Download Closed categories and topological vector spaces

C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
M ICHAEL BARR
Closed categories and topological vector spaces
Cahiers de topologie et géométrie différentielle catégoriques,
tome 17, no 3 (1976), p. 223-234.
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Article numérisé dans le cadre du programme
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(1976)
Vol. XVII-,3
CAHIERS DE TOPOLOGIE
ET GEOMETRIE DIFFERENTIELLE
CLOSED CATEGORIES AND TOPOLOGICAL VECTOR SPACES
by Michael
BARR
I
INTRODUCTION
paper [1],
In the
two
a
duality
theories
discrete field K .
set
on
henceforth referred
category 2
the
They
were
of linear functionals. The
flexive spaces ( i. e.
of
to as
DVS,
topological
considered
we
vector
each described
first, the weak
isomorphic
to
monoidal category when the hom
by a certain topology
dual, led to a category
their second dual) which gave
sets
are
over
spaces
a
the
on
of
re-
closed
topologized by pointwise
con-
vergence.
second, strong duality, was based on uniform convergence on
linearly compact (LC) subspaces. This led to a nicer duality theory (now
the discrete spaces are reflexive) but we did not describe there any closed
The
monoidal category based
It is clear that
haves well
on
on
one cannot
all spaces
of the closed monoidal
that strong hom . In this paper
or even
expect
an
we
fill that gap.
internal hom-functor which be-
all reflexive
ones.
It is
a
consequence
LC spaces
product
must be LC ( see Section 1). Such a product is totally bounded (in a suitable generalized sense which is, together with completeness, equivalent
to linear compactness). This fact suggests looking at a subcategory of spaces which satisfy same completeness condition. If the category is to have
a self-duality theory, a dual condition is imposed as well. When this is done
the result is indeed a closed monoidal category in which every object is
reflexive. The set of morphisms between two spaces is topologized by a
topology finer (possibly) than LC convergence to provide the internal hom .
*
I would like
Canada for
a
to
structure
that the
tensor
of
two
thank the Canada Council and the National Research Council of
supporting
this resarch, and the I.F.M.,
congenial environment.
223
E.T.H., Zurich, for providing
The
dual,
moreover, has the strong
topology.
One word about notation. For the
most
part
adhere
we
that of DVS.
to
significant change. Owing to a lack of enough kinds
of brackets and hieroglyphs, we take advantage of the fact that this paper
concerns exclusively itself with the strong hom to use (-, -) and (- )* to
refer to the strong hom and strong dual, respectively. Similarly, a space
V is called reflexive if it is isomorphic with the strong second dual, here
However there is
one
denoted V**.
1. Preliminaries.
The
equipped
of this paper is the
subject
are
maps U - v
topologized by
subspace
Uo
where
is
spaces,
let
we
LC
subspace
Let U be
with
limits and has
It
denote the
on
DVS,
Specifically,
set
of continuous linear
LC
Sl
spaces.
A basic
in U and
V0
is
an
open
subspace
of V.
V*=(V,K).
1.1.
projective
( U, V )
uniform convergence
P R O P O S IT ION
is in
described in
is
an
We l et
certainly
question. Let
PROOF.
as
with the strong internal hom functor defined there.
if U and V
open
category 2
does
at
fixed space. The functor ( U,- )
an adjoint - 0 U.
a
the
underlying
set
level
so
only
the
commutes
topology
projection. A basic open set in (U , V ) is {f)l f (U0) C V0} where
Uo is an LC subspace of U and Vo an open subspace of V . We can suppose Vo
TT Ww where Ww is open in Vw and is, Vw for all but a finite
set 00 of i ndices. Then
be the
=
There is
no
restriction
on
the other coordinates of
224
f.
Then
I fl f (U0) C V0}
corresponds
the
to
set
The argument for
V is
equalizers is easy and is omitted. In fact, when
subspace of W , ( U, V) has the subspace topology in ( U, W).
Now the existence of the adjoint follows from the special adjoint
a
( cf. DVS, 1.2 -1.4 ).
theorem
This hom is
not
symmetric
and is
not
closed monoidal. A map from
easily seen to be a bilinear map U X V - W which is, for
each u f U , continuous on V , and for each LC subspace C v , an equicontinuous family on U . From this it is easy to see the assymmetry. To
Uh
see
W
to
that
would
can
be
don’t
we
imply
even
that the
a
get
closed monoidal category,
equivalences
we
observe that that
between maps
isomorphism (UXV, W) = (U, (V, W)) (see [2],11.3).
Suppose X and Y are infinite sets, U K X and V Ky . Then assuming
that the above isomorphism held, we would have ( U X V ) *= ( U , V*), which
arise from
natural
a
=
=
can
be
directly
calculated
to
proper when X and Y are
braic
a
tensor
map
This
There
tion. A
which is
has
no
is, however,
an
clearly
space V
is called
KXXY
Let W be the subspace of
infinite, whose elements
product K X XKY .
KX-&#x3E; (KY, W)
topology.
be
are
KXxY
I
those of the
alge-
purely algebraic grounds there is
continuous when W is given the subspace
Then
on
continuous extension
alternative. To
KXxY -&#x3E;
explain
it
(linearly) totally bounded
we
W.
require
a
defini-
if for every open sub-
U , there is a finite number of vectors vl , ... , vn which, together
with U , span V . Equivalently, every discrete quotient is finite dimensional. The obvious analogy of this definition with the usual one is strengthened by the following proposition whose proof is quite easy and is omitted.
space
P RO P O SITION 1.2.
The space V is LC
ded.
225
iff it
is
complete and totally
boun-
PROPOSITION
is
bounded.
totally
Let W be discrete and
P ROO F.
U , ( V, W) and
ted by the images
Each of these in
S-
of
a
and
image
space U is
a
spaces is
denoted CZ .
P R O P O S I T I O N
2.1.
Let V0 -&#x3E;
space V. The
which
we
which
tends, since
The
may
as
is
given,
is
a
finite di-
if every closed
a
(-space iff
every map to U
be extended
can
totally bounded
subcategory of I -
The full
well suppose is U . Now U is
it is
a
to
from
a
the whole space.
complete
is trivial and
2.2.
so
LC, hence is
a
power
uniform space and thus the map
continuous linear function is
converse
PROPOSITION
P RO O F .
finite dimen-
given where Vo is a dense subspace of the LC
image Uo C U is totally bounded and hence has an LC clo-
discrete and LC spaces
a
a
U be
means
a
(-space
a
The space U i s
a LC space to U
the
uniformly
proposition
follows.
a
Let U be
a
(-space.
ex-
continuous.
(*-space provided U* is a (-space. Since
are C-spaces, they are each (*-spaces.
We say that U is
U ** is
is
image f (U X V)
(or, equivalently, complete).
subspace of
PROOF.
K,
W whose
C*-spaces.
is LC
of
map V -
the whole
is genera-
image
subspace of W
subspace
sure
say
a
a
g(u1 ), ... , g(un) .
elements,
defines
turn
We say that
dense
U X V ,-&#x3E; W. Then there corresponds
finite number of
subspace of W, and thus
mensional
2.
f:
the latter space is discrete. Hence the
g:
sional
V be LC. Then UXV
totally bounded and
Let U be
1.3.
Then U* is
a
(*-space;
both
i. e.,
ispace.
Let V0 -&#x3E; V be
we
a
dense inclusion with V
have, using the fact that U - is
gram
226
an
LC space. If
a (-space,
ho -&#x3E;
U **
the commutative dia-
Double dualization
If U is
a
gives
us
required V =
the
space, it has
a
denote the intersection of all the
dently, U
is
a
dense
completion U" ,
uniform
C-subspaces
subspace V C U", let (I V
totally bounded subspaces of V. For
for
and,
limit
a
Let C V be the
is
a
we
let CU
of U" which contain U . Evi-
be the union of the closures
an
ordinal 11,
let
let
ordinal ft ,
union of all the
v
SU = CooU.
PROPOSITION 2.3.
PROO F.
and
subspace of CU.
For any
of the
V** - U** .
It is clear that
SooU
is closed under the
U.
C-space containing
Thus (V C Soo U ,
operation
while the
of
C,
and hence
reverse
inclusion
is obvious.
PROPOSITION
2.4.
The construction U l-&#x3E;
with the inclusion U
C U,
determines
SU is a functor which, together
a left adjoint to the inclusion of
.
P ROO F.
induced
f: U
map f -:
Let
a
- V be
W0
we
see
map. Since it is
uniformly continuous,
U- -&#x3E; V-. It is clearly sufficient
But since the continuous
ed,
a
image
of
that whenever W is
a
a
to
totally bounded
subspace of U"’
show that
space is
with
C W is
that
this
f- ((U )CSv.
totally
bound-
f (W) C S V ,
and
totally bounded, f ( cl ( W0 )) C (Vas well. From this it follows
f(S1W)CSV, and so we see by induction f((W)C’V. Applying
to U , we see that f (SU) C C V.
PROPOSITION 2.5.
Let U be
Then
reflexive.
so
can
there is
be extended
to
The
so
that
diagram
227
is CU.
commutes, the first square
U is dense
Thus’ U
in (U
and the top map is the
Let U be
Let Vo -
V be
a
reflexive ’*-space.
a
C-space,
and since V * is discrete, hence
dual is
The
identity,
dense inclusion with V
a
extends, since U * is
this
the second
a
outer
to a
there is
square and lower
a
complete,
use
is closed in
a
Then
so
is
CU.
LC space. Given
U * . This
this extends
gives
to £ U -
a
map
us
V* whose
X2
if
space and
is
Xi n X2
X1
a
and
so
X2
does the upper
are
triangle.
of X , say
Equivalently
subsets
closed subset of
X2.
closed subset
without
closed in
one.
of
triangle
topological
X’1 C X
We
is the bottom
map
If X is
X1
an
map V-&#x3E;
commute, and since the lower map is 1-1 and onto,
that
by naturality. Since
so
is reflexive.
PROPOSITION 2.6.
P RO O F .
by construction,
X4
proof
implies
such that
the obvious assertion that
X1 n X2
that
Let
{U w }
subspace of TT U. Then
U is
PROPOSITION
2.7.
is closed in
XI closed
X3 n X4.
in
X2
and
X3
family of discrete spaces and U be a
a S-space iff for every choice of a collection
be
a
228
of finite
dimensional
subspaces V w C Uw ,
U is closed in
Suppose the latter condition is satisfied
ly bounded subspace of U . Then the image of
PROOF.
is
totally bounded, hence finite
ly Uo C TTVw, and since Uo is
and
TT Vw .
U0
is
closed in
subspace Vw C Uw . EvidentU , it is closed in U n TT Vw,
TT Vc. Thus Uo is LC. Conversely, if U is C-space,
any collection ( V (L)} of finite dimensional subspaces, U n TT Vw is
then for
closed
in II
closed total-
dimensional
a
which is closed in
a
a
totally
a
bounded subspace of U and hence is LC, hence closed
V (L).
3. The internal hom .
If U and V
( U, V) denotes the set of
continuous linear mappings U - V topologized by taking as a base of open
subspaces {f I {(U0) C V0 I where Uo is an LC subspace of U and Vo
an open subspace of V . An equivalent description is that ( U , V) is topologized as a subspace of TT(UY, V/ Vw) where U Y ranges over the LC subspaces of U
V / V(L)
crete
range
are
and Vw
over
topology.
the
From
spaces,
recall that
subspaces of V . We may consider that
discrete quotients of V. Each factor is given the disthat description and the duality between discrete and
over
LC spaces, the
following
PROPOSITION
3.1.
ence
we
the open
becomes
a
formal exercise.
L et U and V be
reflexive
between maps U , V and V* , U* underlies
an
Suppose U is a reflexive S*-space and V
( U, h) is a £.space.
LEMMA 3.2.
Then
spaces. Then the
P RO O F .
Let {UY}
and
{Vw}
equival-
isomorphism
a
reflexive’ -space.
the LC and open
subspaces, respfinite dimensional subspace of
(UY, V/ Vw) is
range
over
ectively, of U and V. A
spanned by a finite number of maps, each of which has a finite dimensional
range. Thus altogether it is contained in a subspace of the form
229
where
is
U,,,
a
nite dimensional
cofinite dimensional
of
subspace
families of finite dimensional
that for all
and
a
finite dimensional
is closed in fl
cv
To
apply
subspaces
of indices
pairs w ifI
glr , V
each
h/ Vw .
a
U,
and
is
a
fi-
to
consider
us
suppose
dimensional quotient
UY /UYw
have been chosen.
Then for
so
is closed in
1.
is closed in
It follows that
VYw /Vw
2.7 it is sufficient
VYw/w,
(U Y) V)
subspace
and
of the factors. So let
cofinite
VYw/ Vw
of
subspace
Using 3.1,
we
q
have
and
similarly that,
for
each ;&#x26; , (U, V/ V.) is
closed in J
is closed in
so
is closed in
A collection of maps
a
U4 -+
collection of maps U -
map in both
TT (UY, V)
The top map is
fill-in
is
a
ly
any map
and hence
on
the
onto
and
V is the
,
and
similarly
is
subspace inclusion and hence there
intersection is exactly ( U, V). Clear-
and the lower
diagonal.
map :
equivalent to one U - II V / V úJ. Now a
TT (U , V/ V(L)) corresponds to a commutative square
V/V (L)
a
Thus the
Uy/UY w -&#x3E; V 41 w /Vw
to
same as a
belongs
their intersection. Thus
to
both
(U, V)
230
is closed in
C-space.
and is
a
4. The
category R.
Ye let 31
denote the full
The
PROPOSITION 4. 1.
a
The map
each 1-1 and
map,
we
UF l-&#x3E; 6 U
functor
(((U*))*
=
For any U, 8 U -&#x3E; U is 1-1 and
inclusion R -&#x3E; SB.
are
whose
objects
the
are
S-S*-spaces.
reflexive
P ROO F.
subcategory of b
U* -&#x3E; S U*
onto
and
so
is
a
dense inclusion
their
is
right adjoint
to the
onto.
so
that
is. If h is in N and V-&#x3E;U is
composite
get
Now U* is
a
reflexive ( DVS 4.4 )
and hence its dual is reflexive
as
(-(*-space (2.2),
and
so
I s SU* (2.6),
well. If h is in fl and V -&#x3E;
U,
we
get
and then
The other direction
We
now
comes
from
define, for U, h in R, [U, V] = 6 (U, V)
consists of the continuous maps U - V with
than that of uniform convergence
U*
[U, K]
If
is
U, V
on
LC
a
(cf. 3.2). It
topology (possibly)
subspaces. Note,
finer
of course, that
unchanged.
are
in 91 and
W0
is
its closure W is LC. Then 8 W = W is
a
totally
an
LC
bounded subset of
subspace
of
[U, V]
( ll , V) ,
and
con-
same W0 . Thus Wo is totally bounded in [U, V] . The converse
being clear, we see that ( U , V) and [U, h] have the same totally bounded
subspaces.
tains the
LEMMA 4.2.
is
,Suppose U, V in 91. Any totally
equicontinuous.
231
bounded
subspace of [U, V]
(U, V ) be totally bounded. Corresponding to W -&#x3E; (U, V )
we have Jl’ - (V*, U*) (3.1), and thus WX V* -&#x3E; U * . If Vo
is an open subspace of V , its annihilator ann Vo in V * is LC . This follows from the reflexivity of V and the definition of the topology V** . Then WX(ann V0 ) is
totally bounded (1.2 ) and hence so is its image in U * . The closure of that
image is an LC subspace of U * which we can call ann Uo , with Uo open
in U ( same reason as above). From this it is clear that the image of W X U0
is in Vo , which means that W is equicontinuous.
PROO F.
Let W C
COROLLARY
between maps
PROOF.
U, V, W be in R. There
U , [ V, W] and V - [ U, W] .
L et
4.3.
U-&#x3E;[V,W] gives U -&#x3E; (V, W)
A map
is
a
1-1
correspondence
and UXV -&#x3E; W. To any LC
equicontinuous family V , W . Certainly any
vc V gives a continuous map U , W and thus, by the discussion in Section
1, we get V -+ (U, W) and then V -&#x3E; [U,W] .
subspace
of U
PROPOSITION
corresponds
4.4.
an
[U, V]= [V*, U*] by
Let U and V be in R. Then
the natural map.
PROOF.
Apply 6
Now
we
P R O P O S IT IO N
to
define, for U , h in R,
4.5.
dence between maps
PROOF.
PROPOSITION
an
a
1-1
a
1-1 correspon-
correspondence
For any i
4.7.
Let
U, V be LC
spaces. Then U0V
=
S(U XV)
and
L C space.
PROOF,
W is
U, V, W be in 31. Then there is
U X V -+ W and maps U -&#x3E; I V , W].
Let
Each of the transformations below is
COROLLARY 4.6.
is
both side s in 3.1.
We know it is
in 31 ,
totally
bounded (1.2)
so
each of the transformations below is
232
that
a
S( U X V)
1-1
is LC. When
correspondence:
Let U, V and W’
belong
to
tion of maps (V, W) x (U, V) -&#x3E; (U, W)
arises
from
PROPOSITION
PROOF.
and
Wo
If G is
an
tion map
is
an
LC
4.8.
open
of
(U,. V ), U0
a
map
an
the closure of the
LC
subspace
image
of
U,
of the evalua-
-
subspace
is transformed
in V. Then the basic open
set
in
(V, W),
by G into
Thus G determines
so we
LC
subspace
subspace of W,
an
m. The natural composi-
an
equicontinuous family
of maps
( V, W) -&#x3E; (U, W)
and
have the indicated map.
PROPOSITION 4.9.
tion arises
P RO O F .
from
Let
U, V and W belong to 91. Then natural composi-
map
FC[V, W] and GC[U, V]
and the fact that
and then,
a
Let
(U, W)
is
a (-space,
we
be LC
subspaces.
From
have
by adjointness,
gives us G -&#x3E; [F, [U, W]]] . Now G is an LC space and hence defines an equicontinuous family. Each fc F gives, by composition, a continuous function to -: ( U, V ) -&#x3E; ( U, W) which extends by functoriality to
a continuous function [ U, V] -&#x3E; [U, W]. Hence we have a bilinear map:
F x [ U, V] -&#x3E; [U, W] which is, for all ff F, continuous on [U, V], and for
all LC subspaces G C [U, V] , equicontinuous on F . There results a map
This
233
F X[U, V] - [U, W] which gives F -&#x3E; ([ U, V), [ U, W] ) and, by adjointness, F -&#x3E; [[ U, V], [ U, W]] . Then F determines an equicontinuous family,
so that we have a bilinear map [ V, W] X [U , V]-&#x3E; , [ U, W] with the property
that every LC subspace of [ V, W] determines an equicontinuous family on
[U, V] . Repeating the argument used above, we see that every element
of [ U , V] determines a continuous map on [V, W] . Hence we have
THEOREM 4-io.
The
monoidal category in
category N, equipped with - @ - and [ - , - ] ,
which every object is reflexive.
isaclosed
REFERENCES.
1. M.
BARR, Duality of
vector
spaces,
Cahiers
Topo.
et
Géo.
Diff.
XVII- 1 (1976),
3- 14.
2. S. EILENBERG
Algebra
and G. M.
( L a Jolla, 1965),
KELLY, Closed categories, Proc.
Springer, 1966.
pp. 421 - 562.
Mathem atics Department
Me Gill
University
P . O. Bo x 6070, Station A
MONTREAL, P. Q. H3C - 3GI
CANADA
234
Conf. Categorical