C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
J OAN W ICK P ELLETIER
R OBERT D. R OSEBRUGH
The category of Banach spaces in sheaves
Cahiers de topologie et géométrie différentielle catégoriques, tome 20, n
o4 (1979), p. 353-372
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THE CATEGORY OF BANACH SPACES IN SHEAVES *)
by Joan Wick PELLETIER
andRobert D. ROSEBRUGH
CAHIERS DE TOPOLOGIEET GEOMETRIE DIFFERENTIELLE
Vol. XX-4 (1979)
1. INTRODUCTION.
The notion of a Banach space in the topos
Sh(X)
of sheaves over atopological space X
was first introducedby Mulvey [11].
Interest in the category of Banach spaces inSh ( X ) ,
which we hereafter denoteby ban(X),
hasconverged
from manysides,
due toI Mulveyts
realization thatban(X)
isequivalent
both to the category of Banach fibre spacesover X ,
studiedby
Hofmann[5 ,6] ,
and to the category of Banach(Q-
orapproximation)
sheaves onX ,
studiedby Auspitz [1] and Banaschewski
[2]. Thus,
the results obtained inban(X) by using
thetechniques
ofintuitionistic mathematics available in
Sh(X)
can be transferredfreely
to these other
categories. Important
among these results is the Hahn-Ba- nach Theorem for*R-valued
functionals inSh(X)
due toBurden [3],
where
*R
denotes the ordercompletion
of the Dedekind reals.Our
object
in this paper is tobegin
aninvestigation
of the categoryban(X).
We are interested in the existence of internal and external limits inban(X) .
Westudy
the construction of dual spaces andapplications
of the Hahn-Banach Theorem. ’ibe calculate dual spaces of
quotients
andsubspaces.
It evolves thatduality
behaves best when restricted to a sub- category ofban(X) consisting
of«*normed »
Banach spaces. We examine*closure
inban(X), describing
it in terms of annihilators.Finally,
inbringing together
many of the abovenotions,
we show that*normed, *com- plete
Banach spaces can be characterized in terms of the behaviour of their duals. We see that this lattersubcategory
arisesnaturally
and is re-flective in
ban ( X) .
* ) This
work wassupported
in part by the National Council of Canada under Grant A 9134.We recall for the convenience of our readers that the Dedekind real numbers
object
inSh(X),
denotedR ,
is the sheaf of continuous real- valued functionson X .
A normed spaceB
inSh( X)
is a sheaf of R-mod- ulesequipped
with amorphism (the «norm ») N : R+ -> QB satisfying (in
the internal
language
ofSh(X),
see[7]
or[10]):
A
Cauchy approximation,
on a normedspace B ,
is amorphism C : N -> QB ,
where N denotes the natural numbersobjects
inSh(X) (the locally
constant natural numbers-valued functionson X ) satisfying:
The
Cauchy approximation C
is said to converge ifA Banach space in
Sh(X)
is a normed space B for which(in
theintemal language
ofSh(X) )
everyCauchy approximation
converges.If
B’
andB "
are Banach spaces, a linear mapf : B’ 4 B"
is saidto be continuous if:
f
is said to be bounded if:Both the Dedekind reals R and the Dedekind-Mac Neille reals
*R
will be used in this paper. The former
object
isquite
standardby
now, while the latter hasonly
morerecently
become anobject
ofstudy. *R
isthe
(internal)
Dedekind-Mac Neillecompletion
of the rational numbersobject Q ;
it is also known as the extended reals(Burden [3] ). Formally,
it is defined to be all
pairs (L , U)
ofsubobjects
ofQ ,
the sheaf of loc-ally
constant rational-valued functions onX , satisfying:
Mulvey
has identified the sections of*R
atpairs
of functions(f,f)
wheref
is lowersemicontinuous, i
is upper semicontinuous and/
is the least upper semicontinuous function greater thanf ,
anddually.
2. LIMITS IN
ban(X).
In this section we consider internal and external limits in
ban (X).
To
study
internal limits it will be necessary to endowban(X)
withthe structure of a
Sh (X)-indexed
category.Recall
that thisrequires that,
for each
object
I inSh(X) ,
there begiven
acategory ban(X)I,
of el-indexed families » of
objects
ofban(X)
and for eachmorphism
a:J ->
Iin
Sh(X) ,
there be a functora*: ban(X)I-> ban(X)J ,
called (csubstitu- tionalong
a of I -indexed families in7-indexed
families ». Substitution isrequired
to becompatible
withcomposition
and identities inSh(X)
upto canonical
isomorphism. (For details,
see Pare and Schumacher[12].)
For
example, Sh(X)
is itself aSh ( X )-indexed
category withand the substitution functors defined
by pulling
back. We note further that theunique morphism
I -> I is denoted I and substitutionalong
itby I * .
For I an
object
ofSh(X),
we defineban(X)I
to be the category of Banach spaces in the toposSH(X)/I = Sh(Xl
so that an I -indexedfamily
of Banach spaces isjust
a «Banach space in 1-indexed families »(of objects).
Next suppose thata: j -> I
is amorphism
inSh(X)
andB is an
object
ofban(X)I
with normNB .
We wish to describe the ob-j ect a * B
inban(X)J.
Since a * is alogical
functor- fromSh(X)I
toSh( X)J
it preserves natural and rational numbersobjects
as well as thesentences in the
language
ofSh(X)I defining
norm andcompleteness
of a norm.
Thus, denoting
the Dedekind reals inSh(X)I by Ri
and theunderlying object
of Bby B ,
we havea*RI = Rj
and so a* B is aR j-
module in
Sh(X)J.
a* B comesequipped with,
and iscomplete
with res-pect to, the norm
Na*B = a * NB . C learly, morphisms
ofban(X)I
arepreserved by
a* aswell,
and the substitution functors so defined inheritcompatibility
withcomposition
and identities from theSh ( X )-indexed
cat-egory
Sh(X). Thus, ban(X)
isa Sh(X)-indexed
category.To see that
ban (X)
isinternally complete
we needonly
show thatit and
ban(X)I
have stable finite limits(finite
limitspreserved by a * )
and internal
products ( see [12] ).
2.1.
LEMMA.ban(X)I
hasstable finite limits.
PROOF. Finite limits will be constructed in
ban(X)
and then those inban(X)I
existby
localization. Theirstability
is clear from the construc-tions
given. First,
letB’
andB"
be inban(X)
with normsNB,
andNB" .
Theunderlying
R-module ofB’ X B"
is the R-moduleB’ X B" .
Thenorm
NB
’X B" is definedby
To see that
NB’x B"
is a norm onB’x B " requires
some verification.For
example,
to show thatlVB
’XB" satisfies(2N ),
we use the fact( about R+)
thatThen
Verification of the other
properties
ofNB ’XB " is
similar and is left tothe reader.
Moreover, showing
thatB’ X B"
iscomplete
with respect toNB’x B "
also follows the standard argument and is omitted.To construct the
equalizer
of apair
of mapsin
ban(X),
wesimply
form theequalizer
e :B0>->B’
inR-modules,
andendow it with norm
NB,,
0which is the
pullback
ofNB, along
This makes e a
morphism
of Banach spaces. Verification is omitted sincee is
equally
well defined as the kernel off- g (Burden [4] ). Now,
sinceban(X)
has finiteproducts
andequalizers,
the lemma isproved.
Before
constructing arbitrary
internalproducts
inban(X),
we re-call the construction of
products
inban ,
the category of Banach spaces and normdecreasing
linear maps in set . If(Bi )ifl
is afamily
of Banachspaces, their
product
is the setwith coordinatewise
operations
andRecall that what is
required
for the construction of internal pro- ducts inban(X)
is aright adjoint
tofor each I in
Sh(X).
Nowsuppose B
is inban (X)I
with normNB .
Letn be the front
adjunction
forI * -| III: Sh (X)I -> Sh (X )
and( Sh(X)
is itselfinternally complete, being
atopos. )
By using exponential adjointness
and the fact that Q classifiessubobjects,
we may writeNB
as asubobject
ofB X RI (as will be con-
venient for the remainder of this
section ).
Let the top square in the fol-lowing diagram
be apullback
inSh (X):
where v ,
which isIII applied
to the inclusionNg
>->B X R+ ,
is a mono-morphism
sinceMr
is aright adjoint.
B is theimage
inIII B
ofv’
fol-lowed
by
theprojection PIon III B .
lke then have thefollowing
lemma :2.2.
L EMMA. B is in ban (X) with
normNB and is
theinternal product of B.
P ROO F. We first note that
III B
inSh(X)
is an R-module since B is anR-module and the
forgetful
functor from R-modules toSh(X)
creates in-ternal
products [12]. Consequently,
the backadjunction e: I *III B->
Bis an R-module
morphism
which will be used below. Beforeshowing
thatB is a submodule of
III B ,
we will need someproperties
ofNB . By
thedefinition of
An
as apullback
inSh (X),
for(b, r) eN B (U) ( U
openin
X )
we have:where the second
bijection
usesI*-| III,
and so elements(b, r) e N-B (U)
correspond bijectively
with elements(c, s)e NB(I*U)
such thatWith this we can show that
properties (4N)
and(3N)
of a norm onIII B
hold for
N-B.
Forexample, using
the transpose of(4N)
for the normNB,
if
( b, r) e N-B
anda | s ,
thenusing
that E is an R-modulehomomorphism,
so( a b, rs )
cNg . Similarly,
U sing these,
we can showthat B
is a submodule ofIII B .
Notice firstthat
( 1N )
for a normon B
holds forlVB by
the definition ofB . Now,
ifb E B and s e R
we haveThus B
is closed under scalarmultiplication
and an evensimpler
argu-ment shows it closed under addition. The arguments above show that
N-B
satisfies
(IN), (3N)
and(4N)
in the definition of norm with respect toB.
Theremaining
two conditions may be shownsimilarly
and are left tothe reader. We conclude
that B
is a normed space.Suppose
that C: N ->Q 9
is aglobal Cauchy approximation
inB .
Then
I *C
is aCauchy approximation
inI *B
and theRI-module morphism
is
clearly norm-decreasing.
Thusf o I* C
is aCauchy approximation
inB and so there exists b c B to which it converges. Now b is
unique
soit exists
globally
and hencecorresponds
toI
cIII B ,
to which C obvious-ly
converges.That 1
liesin B
follows sinceI*C(n)
is boundedby
aconstant
implying that b and 1
are also. This argument can be localizedto any open set in X and so
( in
the internalsense)
everyCauchy
appro-ximation
in B
converges.Finally,
it remains to show that B acts like aproduct for B .
LetO: B’ -> -B
be amorphism
inban(X).
Fromit, using
thef
in thepreced- ing paragraph,
we obtain amorphism
in
On the other
hand,
ifw: I* B’ -> B
is amorphism
inban(X)I ,
we obtaina
morphism
ofR-modules 0: B’ -> III B ,
and we wish to show thatw
fac-tors
through B .
Nowand
(b’, r) e N B’ implies (I * b’, I *r) e NI * B’ . Hence, (w I*b’, I*r) e NB
since X
isnorm-decreasing.
Butusing I*-| III
by naturality (
seediagram below ) by
definitionof w.
Thus
by
the characterization at thebeginning
of thisproof, (w b’, r) e NB .
So we have
Moreover, w
is alsoclearly norm-decreasing by
thepreceding
remarks.That the
correspondences qb F qb and w |->w
aremutually
inverse is leftto the reader. The
stability
of the construction of B follows from the con-2.3. PROPOSITION.
ban(X) is internally complete..
We wish to show also that
ban(X)
isexternally complete. Sup-
pose I is a set and
( Bi )ifI
is afamily
ofobjects
ofban(X).
The(ex-
ternal) product
inSh ( X)
of( Bi )ieI
is an R-module and we denote itby
IT iff B i.
To define theproduct
of theBi
inban(X)
consider thediagram
in which the top square is a
pullback and -B
is theimage
ofNB
inIIieI Bi.
Notice
that g
is thesubobject
ofllifI B
i definedby
the formulagiving
the usual
product
inban. B
is theproduct
of theBi
inban(X)
withnorm
Ng .
Theproof proceeds
in much the same way as that of Lemma 2.2. It isperhaps
worthwhile to note that the elements ofNB(U)
arepairs ((bi),r),
wherewhich is similar to the characterization of such elements obtained
early
in Lemma 2.2.
Again combining
with2.1,
we obtain the desired result.2.4.
PROPOSITION.ban(X) is externally complete. a
In
closing
this section we remark that the definition of a Banach space makes sense in any toposhaving
a natural numbersobject, although
to date
spatial topoi
haveprovided
the customarysetting. Moreover,
the constructions used in2.1-2.3
do notdepend
upon the nature of the toposconsidered,
nor do the results(except
torequire
the existence of the nat-ural numbers
object).
It is also worthnoting
that the constructionsgiven
here show that the category of normed spaces in a topos
(of sheaves)
isinternally (and externally) complete.
3. TH E DU AL SPACE.
In
analogy
toordinary
Banach spacetheory,
we could consider the dual space of the Banach space B to be the sheaf of R- or*R-valued
continuous linear functionals
on B .
We choose the latter in view of the fact that Burden[3]
hasproved
a Hahn-Banach Theorem for*R-valued
functionals. The results which follow from theadoption
of this definitionare stated below.
They
have beenproved by
Burden[4]
andindependently by
the present authors.Given
B1, B2 E ban(X),
it iseasily
seen thatban(X)(B1, B2)
is the unit ball of a normed space
HOM ( B1 , B2 )
which consists of all bounded linear transformations fromB1
toB2 .
The normN B1, B2
onHOM(B1, B2 )
isgiven by
One can
readily
prove that thecompleteness
ofHOM ( B1, B2 ) depends
only
on thecompleteness
ofB2 ,
so thatHOM(BZ,B2)f ban(X).
IfB3
is another Banach space, we also have
where we understand
by =
an isometricisomorphism,
i. e. a R-moduleisomorphism 0: A -> A2
such thatWe denote
by B *
the spaceHOM (B, *R). Clearly, ( )*
is a con-travariant endofunctor on
ban(X) ,
withf*: B54 BI
definedby
for
and we denote
by iB :
B -B **
themorphism
whichcorresponds
to theidentity
onB * .
Usual Banach spacetheory
reliesstrongly
on the factthat
iB
is an isometricinclusion,
which fact itself uses theHahn-Banach
Theorem for itsproof.
In oursetting
the Hahn-Banach Theorem is insuf- ficient to obtain this result and there are non-trivial Banach spaces hav-ing
trivial dual spaces. Thefollowing example
illustrates ourpredicament.
Let X
denote theSierpinski
space( X
is atwo-point
space withone
point open).
ThenSh( X)= s et2
and a Banachspace B
=(Bl, B2, f)
in
Sh(X)
is apair
of(ordinary)
Banach spacestogether
with a norm-decreasing
linear transformationf: B1 -> B2 . Since X
isextremally
dis-connected, *R
and R agree onsel2 ( Johnstone [8] ).
We caneasily
cal-culate that this
object
is1 R : R - R
whereR
denotes theordinary
real numbers. Then the dualB* = (A1, A2, g)
of the Banachspace B
con-sists of
HOM((B1, B2,f), (R,R,1R)). Thus
we see thatHence,
inparticular,
the dual of( B1, B2, f) is 0
wheneverB2
is.We remark that the above discussion characterizes dual spaces in
set 2
asordinary
dual spacesequipped
with theidentity
map.The condition which
salvages
the situation and ensures thatiB
is an isometric inclusion is the existence of a
*norm
onB ,
so namedby
Burden [4],
who hasexplored
the relation betweennormed, *normed
andco-normed spaces.
lA norm N on the
space B
is said to be a*norm
if it satisfies:Burden [4] justifies
histerminology
inshowing
that the existence of a*norm
N on B isequivalent
to the existence of amorphism 11 11 :
B ->*R
with the usual
properties
of a norm(we
defineThe
following
result summarizes theimportance
of the*norm.
3.1.
PROPOSITION.(1) The dual
norm onB * is
a*norm for
anyB
inban(X).
(2)
B is*nonned iff iB :
B 4B
** is an isometricinclusion.
’We remark that R and
*R
areexamples
of*normed
spaces inSh(X)
forany X ,
but that theirquotient,
which is defined and discussed later on, need not be*normed,
the latter factbeing
an observation of Mul- vey.We further remark that the full
subcategory
ofban(X) consisting
of
*normed
spaces is reflective. We refer to3.12
for ananalogous
result.We now
investigate
someexamples
of Banach spaces and their duals.First,
recall that inset2
a Banach space isgiven by
apair
ofBanach spaces
B1 , B2
and anorm-decreasing
linearmap f: B1 -> B2 .
The Banach space
(Bi, B2, f)
will be*normed
iff there is a mapsatisfying
theproperties
of a norm. Such a mapclearly corresponds
to theexistence of norms
such that
Thus, ( B 1 ’ 3’ B2, f)
is*normed
ifff
is an isometric inclusion.Let X be the
three-point
space with two discretepoints.
ThenSh(X) = s etA ,
whereIn this topos, R is
but
R *
isA Banach space consists of a
triple
of(ordinary)
Banach spaces withtwo
norm-decreasing
linear mapsThe dual space of
( A , B, C, f , g)
w ill be(A’, B’l C’9 f’,g’)
wheresatisfying
Hence,
dual spaces areproduct diagrams
ofordinary
dual spaces. Since*normed
spaces must beisometrically
embedded in their secondduals,
we see that
(A, B, C, f, g)
is*normed
iffVirtually
all results on Banach spaces and their dualsrequire
the H ahn-B anach Theorem. The Hahn-Banach Theorem inSh(X)
asproved by
Burden[3]
states that every*R-valued
functional defined on a sub- space of a normedspace B
can be extended to all of B without increas-ing
the norm. Inordinary analysis
oneimmediately
wouldapply
this theo-rem to show that the dual space separates the
points
of theoriginal
space.However,
wealready
observed anexample
in whichB*
is trivial and B isnon-trivial,
so dual spacescertainly
do not ingeneral
separatepoints.
On the other
hand,
we can obtain a usefulcorollary
of the Hahn-Banach Theorem which is akin to theordinary separation
ofpoints
result.To do this it will be necessary to introduce a new notion of clo-
sure for a
subobject
of a Banach space which was introducedby
Burden[3].
Asubobject
A of a Banachspace B
is said to be*closed
ifBurden has shown that the
*closure o f A ,
denoted*Cl ( A ) ,
consists ofthose elements o f B to which so me
Cauchy *approximation
converges. ACauchy *approximation
in B is amorlhism C : N -> QB satisfying
condition( 2 A )
above o f aCauchy approximation
and o f3.2.
THEOREM (Corollary of Hahn-Banach Theorem). Let A,
B be inban (X ) and j : A -
B be an isometric inclusion.Then
The
proof requires
apreliminary
lemma. For a Banach spaceB,
asubspace j : A->B ,
andb e B ,
we define3.3. LEMMA.
PROOF.
Rewriting
K asit is easy to see that K is clo sed under
larger
elements.Thus,
entails both
r E K and - (r E K ).
Hence we haveso
PROOF of 3.2. Let
Ab
be the Banachsubspace
of B definedby
Define
f : Ab -> *R by
where We observe
by
Lemma 3.3that - (in f K )
= 0 sinceSeparation
ofpoints
of a Banach spaceby
functionals does not fol- low from 3.2 because of thestrength
of the statementwhich
requires
that b remain non-zero when restricted to any open set. Forexample,
inset2
any Banach space of the form( B’, 0, f )
has no elementsb
satisfying ï ( b = 0)
since all non-zeroglobal
elements are zero whenrestricted to the other open.
We remark that Burden
([4] 4.7) gives
acorollary
to the Hahn-Ba- nach Theorem very similar to our own in which aslightly
strongerhypothesis
is used and a
slightly
stronger result is deduced.For a linear
subspace
A C B we define the annihilatorA-,
ofA,
as fo llo ws :
For a linear
subspace
M o f B * we define the annihilator -M o f Mby
Clearly, A 1 and !M
are closed linearsubspaces
ofB*
andB ,
respec-tively.
As an
application
o f3.2,
we shall show that the*clo sure
o f a sub- space o f B is its double annihilator. Thefollowing
lemma due to Burden([4] 3.14 )
will be used.3.4. LEMMA.
I f f: B1 -> B2 is
a bo unded linear mapo f Banach
spaces,where B2 is *normed,
thenis
*clo
sed.3 .5. PROPOSITION.
Let
Aand M
be linearsubspaces of
B andB*,
res-pectively.
ThenA-
and1M
are*clo sed linear subspaces of B*
andB, resp ectively.
PROOF. Lemma
3.4 gives
us our resultimmediately
if we note thatand
where
iB ; B -> B **
is the map described above and the(internal)
inter-sections exist
by
2.3.3.6.
THEOREM.For
anylinear subspace
Aof
aBanach
spaceB,
PROOF. It is clear
that A
C1( A1). Thus,
since-(A-)
is*closed by 3.5
we have *Cl(A)C -(A-).
To show the
o ppo site inclusion,
we noteimplies by 3.2
thatHowever, by
definitionof -(A-),
Thus,
we may concludethat 77 (b e *Cl (A ).
Since*Cl(A)
is*closed, we
have
i. e., *Cl (A)
isdo uble-ne gation closed,
whichcompletes
theproof.
3.7.
COROLLARY. -(R- )= *R .
·Given a closed linear
subspace A of B ,
we can define thequotient
space
B/ A
and thequotient
map II:B - B/A
in the o bvious way. W e normB/A
to make it into a Banach space as follows:*normed spaces, is not
*normed.
One can see this because there are manynon-zero elements x in
*R/R
for which the statementholds when the base
space X ,
say, is[0,1].
The annihilator
helps
to describe the duals o f thesubspace
A andthe
quotient
spaceB/ A .
3.8. P ROPOSITION.
If A is
a closed linearsubspace of
a Banach spaceB,
then
and
P ROO F .
(1)
This has beenproved by
Burden([4] 4.9).
( 2 )
Wedefine Vf : (BI A)* -> A- by Vf (g )
= go II , where ?7 is the quo- tient map.Clearly, w(g)
cA J,
andw
is linear. Givenf e Ai ,
since A iscontained in
ker f
we have a well-defined linear mapg : B/A -> *R given by
where If x c
NBI A (r),
thenHence,
iff
is boundedby
y , thenf ( b ) = g (x ) e N*R(y s )
so g is also bo undedby
y and g e( B/ A )* . Thus, w
is o nto .To show
that 0
is anisometry,
iff =w (g )e NB*(y),
thenIf x c
NB /A ( r ) ,
th enHence,
so
Since w
isclearly norm-decreasing,
we havethat Vf
is aniso morphism, s
We conclude
by investigating *complete
Banach spaces, i. e. normed linear spaces in which each*approximation
converges. As we have seenabove,
the*closure of
asubspace
of a Banach space is its double annihil-ator. One can
easily
see that if A is a*closed subspace
of a*complete
space, then A is
*co mplete. Moreover,
it is not difficult to show that B * is*complete
for all B Eban (X ).
"BtIe remark that
*co mpleteness
does notimply *normed
nor the con-verse. In
set2 , 1/2 :
R - R is*complete
but not*no rmed (recall that *norm-
ed spaces arepairs
of Banach spaces with isometricinclusions ).
In anyspatial
topos in whichR # *R ,
R is anexample
of a*no rmed
space which is no t*co mplete
since its*closure
is*R .
We will denote the full
subcategory
ofban (X ) consisting
of all*co mplete, *normed
spacesby *ban ( X ) .
Thefollo wing theorem,
based onthe (c Pullback Lemmas o f Linton
[9],
leads us to a characterization o f the elements of*ban(X).
3.9. THEOREM.
Let j : A -> B
be an isometricinclusion o f Banach
spaces, where A is*co mplete. Then
thefo
llowing diagram
is apullback
in ban(X J
P ROOF. Let
a e A **, b f B
be suchthat j **( a) = iB ( b ). Suppose that
7 ( b e j (A)). By 3.2
since A is
*clo sed.
Thenwhich
contradicts 7 (f(b)
=0). Hence, 77 ( b 6 j (A)). Since A is *com- plete,
it isdouble-negation closed,
as we showed in theproof
of3.6,
sob
e j(A ) . Finally letting
b= j (a) (a
isunique, since j
is anisometry),
we have
in ban
(X) :
PROOF. One direction follows
immediately
from 3.1 and 3.9 In the o therdirection, if
the abovediagram
is apullback,
thenHence, iA
is anisometry
andby 3.I,
A is*normed. lliloreover,
A is*clo s-
ed in A**by 3.6 and,
since A** is*complete7
so is A .Finally,
wepoint
out that the abovecorollary actually
shows thatthe functor
Q :
L4|->*Cl(iA(A))
is theidempotent triple
associated to the double dualizationtriple
o nban(X)
o btainedby
localization.( For
similarexamples
oflocalizations, see [13].)
Thenand, by general
results onlocalizations, ban (X )
is a reflective subcat- egory ofban(X).
ACKNOWLEDGMENT.
We would like to express our
appreciation
to Charles Burden and ChrisMulvey
forstimulating
and informative conversations and correspo n-dence. To Chris
Mulvey
go ourspecial
thanks for the detailedsuggestions
made on an earlier draft of this paper.
REFERENCES.
1. N.
AUSPITZ, Q-Sheaves of
Banach spaces,Dissertation,
Univ. of Waterloo, 1975.2. B. BANASCHEWSKI, Sheaves of Banach spaces,
Quaest.
Math. 1(1977),
1- 22.3. C. BURDEN, The Hahn-Banach Theorem in a category of sheaves, J. Pure and
Applied Algebra,
to appear.4. C. BURDEN, Normed and Banach spaces in
categories of
sheaves, Dissertation Un iv. of Sussex 1978.5. K. H. HOFMANN, Bundles
o f
Banach spaces, Sheavesof
Banach spaces,C(B)- modules,
Lectures at the TH Darmstadt, 1974.6. K. H. HOFMANN, Sheaves and bundles
o f
Banach spaces,Preprint,
1976.7. P. T. JOHNSTONE,
Topos Theory,
Academic Press, 1977.8. P. T. JOHNSTONE, Conditions related to De
Morgan’s
law,Preprint,
1977.9. F. E. J. LINTON, On a
pullback
Lemmafor
Banach spaces and thefunctorial
semantics
of double dualization, Preprint
withaddendum,
1970.10. C. J. MULVEY, Intuitionistic
Algebra
andrepresentation
ofrings,
Memoirs A.M. S. 14 8 (1974).
11.
C. J.
MULVEY, Banach sheaves, J. Pure andApplied Algebra,
to appear.12. R. PARE & D. SCHUMACHER, Abstract families and the
adjoint
functor Theo- rems, Lecture Notes in Math.661, Springer
( 1978).13.
J.
WICK PELLETIER,Examples
oflocalizations,
Comm. inAlg.
3 ( 1975), 81- 93.Joan Wick Pelletier :
Department
of.Mathematics, YorkUniversity
and
University
of Massachussetts Robert D.Rosebrugh :
Department o f
Mathematic s,
Mc Gill