<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
branch	O
of	O
mathematics	O
,	O
the	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
,	O
often	O
abbreviated	O
SOT	O
,	O
is	O
the	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
the	O
set	O
of	O
bounded	O
operators	O
on	O
a	O
Hilbert	O
space	O
H	O
induced	O
by	O
the	O
seminorms	O
of	O
the	O
form	O
,	O
as	O
x	O
varies	O
in	O
H	O
.	O
</s>
<s>
Equivalently	O
,	O
it	O
is	O
the	O
coarsest	O
topology	B-Architecture
such	O
that	O
,	O
for	O
each	O
fixed	O
x	O
in	O
H	O
,	O
the	O
evaluation	O
map	O
(	O
taking	O
values	O
in	O
H	O
)	O
is	O
continuous	O
in	O
T	O
.	O
The	O
equivalence	O
of	O
these	O
two	O
definitions	O
can	O
be	O
seen	O
by	O
observing	O
that	O
a	O
subbase	O
for	O
both	O
topologies	B-Architecture
is	O
given	O
by	O
the	O
sets	O
(	O
where	O
T0	O
is	O
any	O
bounded	O
operator	O
on	O
H	O
,	O
x	O
is	O
any	O
vector	O
and	O
ε	O
is	O
any	O
positive	O
real	O
number	O
)	O
.	O
</s>
<s>
In	O
concrete	O
terms	O
,	O
this	O
means	O
that	O
in	O
the	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
if	O
and	O
only	O
if	O
for	O
each	O
x	O
in	O
H	O
.	O
</s>
<s>
The	O
SOT	O
is	O
stronger	O
than	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
and	O
weaker	O
than	O
the	O
norm	O
topology	B-Architecture
.	O
</s>
<s>
The	O
SOT	O
lacks	O
some	O
of	O
the	O
nicer	O
properties	O
that	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
has	O
,	O
but	O
being	O
stronger	O
,	O
things	O
are	O
sometimes	O
easier	O
to	O
prove	O
in	O
this	O
topology	B-Architecture
.	O
</s>
<s>
It	O
can	O
be	O
viewed	O
as	O
more	O
natural	O
,	O
too	O
,	O
since	O
it	O
is	O
simply	O
the	O
topology	B-Architecture
of	O
pointwise	O
convergence	O
.	O
</s>
<s>
The	O
SOT	O
topology	B-Architecture
also	O
provides	O
the	O
framework	O
for	O
the	O
measurable	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
,	O
just	O
as	O
the	O
norm	O
topology	B-Architecture
does	O
for	O
the	O
continuous	O
functional	O
calculus	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
SOT	O
are	O
precisely	O
those	O
continuous	O
in	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
(	O
WOT	O
)	O
.	O
</s>
