<s>
In	O
functional	B-Application
analysis	I-Application
,	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
,	O
often	O
abbreviated	O
WOT	O
,	O
is	O
the	O
weakest	O
topology	B-Architecture
on	O
the	O
set	O
of	O
bounded	O
operators	O
on	O
a	O
Hilbert	O
space	O
,	O
such	O
that	O
the	O
functional	O
sending	O
an	O
operator	O
to	O
the	O
complex	O
number	O
is	O
continuous	O
for	O
any	O
vectors	O
and	O
in	O
the	O
Hilbert	O
space	O
.	O
</s>
<s>
The	O
WOT	O
is	O
the	O
weakest	O
among	O
all	O
common	O
topologies	B-Architecture
on	O
,	O
the	O
bounded	O
operators	O
on	O
a	O
Hilbert	O
space	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
,	O
or	O
SOT	O
,	O
on	O
is	O
the	O
topology	B-Architecture
of	O
pointwise	O
convergence	O
.	O
</s>
<s>
The	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
the	O
set	O
of	O
bounded	O
operators	O
on	O
a	O
Hilbert	O
space	O
that	O
are	O
continuous	O
in	O
the	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
are	O
precisely	O
those	O
that	O
are	O
continuous	O
in	O
the	O
WOT	O
(	O
actually	O
,	O
the	O
WOT	O
is	O
the	O
weakest	O
operator	B-Algorithm
topology	I-Algorithm
that	O
leaves	O
continuous	O
all	O
strongly	O
continuous	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
the	O
set	O
of	O
bounded	O
operators	O
on	O
the	O
Hilbert	O
space	O
H	O
)	O
.	O
</s>
<s>
The	O
predual	O
of	O
B(H )	O
is	O
the	O
trace	O
class	O
operators	O
C1(H )	O
,	O
and	O
it	O
generates	O
the	O
w*	O
-topology	O
onB(H )	O
,	O
called	O
the	O
weak-star	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
or	O
σ-weak	O
topology	B-Architecture
.	O
</s>
<s>
The	O
weak-operator	O
and	O
σ-weak	O
topologies	B-Architecture
agree	O
on	O
norm-bounded	O
sets	O
inB(H )	O
.	O
</s>
<s>
A	O
net	O
 { Tα } 	O
⊂	O
B(H )	O
converges	O
to	O
T	O
in	O
WOT	O
if	O
and	O
only	O
Tr(TαF )	O
converges	O
to	O
Tr(TF )	O
for	O
all	O
finite-rank	O
operator	O
F	O
.	O
Since	O
every	O
finite-rank	O
operator	O
is	O
trace-class	O
,	O
this	O
implies	O
that	O
WOT	O
is	O
weaker	O
than	O
the	O
σ-weak	O
topology	B-Architecture
.	O
</s>
<s>
Therefore	O
every	O
norm-bounded	O
set	O
is	O
compact	O
in	O
WOT	O
,	O
by	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
(	O
Because	O
WOT	O
coincides	O
with	O
the	O
σ-weak	O
topology	B-Architecture
on	O
bounded	O
sets	O
,	O
multiplication	O
is	O
not	O
jointly	O
continuous	O
in	O
the	O
σ-weak	O
topology	B-Architecture
.	O
)	O
</s>
<s>
The	O
resulting	O
family	O
of	O
seminorms	O
generates	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
on	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
on	O
is	O
generated	O
by	O
the	O
family	O
of	O
seminorms	O
via	O
the	O
rules	O
.	O
</s>
<s>
The	O
different	O
terminology	O
for	O
the	O
various	O
topologies	B-Architecture
on	O
can	O
sometimes	O
be	O
confusing	O
.	O
</s>
<s>
The	O
weak	O
topology	B-Architecture
on	O
a	O
normed	O
space	O
is	O
the	O
coarsest	O
topology	B-Architecture
that	O
makes	O
the	O
linear	B-Algorithm
functionals	I-Algorithm
in	O
continuous	O
;	O
when	O
we	O
take	O
in	O
place	O
of	O
,	O
the	O
weak	O
topology	B-Architecture
can	O
be	O
very	O
different	O
than	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
And	O
while	O
the	O
WOT	O
is	O
formally	O
weaker	O
than	O
the	O
SOT	O
,	O
the	O
SOT	O
is	O
weaker	O
than	O
the	O
operator	O
norm	O
topology	B-Architecture
.	O
</s>
<s>
The	O
WOT	O
on	O
is	O
a	O
formally	O
weaker	O
topology	B-Architecture
than	O
the	O
SOT	O
,	O
but	O
they	O
nevertheless	O
share	O
some	O
important	O
properties	O
.	O
</s>
