<s>
In	O
the	O
mathematical	O
field	O
of	O
functional	B-Application
analysis	I-Application
there	O
are	O
several	O
standard	O
topologies	B-Architecture
which	O
are	O
given	O
to	O
the	O
algebra	O
of	O
bounded	O
linear	O
operators	O
on	O
a	O
Banach	O
space	O
.	O
</s>
<s>
If	O
,	O
that	O
is	O
,	O
the	O
operator	O
norm	O
of	O
(	O
the	O
supremum	O
of	O
,	O
where	O
ranges	O
over	O
the	O
unit	O
ball	O
in	O
)	O
converges	O
to	O
0	O
,	O
we	O
say	O
that	O
in	O
the	O
uniform	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
If	O
for	O
all	O
,	O
then	O
we	O
say	O
in	O
the	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
Finally	O
,	O
suppose	O
that	O
for	O
all	O
we	O
have	O
in	O
the	O
weak	O
topology	B-Architecture
of	O
.	O
</s>
<s>
In	O
this	O
case	O
we	O
say	O
that	O
in	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
There	O
are	O
many	O
topologies	B-Architecture
that	O
can	O
be	O
defined	O
on	O
besides	O
the	O
ones	O
used	O
above	O
;	O
most	O
are	O
at	O
first	O
only	O
defined	O
when	O
is	O
a	O
Hilbert	O
space	O
,	O
even	O
though	O
in	O
many	O
cases	O
there	O
are	O
appropriate	O
generalisations	O
.	O
</s>
<s>
The	O
topologies	B-Architecture
listed	O
below	O
are	O
all	O
locally	O
convex	O
,	O
which	O
implies	O
that	O
they	O
are	O
defined	O
by	O
a	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
In	O
analysis	O
,	O
a	O
topology	B-Architecture
is	O
called	O
strong	O
if	O
it	O
has	O
many	O
open	O
sets	O
and	O
weak	O
if	O
it	O
has	O
few	O
open	O
sets	O
,	O
so	O
that	O
the	O
corresponding	O
modes	O
of	O
convergence	O
are	O
,	O
respectively	O
,	O
strong	O
and	O
weak	O
.	O
</s>
<s>
(	O
In	O
topology	B-Architecture
proper	O
,	O
these	O
terms	O
can	O
suggest	O
the	O
opposite	O
meaning	O
,	O
so	O
strong	O
and	O
weak	O
are	O
replaced	O
with	O
,	O
respectively	O
,	O
fine	O
and	O
coarse	O
.	O
)	O
</s>
<s>
If	O
is	O
a	O
vector	O
space	O
of	O
linear	O
maps	O
on	O
the	O
vector	O
space	O
,	O
then	O
is	O
defined	O
to	O
be	O
the	O
weakest	O
topology	B-Architecture
on	O
such	O
that	O
all	O
elements	O
of	O
are	O
continuous	O
.	O
</s>
<s>
The	O
norm	O
topology	B-Architecture
or	O
uniform	O
topology	B-Architecture
or	O
uniform	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
is	O
defined	O
by	O
the	O
usual	O
norm	O
||x||	O
on	O
.	O
</s>
<s>
It	O
is	O
stronger	O
than	O
all	O
the	O
other	O
topologies	B-Architecture
below	O
.	O
</s>
<s>
The	O
weak	O
(	O
Banach	O
space	O
)	O
topology	B-Architecture
is	O
,	O
in	O
other	O
words	O
the	O
weakest	O
topology	B-Architecture
such	O
that	O
all	O
elements	O
of	O
the	O
dual	O
are	O
continuous	O
.	O
</s>
<s>
It	O
is	O
the	O
weak	O
topology	B-Architecture
on	O
the	O
Banach	O
space	O
.	O
</s>
<s>
It	O
is	O
stronger	O
than	O
the	O
ultraweak	O
and	O
weak	B-Algorithm
operator	I-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
(	O
Warning	O
:	O
the	O
weak	O
Banach	O
space	O
topology	B-Architecture
and	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
and	O
the	O
ultraweak	B-Algorithm
topology	I-Algorithm
are	O
all	O
sometimes	O
called	O
the	O
weak	O
topology	B-Architecture
,	O
but	O
they	O
are	O
different	O
.	O
)	O
</s>
<s>
The	O
Mackey	B-Algorithm
topology	I-Algorithm
or	O
Arens-Mackey	B-Algorithm
topology	I-Algorithm
is	O
the	O
strongest	O
locally	O
convex	O
topology	B-Architecture
on	O
such	O
that	O
the	O
dual	O
is	O
,	O
and	O
is	O
also	O
the	O
uniform	O
convergence	O
topology	B-Architecture
on	O
,	O
-compact	O
convex	O
subsets	O
of	O
.	O
</s>
<s>
It	O
is	O
stronger	O
than	O
all	O
topologies	B-Architecture
below	O
.	O
</s>
<s>
The	O
σ-strong-*	O
topology	B-Architecture
or	O
ultrastrong-*	O
topology	B-Architecture
is	O
the	O
weakest	O
topology	B-Architecture
stronger	O
than	O
the	O
ultrastrong	B-Algorithm
topology	I-Algorithm
such	O
that	O
the	O
adjoint	O
map	O
is	O
continuous	O
.	O
</s>
<s>
It	O
is	O
stronger	O
than	O
all	O
topologies	B-Architecture
below	O
.	O
</s>
<s>
The	O
σ-strong	O
topology	B-Architecture
or	O
ultrastrong	B-Algorithm
topology	I-Algorithm
or	O
strongest	O
topology	B-Architecture
or	O
strongest	O
operator	B-Algorithm
topology	I-Algorithm
is	O
defined	O
by	O
the	O
family	O
of	O
seminorms	O
for	O
positive	O
elements	O
of	O
.	O
</s>
<s>
It	O
is	O
stronger	O
than	O
all	O
the	O
topologies	B-Architecture
below	O
other	O
than	O
the	O
strong*	O
topology	B-Architecture
.	O
</s>
<s>
Warning	O
:	O
in	O
spite	O
of	O
the	O
name	O
"	O
strongest	O
topology	B-Architecture
"	O
,	O
it	O
is	O
weaker	O
than	O
the	O
norm	O
topology	B-Architecture
.	O
)	O
</s>
<s>
The	O
σ-weak	O
topology	B-Architecture
or	O
ultraweak	B-Algorithm
topology	I-Algorithm
or	O
weak-*	O
operator	B-Algorithm
topology	I-Algorithm
or	O
weak-*	O
topology	B-Architecture
or	O
weak	O
topology	B-Architecture
or	O
)	O
topology	B-Architecture
is	O
defined	O
by	O
the	O
family	O
of	O
seminorms	O
|( w	O
,	O
x	O
)	O
|	O
for	O
elements	O
w	O
of	O
.	O
</s>
<s>
It	O
is	O
stronger	O
than	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
(	O
Warning	O
:	O
the	O
weak	O
Banach	O
space	O
topology	B-Architecture
and	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
and	O
the	O
ultraweak	B-Algorithm
topology	I-Algorithm
are	O
all	O
sometimes	O
called	O
the	O
weak	O
topology	B-Architecture
,	O
but	O
they	O
are	O
different	O
.	O
)	O
</s>
<s>
The	O
strong-*	O
operator	B-Algorithm
topology	I-Algorithm
or	O
strong-*	O
topology	B-Architecture
is	O
defined	O
by	O
the	O
seminorms	O
||	O
x(h )	O
||	O
and	O
||x*( h	O
)	O
||	O
for	O
.	O
</s>
<s>
It	O
is	O
stronger	O
than	O
the	O
strong	O
and	O
weak	B-Algorithm
operator	I-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
(	O
SOT	O
)	O
or	O
strong	O
topology	B-Architecture
is	O
defined	O
by	O
the	O
seminorms	O
||	O
x(h )	O
||	O
for	O
.	O
</s>
<s>
It	O
is	O
stronger	O
than	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
The	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
(	O
WOT	O
)	O
or	O
weak	O
topology	B-Architecture
is	O
defined	O
by	O
the	O
seminorms	O
|	O
(x(h1 )	O
,	O
h2	O
)	O
|	O
for	O
.	O
</s>
<s>
(	O
Warning	O
:	O
the	O
weak	O
Banach	O
space	O
topology	B-Architecture
,	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
,	O
and	O
the	O
ultraweak	B-Algorithm
topology	I-Algorithm
are	O
all	O
sometimes	O
called	O
the	O
weak	O
topology	B-Architecture
,	O
but	O
they	O
are	O
different	O
.	O
)	O
</s>
<s>
The	O
continuous	O
linear	O
functionals	O
on	O
for	O
the	O
ultraweak	O
,	O
ultrastrong	O
,	O
ultrastrong*	O
and	O
Arens-Mackey	B-Algorithm
topologies	I-Algorithm
are	O
the	O
same	O
,	O
and	O
are	O
the	O
elements	O
of	O
the	O
predual	O
.	O
</s>
<s>
By	O
definition	O
,	O
the	O
continuous	O
linear	O
functionals	O
in	O
the	O
norm	O
topology	B-Architecture
are	O
the	O
same	O
as	O
those	O
in	O
the	O
weak	O
Banach	O
space	O
topology	B-Architecture
.	O
</s>
<s>
On	O
norm	O
bounded	O
sets	O
of	O
,	O
the	O
weak	O
(	O
operator	O
)	O
and	O
ultraweak	B-Algorithm
topologies	I-Algorithm
coincide	O
.	O
</s>
<s>
This	O
can	O
be	O
seen	O
via	O
,	O
for	O
instance	O
,	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
topology	B-Architecture
is	O
the	O
same	O
as	O
the	O
strong	O
topology	B-Architecture
on	O
any	O
(	O
norm	O
)	O
bounded	O
subset	O
of	O
.	O
</s>
<s>
Same	O
is	O
true	O
for	O
the	O
Arens-Mackey	B-Algorithm
topology	I-Algorithm
,	O
the	O
ultrastrong*	O
,	O
and	O
the	O
strong*	O
topology	B-Architecture
.	O
</s>
<s>
for	O
all	O
,	O
has	O
closed	O
intersection	O
with	O
the	O
closed	O
ball	O
of	O
radius	O
in	O
the	O
strong*	O
,	O
strong	O
,	O
or	O
weak	O
(	O
operator	O
)	O
topologies	B-Architecture
.	O
</s>
<s>
The	O
norm	O
topology	B-Architecture
is	O
metrizable	O
and	O
the	O
others	O
are	O
not	O
;	O
in	O
fact	O
they	O
fail	O
to	O
be	O
first-countable	O
.	O
</s>
<s>
However	O
,	O
when	O
is	O
separable	O
,	O
all	O
the	O
topologies	B-Architecture
above	O
are	O
metrizable	O
when	O
restricted	O
to	O
the	O
unit	O
ball	O
(	O
or	O
to	O
any	O
norm-bounded	O
subset	O
)	O
.	O
</s>
<s>
The	O
most	O
commonly	O
used	O
topologies	B-Architecture
are	O
the	O
norm	O
,	O
strong	O
,	O
and	O
weak	B-Algorithm
operator	I-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
The	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
is	O
useful	O
for	O
compactness	O
arguments	O
,	O
because	O
the	O
unit	O
ball	O
is	O
compact	O
by	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
The	O
norm	O
topology	B-Architecture
is	O
fundamental	O
because	O
it	O
makes	O
into	O
a	O
Banach	O
space	O
,	O
but	O
it	O
is	O
too	O
strong	O
for	O
many	O
purposes	O
;	O
for	O
example	O
,	O
is	O
not	O
separable	O
in	O
this	O
topology	B-Architecture
.	O
</s>
<s>
The	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
could	O
be	O
the	O
most	O
commonly	O
used	O
.	O
</s>
<s>
The	O
ultraweak	O
and	O
ultrastrong	B-Algorithm
topologies	I-Algorithm
are	O
better-behaved	O
than	O
the	O
weak	O
and	O
strong	B-Algorithm
operator	I-Algorithm
topologies	I-Algorithm
,	O
but	O
their	O
definitions	O
are	O
more	O
complicated	O
,	O
so	O
they	O
are	O
usually	O
not	O
used	O
unless	O
their	O
better	O
properties	O
are	O
really	O
needed	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
dual	O
space	O
of	O
in	O
the	O
weak	O
or	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
is	O
too	O
small	O
to	O
have	O
much	O
analytic	O
content	O
.	O
</s>
<s>
The	O
adjoint	O
map	O
is	O
not	O
continuous	O
in	O
the	O
strong	O
operator	O
and	O
ultrastrong	B-Algorithm
topologies	I-Algorithm
,	O
while	O
the	O
strong*	O
and	O
ultrastrong*	O
topologies	B-Architecture
are	O
modifications	O
so	O
that	O
the	O
adjoint	O
becomes	O
continuous	O
.	O
</s>
<s>
The	O
Arens	O
–	O
Mackey	B-Algorithm
topology	I-Algorithm
and	O
the	O
weak	O
Banach	O
space	O
topology	B-Architecture
are	O
relatively	O
rarely	O
used	O
.	O
</s>
<s>
To	O
summarize	O
,	O
the	O
three	O
essential	O
topologies	B-Architecture
on	O
are	O
the	O
norm	O
,	O
ultrastrong	O
,	O
and	O
ultraweak	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
The	O
weak	O
and	O
strong	B-Algorithm
operator	I-Algorithm
topologies	I-Algorithm
are	O
widely	O
used	O
as	O
convenient	O
approximations	O
to	O
the	O
ultraweak	O
and	O
ultrastrong	B-Algorithm
topologies	I-Algorithm
.	O
</s>
<s>
The	O
other	O
topologies	B-Architecture
are	O
relatively	O
obscure	O
.	O
</s>
