<s>
The	O
strongest	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
(	O
TVS	O
)	O
topology	O
on	O
the	O
tensor	O
product	O
of	O
two	O
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
,	O
making	O
the	O
canonical	O
map	O
(	O
defined	O
by	O
sending	O
to	O
)	O
continuous	O
is	O
called	O
the	O
projective	O
topology	O
or	O
the	O
π-topology	O
.	O
</s>
<s>
Throughout	O
let	O
and	O
be	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
and	O
be	O
a	O
linear	O
map	O
.	O
</s>
<s>
denotes	O
the	O
topology	O
of	O
bounded	O
convergence	O
on	O
or	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
on	O
and	O
or	O
denotes	O
endowed	O
with	O
this	O
topology	O
.	O
</s>
<s>
In	O
particular	O
,	O
this	O
allows	O
us	O
to	O
identify	O
the	O
algebraic	O
dual	O
of	O
with	O
the	O
space	O
of	O
bilinear	O
forms	O
on	O
Moreover	O
,	O
if	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
(	O
TVSs	O
)	O
and	O
if	O
is	O
given	O
the	O
-topology	O
then	O
for	O
every	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
this	O
map	O
restricts	O
to	O
a	O
vector	O
space	O
isomorphism	O
from	O
the	O
space	O
of	O
continuous	O
linear	O
mappings	O
onto	O
the	O
space	O
of	O
bilinear	O
mappings	O
.	O
</s>
<s>
Throughout	O
we	O
will	O
let	O
and	O
be	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
(	O
local	O
convexity	O
allows	O
us	O
to	O
define	O
useful	O
topologies	O
)	O
.	O
</s>
<s>
then	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
whose	O
topology	O
is	O
given	O
by	O
the	O
family	O
of	O
all	O
possible	O
tensor	O
products	O
of	O
the	O
two	O
families	O
(	O
i.e.	O
</s>
<s>
In	O
particular	O
,	O
if	O
and	O
are	O
seminormed	O
spaces	O
with	O
seminorms	O
and	O
respectively	O
,	O
then	O
is	O
a	O
seminormable	O
space	O
whose	O
topology	O
is	O
defined	O
by	O
the	O
seminorm	O
If	O
and	O
are	O
normed	O
spaces	O
then	O
is	O
also	O
a	O
normed	O
space	O
,	O
called	O
the	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
of	O
and	O
where	O
the	O
topology	O
induced	O
by	O
is	O
the	O
same	O
as	O
the	O
π-topology	O
.	O
</s>
<s>
If	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
topology	O
on	O
(	O
with	O
this	O
topology	O
will	O
be	O
denoted	O
by	O
)	O
,	O
then	O
is	O
equal	O
to	O
the	O
π-topology	O
if	O
and	O
only	O
if	O
it	O
has	O
the	O
following	O
property	O
:	O
</s>
<s>
Let	O
and	O
be	O
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
.	O
</s>
<s>
However	O
,	O
can	O
always	O
be	O
linearly	O
embedded	O
as	O
a	O
dense	O
vector	O
subspace	O
of	O
some	O
complete	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
,	O
which	O
is	O
generally	O
denoted	O
by	O
via	O
a	O
linear	O
topological	O
embedding	O
.	O
</s>
<s>
Any	O
continuous	O
map	O
on	O
can	O
be	O
extended	O
to	O
a	O
unique	O
continuous	O
map	O
on	O
In	O
particular	O
,	O
if	O
and	O
are	O
continuous	O
linear	O
maps	O
between	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
then	O
their	O
tensor	O
product	O
which	O
is	O
necessarily	O
continuous	O
,	O
can	O
be	O
extended	O
to	O
a	O
unique	O
continuous	O
linear	O
function	O
which	O
may	O
also	O
be	O
denoted	O
by	O
if	O
no	O
ambiguity	O
would	O
arise	O
.	O
</s>
<s>
Note	O
that	O
if	O
and	O
are	O
metrizable	B-Algorithm
then	O
so	O
are	O
and	O
where	O
in	O
particular	O
will	O
be	O
an	O
F-space	B-Algorithm
.	O
</s>
<s>
This	O
topology	O
is	O
coarser	O
than	O
the	O
strong	B-Algorithm
topology	I-Algorithm
and	O
in	O
,	O
Alexander	O
Grothendieck	O
was	O
interested	O
in	O
when	O
these	O
two	O
topologies	O
were	O
identical	O
.	O
</s>
<s>
Grothendieck	O
proved	O
that	O
these	O
topologies	O
are	O
equal	O
when	O
and	O
are	O
both	O
Banach	O
spaces	O
or	O
both	O
are	O
DF-spaces	B-Algorithm
(	O
a	O
class	O
of	O
spaces	O
introduced	O
by	O
Grothendieck	O
)	O
.	O
</s>
<s>
They	O
are	O
also	O
equal	O
when	O
both	O
spaces	O
are	O
Fréchet	O
with	O
one	O
of	O
them	O
being	O
nuclear	B-Algorithm
.	O
</s>
<s>
Given	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
assumed	O
to	O
have	O
the	O
strong	B-Algorithm
topology	I-Algorithm
(	O
so	O
)	O
and	O
unless	O
stated	O
otherwise	O
,	O
the	O
same	O
is	O
true	O
of	O
the	O
bidual	O
(	O
so	O
Alexander	O
Grothendieck	O
characterized	O
the	O
strong	B-Algorithm
dual	I-Algorithm
and	O
bidual	O
for	O
certain	O
situations	O
:	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
two	O
linear	O
maps	O
between	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
The	O
π-topology	O
is	O
finer	O
than	O
the	O
ε-topology	B-Algorithm
(	O
since	O
the	O
canonical	O
bilinear	O
map	O
is	O
continuous	O
)	O
.	O
</s>
<s>
If	O
and	O
are	O
Frechet	B-Algorithm
spaces	I-Algorithm
then	O
is	O
barelled	O
.	O
</s>
<s>
If	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
then	O
the	O
canonical	O
map	O
is	O
a	O
TVS-isomorphism	O
.	O
</s>
<s>
If	O
and	O
are	O
Frechet	B-Algorithm
spaces	I-Algorithm
and	O
is	O
a	O
complete	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
then	O
the	O
canonical	O
vector	O
space	O
isomorphism	O
becomes	O
a	O
homeomorphism	O
when	O
these	O
spaces	O
are	O
given	O
the	O
topologies	O
of	O
uniform	O
convergence	O
on	O
products	O
of	O
compact	O
sets	O
and	O
,	O
for	O
the	O
second	O
one	O
,	O
the	O
topology	O
of	O
compact	O
convergence	O
(	O
i.e.	O
</s>
<s>
Suppose	O
and	O
are	O
Frechet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
nuclear	B-Algorithm
then	O
and	O
are	O
nuclear	B-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
the	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
does	O
not	O
respect	O
subspaces	O
(	O
e.g.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
The	O
range	O
of	O
this	O
map	O
is	O
denoted	O
by	O
and	O
its	O
elements	O
are	O
called	O
nuclear	B-Algorithm
operators	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
convex	O
balanced	O
closed	O
neighborhood	O
of	O
the	O
origin	O
in	O
and	O
is	O
a	O
convex	O
balanced	O
bounded	O
Banach	B-Algorithm
disk	I-Algorithm
in	O
with	O
both	O
and	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
One	O
can	O
define	O
the	O
auxiliary	B-Algorithm
Banach	I-Algorithm
space	I-Algorithm
with	O
the	O
canonical	O
map	O
whose	O
image	O
,	O
is	O
dense	O
in	O
as	O
well	O
as	O
the	O
auxiliary	O
space	O
normed	O
by	O
and	O
with	O
a	O
canonical	O
map	O
being	O
the	O
(	O
continuous	O
)	O
canonical	O
injection	O
.	O
</s>
<s>
Let	O
and	O
be	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
When	O
and	O
are	O
Banach	O
spaces	O
,	O
then	O
this	O
new	O
definition	O
of	O
nuclear	B-Algorithm
mapping''	O
is	O
consistent	O
with	O
the	O
original	O
one	O
given	O
for	O
the	O
special	O
case	O
where	O
and	O
are	O
Banach	O
spaces	O
.	O
</s>
<s>
Every	O
nuclear	B-Algorithm
operator	O
is	O
an	O
integral	O
operator	O
but	O
the	O
converse	O
is	O
not	O
necessarily	O
true	O
.	O
</s>
<s>
However	O
,	O
every	O
integral	O
operator	O
between	O
Hilbert	O
spaces	O
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
Assuming	O
that	O
and	O
are	O
Banach	O
spaces	O
,	O
then	O
the	O
map	O
has	O
norm	O
so	O
it	O
has	O
a	O
continuous	O
extension	O
to	O
a	O
map	O
The	O
range	O
of	O
this	O
map	O
is	O
denoted	O
by	O
and	O
its	O
elements	O
are	O
called	O
nuclear	B-Algorithm
bilinear	O
forms	O
.	O
</s>
<s>
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
nuclear	B-Algorithm
norm	O
of	O
is	O
:	O
</s>
<s>
Note	O
that	O
if	O
is	O
a	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
at	O
most	O
countably	O
many	O
terms	O
in	O
an	O
absolutely	O
summable	O
family	O
are	O
non-0	O
.	O
</s>
<s>
A	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
nuclear	B-Algorithm
if	O
and	O
only	O
if	O
every	O
summable	O
sequence	O
is	O
absolutely	O
summable	O
.	O
</s>
<s>
This	O
is	O
a	O
common	O
occurrence	O
when	O
studying	O
the	O
injective	O
and	O
projective	B-Algorithm
tensor	I-Algorithm
products	I-Algorithm
of	O
function/sequence	O
spaces	O
and	O
TVSs	O
:	O
the	O
"	O
natural	O
way	O
"	O
in	O
which	O
one	O
would	O
define	O
(	O
from	O
scratch	O
)	O
a	O
topology	O
on	O
such	O
a	O
tensor	O
product	O
is	O
frequently	O
equivalent	O
to	O
the	O
projective	O
or	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
topology	O
.	O
</s>
<s>
For	O
any	O
such	O
and	O
any	O
let	O
where	O
defines	O
a	O
seminorm	O
on	O
The	O
family	O
of	O
seminorms	O
generates	O
a	O
topology	O
making	O
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
