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C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
M ICHAEL BARR
Duality of vector spaces
Cahiers de topologie et géométrie différentielle catégoriques,
tome 17, no 1 (1976), p. 3-14.
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Vol. XVII-1 (1976)
CAHIERS DE TOPOLOGIE
ET GEOMETRIE DIFFERENTIELLE
DUALITY OF VECTOR SPACES
*
by Michael BARR
duality theory for vector spaces
by introducing a topology into the dual is old - it apparently goes back to
Lefschetz ([3], pp. 78-83) - but is surprisingly little known. Briefly, if
The notion that
V is
an
many
more
infinite dimensional
functionals
less those that
continuous
ground
on
field
space and V* its
dual, then there
are
are
induced
topologized by pointwise convergence, the
discrete.
we
over a
spaces
V* than
nice
a
vector
V* when it is
being
get
by elements of V . Nonetheidentified are precisely the functionals which are
on
be
can
In this paper
vector
one can
extend this
duality
discrete field. The
to
the category of
structure
involved
topological
turns out to
be
rather well-behaved.
subsequent paper(1) I will extend many of these ideas to the
category of (real or complex) Banach spaces. In a few cases in the present
paper, the desire to smooth out this extension has led to a slightly more
cumbersome exposition.
In
a
In accordance with the doctrine of
don’t
itch&#x3E;),
we
leave aside all
questions
Reyes («Don’t
scratch if you
of coherence. The
concreteness
of the constructions guarantees that all necessary coherence conditions
are
satisfied.
* N.D. R.
Cet
article
donnees par M. BARR
conferences
logie
et
ont
et
le suivant
aux
Journees
sont
le
same
Chantilly
o
des
deux conférences
( Septembre 1975 ). Ces
ete r6sum6es dans le volume XVI - 4 ( 1975 ) des «Cahiers de Topo-
Géométrie Differentielle».
( 1) In this
développement
T.A.C. de
issue.
3
I would like
to
Council of Canada for
thank the Canada Council and the National Research
supporting
this work. In addition I would like
to
thank
University and the Forschungsinstitut f.
Mathematik, E.T.H., Zurich, for providing stimulating environments for carrying it out, in particular the former for the use of their magnificient library
the Mathematisk Institut of Aarhus
facilities.
1. Preliminaries.
Let K be
( i. e. discrete) field. By
abstract
an
separated linearly
a
equipped with a topotopologized K-vector space
logy for which addition V X V - V and scalar multiplication K X V - V are
continuous. In addition we require that for any v # 0 , there be an open subwe mean a vector
U
space
veU.
with
By
a
linear map.
The category
understood.
The
described
( linear ) functional
is denoted
joint
V - )
on
If
V
where W is
the
E B,
a
linear
if the
family
( see
[1] , [3] ,
following
analogues.
same vector
a
if K is
whose
The
are’the discrete spa-
inclusion D -&#x3E; B
space with the discrete
subset A C V is called
subspace
objects
a
linear
of V . We say that V
has
a
left ad-
topology.
subvariety if A v + W
is linearly compact (LC)
=
of closed linear subvarieties has the finite intersection property
for details
assertions
PROPOSITION
1.
are
as
well
as
easily proved
proofs
of the assertions
in ways similar
to
below). The
their
topological
A closed
LC space is LC; the
subspace and a (separated) quotient space o f an
topological product of LC spaces is LC; a morphism
whose domain is LC is closed and in
onto
simply 2
or
V .
subcategory of B(K)
by D(K) or simply .
lVl,
call 2 ( K)
continuous
a
equipped -necessarily-with the disby K . A morphism V -K is called a
The full
ces
we
we mean
dimensional space,
one
will also be denoted
topology,
crete
of such spaces
morphism
so
space V
another space is
an
particular
isomorphism.
a
1-1 map
from
A discrete space is LC
an
LC space
iff it
is
finite
dimensional.
The full
subcategory
of LC spaces is
4
denoted G( K ) ,
or
simply S.
The inclusion C -&#x3E; B has
that when K is
a
field,
l-&#x3E;BV.
V
right adjoint,
linear compactness is
It is worth
exactly
the
compactness. Thus the concept of linear compactness
ological
the extension
as
finite
a
field of notions
arbitrary
to an
easily
mentioning
same as
can
top-
be viewed
described for finite
ones.
It is well known
the
that C and D
of all functionals of V
set
product KV. Since
K is LC
are
topologized
(trivial),
so
V E 2, then V* E C is
subset of the topological
dual. If
as
is the
a
product.
For any
v2EV.
vll
is the inverse
image
of the discrete space
and is hence closed. The intersection of these sets, for all v1
closed
and
so
the
functionals
near
is obtained
set
on
of additive maps is closed.
V is closed and is hence LC . The
by sending
the
reverse
set
is
of all li-
equivalence
to
the discrete space of functionals
topologize
the space Hom(V, W) of contin-
the LC space V
V, W E B. We will
Let
linear maps in
uous
by letting
where Wo
as
ways. We
is
an
of
open
Wv .
subspace
in W . In fact,
Since the latter is
A basic open
11,
subspace of [V, ",’ J
Vo C V is an LC
W]
we are
topologized
separated,
topologizing ( V , W )
so
is ( V, W) . It is
unless V is discrete. In
closed
where
let ( V, W) denote that space
subbasic subspace be
a
subspace
a
two
subspace in general
discrete and W is LC, (V, W)
[
Similarly
and v2 ,
it.
on
a
two vectors
subspace
particular,
is also LC . The second
topology
not
if V is
is finer.
is
and Wo C W is open. We call ( V, W) .,,and
then weak and strong internal hom
5
respectively.
The fact that
is continuous follows from the fact that the
ny
We let V * denote
PROPOSITION
need
that
V / U
1.2.
( V , K ) and V " denote [V, K
The space K is
That it is
now
PROPOSITION
first choose
with dense
not
open
the
zero
category 2
In the
image regular
are
an
a
cogenerator
quotient
monos
class. The existence of
maps and
are
such
the requir-
theory.
are
monos
inclusions
are
a cp E V*
we
C V with ve U. Then
subspace 11
follows from standard linear
1.3.
cogenerator in B.
a
if V c 2 and v # 0 in V , then there is
is discrete and v+ U is
ed functional
generator and
a
generator is evident. To show it is
a
only show that
O(v) + 0 . But
ular epis
ma-
is LC .
vectors
PROOF.
subspace spanned by finitely
1-1 maps,-
of
closed
cpis
arc
suhspacr?s,’
maps
reg-
open,. the category i.s well- and cO-l1Jcll..
powered.
PROOF.
This is routine and is left
PROPOSITION
maps
The
category 2
is
complete
and
cocomplepte.
The
are
The
topologized by the
coequalizer of two
is the cokernel of their difference. The cokernel of
tient modulo the closure of the
THEOREM.
ses
the reader.
completeness is routine. Direct sums
topology rendering the injections continuous.
PROOF.
finest
1.4.
to
image .
projective and injective (withrespectto the classubspace iraclusions, respectivPl y ).
T’he space K is
of regul ar epis
P ROO F .
The
and
projectivity
is trivial.
Suppose
V C W’ is
a
subspace
q5: V -&#x3E; K, As usual qb is uniformly continuous and
that ql has a unique extension to the closure of V . Thus
V to be closed. Let Vo
ker O. Then Vo is closed in V
pose
K
=
We have that V/V0 -&#x3E;W/V0 is continuous and V / V 0 is LC
is
an
logy
isomorphism
from
with the
W/ Vo . Thus
we
that V = K v . Now there is
zero on v
map is the quo-
a
and
a
image,
hence V / V 0
some
suitable scalar
functional defined
multiple
6
of it is
is discrete
we
one
on
O(v).
so
may suppose
and hence in W .
so
still has the
may suppose that V is
and sup-
that the map
subspace
dimensional,
topo-
in fact
all of W which is
non-
2. The weak
duality.
For
2. 1.
P ROPOSITION
a
inverse limits. T’hus it has
the
the level of the
At
PROOF.
is
topology
topologized
gized
a
as
issue. But
at
subspace
a
as
of
is
of
spaces, that is clear.
vector
is
a
(II Vi) V = II(VVi)
topologized
if
and
adjoint
with the
VV1,
hence of
of
now
Only
(V, I1 Vi) is
II (V, Vi) is topolo-
V1 C V2
subspace
as a
with
of spaces,
family
Similarly,
commutes
subspace
V2 ,
follows from the
hen-
special
functor theorem.
T’he - -O- described above is
PROPOSITION 2.2.
Both maps U O V -&#x3E; W and V O U -&#x3E; W
PROOF.
U X V -&#x3E; W. It is
bilinear maps
respond
be the
if {Vi}
The existence of the
(V, V2
adjoint
underlying
of the latter.
subspace
topology, ( V , V1)
ce
fixed V E B, the functor ( y’, -)
an adjoint - 0 V -
to
the
same set.
only
linear map. In order that
corresponding
necessary and sufficient that for each u
In order that
inverse
Fl
image
{f
be continuous
on
correspond
to
necessary
Let F : U X V -&#x3E; W be
symmetric.
a
show that
certain
they
set
each
of
cor-
bilinear map and
F1
map
U , F( u, -)
E
to a
I ul
to
( V, W) , it is
be continuous
on
V .
is necessary and sufficient that the
U , it
of
lf( v) W0},
where v
E
c
V,
Wo is
an
open
subspace
of
W,
exactly the condition that for a fixed v , F(-, v )
Thus F corresponds to a map L’ -&#x3E;( V, W) iff it is sepU and V which is evidently the condition to corres-
be open in 11 . But this is
be continuous
arately
pond to
on
a
map V -&#x3E;( U, W ) .
COROLLARY
2.3.
the
on
functional
P ROO F.
U .
continuous in
The natural map
V* evaluation
lVl -&#x3E; V** l
,
is realized
at v,
In view of the symmetry above
maps, the first
being
the
identity :
7
we
by
which
a
have the
assigns
to
v
EV
map V -&#x3E; V **.
following
sequence of
isomorphism we will say that V is weal.
clearly that of the weak hom , simply reflexive.
Yhen this natural map is
reflexive
or, if the context is
THEOREM 2.4.
PROOF.
The
an
The natural map V -&#x3E; V** is
of
hypothesis
show it is onto, observe that
always
1-1 and onto.
separation is equivalent
we have topologized V *
to
its
as a
being
of
KV.
functional
since K is
is open,
To
subspace
injective, any functional V* t-+ K extends to a
f : KV -&#x3E; K. Continuity together with discreteness of K implies that
tors through the projection onto a finite number of factors. For
Thus,
1-1 .
and any open subset of the
(necessarily open)
set
under the
is the inverse
product
projection
this fac-
to a
finite
image of some
product. So f has a
factorization
The second,
being
a
functional
finite dimensional space, is
a
on
a
linear
combination of evaluations, say
Applied
att
to
a O E V*,
that
is,
to a
linear map, this is the
same as
evaluation
a1 v1 +... +an vn.
Hence
whether
or not
reflexiveness reduces
the
purely topological question of
open).
example, an immediate conto
V -&#x3E; V** is closed ( or
For
sequence is:
COROLLARY 2.5.
If
V is LC,
COROLLARY 2.6. For any
PROOF.
which is
right
that is 1-1 and onto,
they
lVl -&#x3E;
onto
reflexive.
and 1-1.
as
always -
are
inverse
A space V is called
map
is reflexive.
V, V* is
We have V -V** is
inverse -
V
to
Dualizing gives
V** * -&#x3E; V*,
the natural map V* - V*** . Since
isomorphisms.
representationally
K is continuous.
us
Evidently
8
a
discrete ( RD ) if every linear
discrete space is RD, but the
con-
Clearly the condition is equivalent to every cofinite dimensional
subspace being open and that, in turn, implies that every subspace - being
the intersection of the hyperplanes containing it-is closed. The minimal RD
( MRD ) topology is the one in which the open subspaces are precisely the
cofinite dimensional ones. That clearly defines a topology, which, just as
clearly, is discrete only in the finite dimensional case.
fails.
verse
is MRD; an RD space is
PROOF.
iff its dual is LC;
reflexive iff it is MRD.
A space is RD
THEOREM 2.7.
the dual
of
an
LC space
RD, then V* is closed in K V (see the discussion follow-
If V is
above) and hence is LC. If V is LC, then from the known duality
V*l*, whence the natural map V** -&#x3E; I V*I * is 1-1 and onto. Thus V*
ing
1-1
V 3’ )
l
is RD.
there is another RD
the
of
points as
V *. Call the resultant topology W . We can assume, by taking their in f (which
can be immediately seen to give an RD topology) that the topology on W is
coarser than that on V * , which means that V* - W is continuous. Both V**
and W* are LC so that the map W* - V** , which is clearly 1-1 and onto
(they are both the set of all functionals on IV *I - lWl), is an isomorphism.
Suppose
topology
V*** -+ W** is also. The result
Thus
can now
on
be
same set
seen
from the commutative
square
in which the bottom and left hand map
is RD and V - V** continuous
ently
RD.
RD, V -V** with
reflexive, then V** = V
If V is
If V is RD and
P ROPOSITION 2.8.
PROO F.
ce W
means
isomorphisms. If V* is LC, V**
V has a finer topology which is evidV** MRD implies they are the same.
If
W C V and V is
are
is RD.
reflexive,
so is W.
V * * - V , the points of W ** are carried into W. Sinsubspace topology, that defines a continuous map W** - W
Under the map
has the
which is inverse
to
COROLLARY 2.9.
the natural map.
If V
is
reflexive,
every RD
9
subspace o f
V must be MRD .
The space V is
TH E O R E M 2. 10.
finite
subspace
is
that V is
not
dimensional.
The
PROOF.
necessity
be
cannot
of this condition is 2.9.
Suppose
now
subspace of V which is not open in I’**.
cofinite dimensional, for then it would be the intersection of
reflexive and let W be
W
every discrete
reflexive iff
of the kernels of
same
functionals,
crete
spaces
an
open
finite number of functionals. But V and V** have the
a
thus W is open in the latter. Now V/ W is discrete. Dis-
are
projective
ensional discrete
subspace.
and
so
V - V/ W splits,
giving
V
an
infinite dim-
3. The weak hom .
PROPOSITION
in the
of [2],
sense
a
composition law (V,W) -&#x3E;((U,V),(U,W))
l. 2.
The function is the obvious
PROOF.
and with the
pointwise
would like
to
one.
The
only question
is
continuity
hom that is easy.
it follows that
From this
we
There is
3.1.
show that the full
we
have
a
subcategory
closed monoidal category. We
of reflexives is likewise. First
need:
PROPOSITION
For any V and any
3.2.
reflexive
W, (V, W) is
rcfl(,xi7,c.
PROOF.
and the result follows from 2.6.
We
now
let
BR(K)
denote the full
subcategory
of
rs ( K) consisting
of the weak reflexive spaces.
THEOREM 3.2.
PROOF.
while
a
The inclusion
If W is
reflexive,
map V** -W
TH EO R E M
3.3.
When
a
gives,
BR(K) CB(K)
map V -&#x3E; W
up on
has
a le ft adjoint
with
equipped with ( - , - ) , BR (K)
10
l-&#x3E; V**.
gives
composition
category.
V
V -V**, a
is
a
map V -&#x3E; W.
closed monoidal
PROOF.
e
so
reflexive spaces,
are
0 V) ** is the adjoint
that
used
If U , V , W
in B (K)
is of
course
to
we
have
( V , - ) . The
same
composition
law
as
is
intended here.
4. The strong dual.
with the idea of better
understanding vector
discrete spaces. Although the theory developed in the prespaces,
ceding two sections is attractive, it has the disadvantage of excluding the
discrete spaces. It is true that the full subcategory of MRD spaces is equivalent to the category of discrete spaces, but nonetheless the theory leaves
something to be desired. Additionally, experience tells us that pointwise
convergence is the wrong topology for function spaces. Thus we are led to
consider the strong dual and strong hom . Recall that [ V, W] is the space
of homomorphisms which has, as a basis of open subspaces,
We
began
including
is
where Vo
V - the
[
context
LC
an
K].
V,
this
theory
subspace
in V and Wo is
an
A space V will be called strong
is clear -
provided
the natural map
open
subspace
of W . Also
reflexive - just reflexive if
l v l -&#x3E; l v "l
comes
from
an
isomorphism V -&#x3E; V ^^. As we will see in the next section, it will be necessary to restrict to a subcategory in order to make [-, -] into an internal hom.
THEOREM
The dual
4.1.
o f an
RD space is compact ; the dual
ofa
compúe
space is discrete,.
The second
PROOF.
know that
But
an
an
statement
RD space
cannot
is obvious. The first is clear
have
an
infinite dimensional LC space
as soon as we
infinite dimensional LC
must
have
subspace.
discontinuous functionals ;
otherwise the associated discrete space would be self dual for the discrete
dual, which is known
to
PROPOSITION
The natural map
open
sets
PROOF.
4.2.
be
impossible.
i7i V "".
Let U C L’ be open. Consider
11
I Vl -1 v--l
takes open
sets
in V to
Thus V/ U is discrete,
so
(V/U) ^
is
is discrete. The
LC, and then (V IV)"’"
kernel of V ^^ -&#x3E;(V/U)^^ is the image of U in V A: The kernel is the inverse
image of the open set n in (V/U)^^ and is thus open.
When U OF,
Th(1
TH EO REM 4. 3.
we
natural map
When V is
PROOF.
let U l C V^ denote the annihilator of
I vil -j, V^l
L.C, the functionals
on
is 1-1 and onto.
V^ are the
and hence the result follows from 2.4. In the
V A --+K must,
U’
the form
O’:
U" -+ K
aluation
in order
for
some
LC
c 11. Then
at a u
subspace
is
(continuity
general
not an
so
U C V . But
a
map.
proof
onto.
V *
on
a
O:
of
subspace
a
functional
issue since U^ is discrete) which is
ev-
is 0.
This is of
course not
a
natural map V^-&#x3E;V
the natural direction for such
permits
an
easy
of
For dny V ,
Since any map from
is
those
functional
induces
But it is the direction in which it is continuous and
CO ROL L ARY 4.4.
one
case a
then O
The result of this and of 2.2 is that there is
which is 1-1 and
same as
be continuous, contain in its kernel
to
U .
an
V^ is
reflexive.
LC space is closed and any map
is open, it follows that discrete and LC spaces
locally
LC if it has
open LC
subspace. In
discrete, hence projective, and thus the quotient
is the direct
reflexive.
sum
To
see
of
a
that
an
discrete and
not
an
are
every reflexive is
discrete
reflexive. A space
that
case
map
splits
LC space. Such
to a
a
locally LC,
the
quotient
is
and the space
space is
we
clearly
consider the
following
EX A M P L E
a
subspace
4. 5.
Let V be the direct
of the direct
product W’ .
sum
of
Ye have
Clearly
12
Xo copies
of K
topologized
as
isomorphisms, V^ =
and I claim that under these
se
V . For V is
evidently
den-
in W , from which W ^ -&#x3E; V ^ is 1-1. It is also onto, since every continuous
homomorphism between topological groups is uniformly continuous and hence
has a unique extension to the closure. V has no infinite dimensional LC
subspace, since V has countable dimension and any infinite dimensional
LC subspace, being the dual of a discrete space, must have dimension at
least
I Vl ^
V =
2
o
.
Thus V A is
is also, it follows
VA
and V ^^ -&#x3E; V is
We say that
THEOREM
4.6.
open subspace
PROOF.
The
isomorphism.
an
subspace
on l V)l
every functional
clear that U is
a
topologized by pointwise convergence
that V A has the subspace topology from l
U C V is
which vanishes
representationally
iff
every RD
equivalence
not
locally
V I A.
Thus
LC.
representationall y open provided
on
U is continuous
on
V . It
is
open iff V/ U is RD .
The space V is strong
is open
Also V i s
and since
of the
reflexive ill
quotient
two
every
representationally
is discrete.
latter
statements
is evident. If V is
strong reflexive, so is every quotient (just turn around the proof of 2.8). But
a reflexive RD must be discrete, since its dual is compact and its second
dual discrete.
Conversely,
if V is
there is
U r V whose
image
a set
V AA have the
I vAAl
same
reflexive, V^^-&#x3E;V
in V AA is open. Since
functionals. Since U is open in
which vanishes
representationally
not
on
open and
U is continuous
not
open.
13
on
V,
VAA, hence
is
not
open,
V^= V AAA, V
any functional
on
so
and
on
V . Thus U is
REFERENCES
1.
J. DIEUDONNE, Introduction
York, 1973.
2. S. EILENBERG
Algebra
the
and G. M. KELLY,
( L a Jolla, 1965),
3. S. LEFSCHETZ;
to
pp. 421-
Theory of formal groups,
Closed categories, Proc.
M.
Dekker, New
Conf. categorical
562 ; Springer, Berlin, 1966.
Algebraic Topology,
A. M. S.
Colloquium Pub. XXVII, New York,
1942.
ADDED IN PROOF.
In
a
subsequent
paper I will discuss
the modifications necessary
to
make
a
Department of Mathematics
Mc Gill
University
P. O. Box 6070, Station A
MONTREAL, P. Q. H3C 3G1
CANADA
14
at
length
the strong hom and
closed monoidal category
using
it.