O
not	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
the	O
weak-*	O
topology	O
,	O
as	O
it	O
has	O
empty	O
interior	O
in	O
the	O
weak*	O
topology	O
,	O
unless	O
the	O
space	O
is	O
finite-dimensional	O
.	O
</s>
<s>
In	O
fact	O
,	O
it	O
is	O
a	O
result	O
of	O
Weil	O
that	O
all	O
locally	O
compact	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
must	O
be	O
finite-dimensional	O
.	O
</s>
<s>
The	O
following	O
elementary	O
proof	O
does	O
not	O
utilize	O
duality	B-Algorithm
theory	I-Algorithm
and	O
requires	O
only	O
basic	O
concepts	O
from	O
set	O
theory	O
,	O
topology	O
,	O
and	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
What	O
is	O
needed	O
from	O
topology	O
is	O
a	O
working	O
knowledge	O
of	O
net	O
convergence	O
in	O
topological	O
spaces	O
and	O
familiarity	O
with	O
the	O
fact	O
that	O
a	O
linear	O
functional	O
is	O
continuous	O
if	O
and	O
only	O
if	O
it	O
is	O
bounded	B-Algorithm
on	O
a	O
neighborhood	O
of	O
the	O
origin	O
(	O
see	O
the	O
articles	O
on	O
continuous	O
linear	O
functionals	O
and	O
sublinear	O
functionals	O
for	O
details	O
)	O
.	O
</s>
<s>
The	O
Cartesian	O
product	O
is	O
usually	O
thought	O
of	O
as	O
the	O
set	O
of	O
all	O
-indexed	O
tuples	B-Application
but	O
,	O
since	O
tuples	B-Application
are	O
technically	O
just	O
functions	O
from	O
an	O
indexing	O
set	O
,	O
it	O
can	O
also	O
be	O
identified	O
with	O
the	O
space	O
of	O
all	O
functions	O
having	O
prototype	O
as	O
is	O
now	O
described	O
:	O
</s>
<s>
Similarly	O
for	O
function	B-Application
composition	I-Application
,	O
if	O
is	O
any	O
function	O
then	O
the	O
net	O
(	O
or	O
sequence	O
)	O
that	O
results	O
from	O
"	O
plugging	O
into	O
"	O
is	O
just	O
the	O
function	O
although	O
this	O
is	O
typically	O
denoted	O
by	O
(	O
or	O
by	O
if	O
is	O
a	O
sequence	O
)	O
.	O
</s>
<s>
It	O
is	O
well	O
known	O
that	O
the	O
product	O
topology	O
is	O
identical	O
to	O
the	O
topology	O
of	O
pointwise	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
This	O
proof	O
will	O
also	O
use	O
the	O
fact	O
that	O
the	O
topology	O
of	O
pointwise	B-Algorithm
convergence	I-Algorithm
is	O
preserved	O
when	O
passing	O
to	O
topological	O
subspaces	O
.	O
</s>
<s>
The	O
essence	O
of	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
can	O
be	O
found	O
in	O
the	O
next	O
proposition	O
,	O
from	O
which	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
follows	O
.	O
</s>
<s>
Unlike	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
,	O
this	O
proposition	O
does	O
require	O
the	O
vector	O
space	O
to	O
endowed	O
with	O
any	O
topology	O
.	O
</s>
<s>
Before	O
proving	O
the	O
proposition	O
above	O
,	O
it	O
is	O
first	O
shown	O
how	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
follows	O
from	O
it	O
(	O
unlike	O
the	O
proposition	O
,	O
Banach	O
–	O
Alaoglu	O
assumes	O
that	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
and	O
that	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
)	O
.	O
</s>
<s>
Observation	O
:	O
If	O
is	O
any	O
set	O
and	O
if	O
is	O
a	O
closed	O
subset	O
of	O
a	O
topological	O
space	O
then	O
is	O
a	O
closed	O
subset	O
of	O
in	O
the	O
topology	O
of	O
pointwise	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
The	O
lemma	O
above	O
actually	O
also	O
follows	O
from	O
its	O
corollary	O
below	O
since	O
is	O
a	O
Hausdorff	O
complete	B-Algorithm
uniform	O
space	O
and	O
any	O
subset	O
of	O
such	O
a	O
space	O
(	O
in	O
particular	O
)	O
is	O
closed	O
if	O
and	O
only	O
if	O
it	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
A	O
closed	O
subset	O
of	O
a	O
complete	B-Algorithm
space	O
is	O
complete	B-Algorithm
,	O
so	O
by	O
the	O
lemma	O
,	O
the	O
space	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
A	O
special	O
case	O
of	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
is	O
the	O
sequential	O
version	O
of	O
the	O
theorem	O
,	O
which	O
asserts	O
that	O
the	O
closed	O
unit	O
ball	O
of	O
the	O
dual	O
space	O
of	O
a	O
separable	O
normed	O
vector	O
space	O
is	O
sequentially	O
compact	O
in	O
the	O
weak-*	O
topology	O
.	O
</s>
<s>
Due	O
to	O
the	O
constructive	O
nature	O
of	O
its	O
proof	O
(	O
as	O
opposed	O
to	O
the	O
general	O
case	O
,	O
which	O
is	O
based	O
on	O
the	O
axiom	O
of	O
choice	O
)	O
,	O
the	O
sequential	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
is	O
often	O
used	O
in	O
the	O
field	O
of	O
partial	O
differential	O
equations	O
to	O
construct	O
solutions	O
to	O
PDE	O
or	O
variational	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
When	O
is	O
the	O
space	O
of	O
finite	O
Radon	O
measures	O
on	O
the	O
real	O
line	O
(	O
so	O
that	O
is	O
the	O
space	O
of	O
continuous	O
functions	O
vanishing	O
at	O
infinity	O
,	O
by	O
the	O
Riesz	O
representation	O
theorem	O
)	O
,	O
the	O
sequential	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
is	O
equivalent	O
to	O
the	O
Helly	O
selection	O
theorem	O
.	O
</s>
<s>
So	O
if	O
is	O
infinite	O
dimensional	O
then	O
its	O
closed	O
unit	O
ball	O
is	O
necessarily	O
compact	O
in	O
the	O
norm	O
topology	O
by	O
F	B-Algorithm
.	I-Algorithm
Riesz	I-Algorithm
's	I-Algorithm
theorem	I-Algorithm
(	O
despite	O
it	O
being	O
weak-*	O
compact	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
reflexive	O
Banach	O
space	O
,	O
then	O
every	O
bounded	B-Algorithm
sequence	O
in	O
has	O
a	O
weakly	O
convergent	O
subsequence	O
.	O
</s>
<s>
(	O
This	O
follows	O
by	O
applying	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
to	O
a	O
weakly	O
metrizable	O
subspace	O
of	O
;	O
or	O
,	O
more	O
succinctly	O
,	O
by	O
applying	O
the	O
Eberlein	O
–	O
Šmulian	O
theorem	O
.	O
)	O
</s>
<s>
In	O
a	O
Hilbert	O
space	O
,	O
every	O
bounded	B-Algorithm
and	O
closed	O
set	O
is	O
weakly	O
relatively	O
compact	O
,	O
hence	O
every	O
bounded	B-Algorithm
net	O
has	O
a	O
weakly	O
convergent	O
subnet	O
(	O
Hilbert	O
spaces	O
are	O
reflexive	O
)	O
.	O
</s>
<s>
As	O
norm-closed	O
,	O
convex	O
sets	O
are	O
weakly	O
closed	O
(	O
Hahn	O
–	O
Banach	O
theorem	O
)	O
,	O
norm-closures	O
of	O
convex	O
bounded	B-Algorithm
sets	I-Algorithm
in	O
Hilbert	O
spaces	O
or	O
reflexive	O
Banach	O
spaces	O
are	O
weakly	O
compact	O
.	O
</s>
<s>
Closed	O
and	O
bounded	B-Algorithm
sets	I-Algorithm
in	O
are	O
precompact	O
with	O
respect	O
to	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
(	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
is	O
weaker	O
than	O
the	O
ultraweak	B-Algorithm
topology	I-Algorithm
which	O
is	O
in	O
turn	O
the	O
weak-*	O
topology	O
with	O
respect	O
to	O
the	O
predual	O
of	O
the	O
trace	O
class	O
operators	O
)	O
.	O
</s>
<s>
Hence	O
bounded	B-Algorithm
sequences	O
of	O
operators	O
have	O
a	O
weak	O
accumulation	O
point	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
has	O
the	O
Heine	O
–	O
Borel	O
property	O
,	O
if	O
equipped	O
with	O
either	O
the	O
weak	O
operator	O
or	O
the	O
ultraweak	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
Most	O
mainstream	O
functional	B-Application
analysis	I-Application
relies	O
on	O
ZF	O
+	O
the	O
axiom	O
of	O
choice	O
,	O
which	O
is	O
often	O
denoted	O
by	O
ZFC	O
.	O
</s>
<s>
In	O
the	O
general	O
case	O
of	O
an	O
arbitrary	O
normed	O
space	O
,	O
the	O
ultrafilter	O
Lemma	O
,	O
which	O
is	O
strictly	O
weaker	O
than	O
the	O
axiom	O
of	O
choice	O
and	O
equivalent	O
to	O
Tychonoff	O
's	O
theorem	O
for	O
compact	O
spaces	O
,	O
suffices	O
for	O
the	O
proof	O
of	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
,	O
and	O
is	O
in	O
fact	O
equivalent	O
to	O
it	O
.	O
</s>
<s>
The	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
is	O
equivalent	O
to	O
the	O
ultrafilter	O
lemma	O
,	O
which	O
implies	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
for	O
real	O
vector	O
spaces	O
(	O
HB	O
)	O
but	O
is	O
not	O
equivalent	O
to	O
it	O
(	O
said	O
differently	O
,	O
Banach	O
–	O
Alaoglu	O
is	O
also	O
strictly	O
stronger	O
than	O
HB	O
)	O
.	O
</s>
<s>
However	O
,	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
is	O
equivalent	O
to	O
the	O
following	O
weak	O
version	O
of	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
for	O
normed	O
space	O
in	O
which	O
the	O
conclusion	O
of	O
compactness	O
(	O
in	O
the	O
weak-*	O
topology	O
of	O
the	O
closed	O
unit	O
ball	O
of	O
the	O
dual	O
space	O
)	O
is	O
replaced	O
with	O
the	O
conclusion	O
of	O
(	O
also	O
sometimes	O
called	O
)	O
;	O
</s>
