<s>
In	O
mathematics	O
,	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
of	O
two	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
was	O
introduced	O
by	O
Alexander	O
Grothendieck	O
and	O
was	O
used	O
by	O
him	O
to	O
define	O
nuclear	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
An	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
is	O
in	O
general	O
not	O
necessarily	O
complete	B-Algorithm
,	O
so	O
its	O
completion	O
is	O
called	O
the	O
.	O
</s>
<s>
Injective	B-Algorithm
tensor	I-Algorithm
products	I-Algorithm
have	O
applications	O
outside	O
of	O
nuclear	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
as	O
described	O
below	O
,	O
up	O
to	O
TVS-isomorphism	O
,	O
many	O
TVSs	O
that	O
are	O
defined	O
for	O
real	O
or	O
complex	O
valued	O
functions	O
,	O
for	O
instance	O
,	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
or	O
the	O
space	O
of	O
continuously	O
differentiable	O
functions	O
,	O
can	O
be	O
immediately	O
extended	O
to	O
functions	O
valued	O
in	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	I-Algorithm
with	O
any	O
need	O
to	O
extend	O
definitions	O
(	O
such	O
as	O
"	O
differentiable	O
at	O
a	O
point	O
"	O
)	O
from	O
real/complex	O
-valued	O
functions	O
to	O
-valued	O
functions	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>
denotes	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
on	O
or	O
the	O
topology	O
of	O
uniform	O
convergence	O
on	O
the	O
convex	O
balanced	O
weakly	O
compact	O
subsets	O
of	O
and	O
or	O
denotes	O
endowed	O
with	O
this	O
topology	O
.	O
</s>
<s>
denotes	O
the	O
Mackey	B-Algorithm
topology	I-Algorithm
on	O
or	O
the	O
topology	O
of	O
uniform	O
convergence	O
on	O
the	O
convex	O
balanced	O
weakly	O
compact	O
subsets	O
of	O
and	O
or	O
denotes	O
endowed	O
with	O
this	O
topology	O
.	O
</s>
<s>
Henceforth	O
,	O
all	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
considered	O
will	O
be	O
assumed	O
to	O
be	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
Every	O
set	O
is	O
bounded	O
,	O
which	O
is	O
necessary	O
and	O
sufficient	O
for	O
the	O
collection	O
of	O
all	O
such	O
to	O
form	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	I-Algorithm
topology	O
on	O
called	O
the	O
-topology	O
.	O
</s>
<s>
A	O
set	O
of	O
continuous	O
linear	O
functionals	O
on	O
a	O
TVS	O
is	O
equicontinuous	O
if	O
and	O
only	O
if	O
it	O
is	O
contained	O
in	O
the	O
polar	B-Algorithm
of	O
some	O
neighborhood	O
of	O
in	O
;	O
that	O
is	O
,	O
</s>
<s>
This	O
fact	O
together	O
with	O
the	O
bipolar	O
theorem	O
means	O
that	O
via	O
the	O
operation	O
of	O
taking	O
the	O
polar	B-Algorithm
of	O
a	O
subset	O
,	O
the	O
collection	O
of	O
all	O
equicontinuous	O
subsets	O
of	O
"	O
encodes	O
"	O
all	O
information	O
about	O
'	O
s	O
given	O
topology	O
.	O
</s>
<s>
Specifically	O
,	O
distinct	O
LCTVS	B-Algorithm
topologies	O
on	O
produce	O
distinct	O
collections	O
of	O
equicontinuous	O
subsets	O
and	O
conversely	O
,	O
given	O
any	O
such	O
collection	O
of	O
equicontinuous	O
sets	O
,	O
the	O
TVS	O
's	O
original	O
topology	O
can	O
be	O
recovered	O
by	O
taking	O
the	O
polar	B-Algorithm
of	O
every	O
(	O
equicontinuous	O
)	O
set	O
in	O
the	O
collection	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
the	O
article	O
now	O
lists	O
some	O
properties	O
of	O
equicontinuous	O
sets	O
that	O
are	O
relevant	O
for	O
dealing	O
with	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
and	O
is	O
locally	B-Algorithm
convex	I-Algorithm
then	O
for	O
any	O
subset	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
If	O
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
then	O
for	O
any	O
subset	O
the	O
following	O
are	O
equivalent	O
:	O
</s>
<s>
We	O
mention	O
some	O
additional	O
important	O
basic	O
properties	O
relevant	O
to	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
:	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
bilinear	O
map	O
where	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
is	O
metrizable	O
,	O
and	O
is	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
If	O
is	O
normed	O
then	O
is	O
in	O
fact	O
a	O
topological	O
vector	O
subspace	O
of	O
And	O
if	O
in	O
addition	O
is	O
Banach	O
then	O
so	O
is	O
(	O
even	O
if	O
is	O
not	O
complete	B-Algorithm
)	O
.	O
</s>
<s>
The	O
canonical	O
map	O
is	O
always	O
continuous	O
and	O
the	O
ε-topology	O
is	O
always	O
finer	O
than	O
the	O
π-topology	B-Algorithm
and	O
coarser	O
than	O
the	O
inductive	B-Algorithm
topology	I-Algorithm
(	O
which	O
is	O
the	O
finest	O
locally	B-Algorithm
convex	I-Algorithm
TVS	I-Algorithm
topology	O
making	O
separately	O
continuous	O
)	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
strongest	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
making	O
the	O
canonical	O
map	O
(	O
defined	O
by	O
sending	O
to	O
the	O
bilinear	O
form	O
)	O
continuous	O
is	O
called	O
the	O
projective	O
topology	O
or	O
the	O
-topology	O
.	O
</s>
<s>
The	O
following	O
definition	O
was	O
used	O
by	O
Grothendieck	O
to	O
define	O
nuclear	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Definition	O
0	O
:	O
Let	O
be	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Then	O
is	O
nuclear	O
if	O
for	O
any	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
the	O
canonical	O
vector	O
space	O
embedding	O
is	O
an	O
embedding	O
of	O
TVSs	O
whose	O
image	O
is	O
dense	O
in	O
the	O
codomain	O
.	O
</s>
<s>
These	O
identifications	O
will	O
be	O
used	O
to	O
define	O
important	O
subspaces	O
and	O
topologies	O
(	O
particularly	O
those	O
that	O
relate	O
to	O
nuclear	B-Algorithm
operators	I-Algorithm
and	O
nuclear	B-Algorithm
spaces	I-Algorithm
)	O
.	O
</s>
<s>
If	O
and	O
are	O
Hilbert	O
spaces	O
then	O
is	O
injective	O
and	O
the	O
dual	O
of	O
is	O
canonically	O
isometrically	O
isomorphic	O
to	O
the	O
vector	O
space	O
of	O
nuclear	B-Algorithm
operators	I-Algorithm
from	O
into	O
(	O
with	O
the	O
trace	O
norm	O
)	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
injective	O
or	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
topology	O
.	O
</s>
<s>
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>
<s>
Throughout	O
,	O
let	O
be	O
an	O
open	O
subset	O
of	O
where	O
is	O
an	O
integer	O
and	O
let	O
be	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
(	O
TVS	O
)	O
.	O
</s>
<s>
One	O
may	O
then	O
define	O
topologies	O
on	O
and	O
in	O
the	O
same	O
manner	O
as	O
the	O
topologies	O
on	O
and	O
are	O
defined	O
for	O
the	O
space	O
of	O
distributions	O
and	O
test	O
functions	O
(	O
see	O
the	O
article	O
:	O
Differentiable	B-Algorithm
vector-valued	I-Algorithm
functions	I-Algorithm
from	I-Algorithm
Euclidean	I-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
All	O
of	O
this	O
work	O
in	O
extending	O
the	O
definition	O
of	O
differentiability	O
and	O
various	O
topologies	O
turns	O
out	O
to	O
be	O
exactly	O
equivalent	O
to	O
simply	O
taking	O
the	O
completed	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
:	O
</s>
<s>
We	O
will	O
now	O
generalize	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
to	O
functions	O
valued	O
in	O
a	O
TVS	O
.	O
</s>
<s>
To	O
generalize	O
the	O
topology	O
of	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
to	O
we	O
give	O
the	O
topology	O
of	O
uniform	O
convergence	O
over	O
of	O
the	O
functions	O
as	O
and	O
vary	O
over	O
all	O
possible	O
pairs	O
of	O
polynomials	O
in	O
variables	O
.	O
</s>
