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SOME EXTREMAL PROPERTIES OF SECTION SPACES
OF BANACH BUNDLES AND THEIR DUALS
D. A. ROBBINS
Received 24 August 2001
WhenXis a compact Hausdorff space andEis a real Banach space there is a considerable literature on extremal properties of the spaceC(X, E)of continuousE-valued functions onX. What happens if the Banach spaces in which the functions onXtake their values vary overX? In this paper, we obtain some extremal results on the section spaceΓ(π )and its dualΓ(π )∗of a real Banach bundleπ:Ᏹ→X(with possibly varying fibers), and point out the difficulties in arriving at totally satisfactory results.
2000 Mathematics Subject Classification: 46B20, 46E40, 46H25.
The present paper is motivated by the considerable literature which exists on vari-ous extremal properties of Banach spaces of the formC(X, E), the continuousE-valued functions from the compact Hausdorff spaceXto the Banach spaceEwith its norm topology. As examples of this literature, we cite for example [3,6] and various of their references. Work has also been done on extremal properties of spaces of the form
C(X, Ew), where in this caseEw is the Banach spaceEwith its weak topology (cf. [7]), and onC(X, Ew∗∗), whereEw∗∗ isE∗with its weak-∗topology (see [12]).
The classic, and first, example of these extremal properties ofC(X, E)is the follow-ing. Recall that ifEis a Banach space and ifA⊂E, then a pointz∈Ais an extreme point of Aif and only ifzis not the midpoint of any line segment with endpoints inA. Denoting the set of extreme points of a setAby extr(A), and the unit ball of a Banach space byB(E), then a functionalφ∈extr(B(C(X, E)∗))if and only if there exists somef∈extr(B(E∗))andx∈Xsuch thatφ=f◦evx, whereevx:C(X, E)→E, evx(σ )=σ (x)is the evaluation map atx∈X. See [14]; a more general version can be found in [8].
Ifπ:Ᏹ→Xis a Banach bundle, then the neighborhood system of a pointz0∈Ex⊂Ᏹ can be described by tubes of the form T =T (V , σ , ε)= {z∈Ᏹ:z−σ (π (z))< ε, π (z)∈V}, whereV varies over the system of open neighborhoods ofx=π (z0),
ε >0 varies, andσ ∈Γ(π )is such that σ (π (z0))=z0. Ifz∈Ex, then there exists σ ∈Γ(π ) such that σ (x)=z and σ = z. The function xσ (x) is upper semicontinuous from X toRfor eachσ ∈Γ(π ). If this function is continuous for eachσ ∈Γ(π ), we callπ:Ᏹ→X acontinuous bundle; in this case, the topology on Ᏹis Hausdorff. We call the bundleπ:Ᏹ→Xseparableif there exists a countable set {σn} ⊂Γ(π )such that{σn(x)}is dense inExfor eachx∈X; this can be interpreted as a sort of uniform separability of the fibersEx.
In thinking about the section spaceΓ(π ), perhaps the most important intuitive point to keep in mind is that whenσ∈Γ(π ), then the valuesσ (x)∈Exvary continuously over (quite possibly very) different spaces as x varies overX. A space of the form
C(X, E)can be regarded as the space of sections of the trivial bundle ρ:᐀=X× E→X, where X×E is given the product topology. Here, the total space ᐀=X×E
can be thought of as a union of copies ofE, and an elementσ ∈C(X, E), which we usually think of as having values which vary continuously over the fixed setE, can be interpreted as a section inΓ(ρ)which varies in a very nice way between copies ofE.
To our knowledge, there is only one extreme point result for Banach bundles to be found in the literature: from [2, Theorem 2], reproven in [1], it can be shown that if
π:Ᏹ→Xis a Banach bundle, thenφ∈extr(B(Γ(π )∗))if and only if there exists some
x∈X andfx∈extr(B(E∗x))such thatφ=fx◦evx, where, again,evx:Γ(π )→Ex, σσ (x)is the evaluation map. This is clearly an analogue of the result forC(X, E), and gives a reasonable first answer to the very loose question “what can we say about extreme points if the spaces, in which our continuous functions take their values, vary?”
With this as an introduction, and in pursuit of the question in the preceding para-graph, we now present some extremal results for section spaces of Banach bundles and their duals. All have analogues to those for spacesC(X, E)and their duals. Be-cause of the varying fibers in a Banach bundle, we have not been able to obtain results as fully descriptive as those for spacesC(X, E); some of the difficulties will be pointed out as the exposition proceeds.
Our first results regard a section spaceΓ(π )itself.
Proposition1(see [6, Theorem 1]). Letπ:Ᏹ→Xbe a Banach bundle. (1)Suppose thatσ ∈extr(B(Γ(π ))). Thenσ (x) =1for allx∈X.
(2)Suppose, in addition, thatπ:Ᏹ→X is a Hausdorff bundle, that is, thatᏱitself is a Hausdorff space. Letσ∈B(Γ(π )), withσ =1, and suppose thatK= {x∈X:
σ (x)∈extr(B(Ex))}is dense inX. Thenσ∈extr(B(Γ(π ))). In particular, this is true ifπ:Ᏹ→Xis a continuous bundle.
Proof. (1) Suppose thatσ∈B(Γ(π ))and that for somex∈Xwe haveσ (x)<1. By upper semicontinuity of the norm, there exists an open neighborhoodVofxsuch that fory∈V,σ (y)< (1+σ (x))/2. Choosez∈Exwithz =(1−σ (x))/2, and a section τ∈Γ(π ) such thatτ(x)=z,τ = z, and such thatτ(y)=0 if
(σ±τ)(y) = σ (y) ≤1; and, ify∈V, we have(σ±τ)(y) ≤ σ (y)+τ(y) ≤ (1+ σ (x))/2+(1− σ (x))/2=1. Thus, σ±τ ∈B(Γ(π )), and so σ is not an extreme point ofB(Γ(π )).
(2) Suppose thatσ =1/2(τ+τ ), whereτ, τ ∈B(Γ(π )). By the hypothesis that
σ (x)be an extreme point inB(Ex)wheneverx∈K, it follows that for eachx∈K we have σ (x)=τ(x)=τ (x). Suppose now that for somey∈X we haveτ(y)≠ τ (y). Becauseπ:Ᏹ→Xis a Hausdorff bundle, there are disjoint tubes aroundτ(y)
and τ (y). In particular, for some neighborhood V of y and some ε >0, we have τ(x)−τ (x) ≥ε for all x∈V. But by the density ofK, this is impossible. The second assertion follows because a continuous bundle is Hausdorff.
Recall that a pointz∈B(E)is called a sequential point of continuity if whenever {zn} ⊂B(E)is a sequence which converges weakly toz, thenzn−z →0. In [3, Theo-rem 3] it is shown thatB(C(X, E))contains no weak sequential points of continuity. The same is true for a large class of bundles.
Proposition2. Letπ:Ᏹ→Xbe a continuous and separable bundle, and suppose thatXis infinite. ThenB(Γ(π ))has no sequential points of continuity.
Proof. We follow the outline of [3, Theorem 3], with an adjustment to take care of the bundle situation.
Let σ ∈B(Γ(π )), and letU = {x:σ (x) <2/3}and V = {x :σ (x)>1/3}. By continuity of the norm, bothU and V are open, and their union is X. At least one ofU orV is infinite, sayUfor the moment. We can find open, pairwise disjoint, nonemptyUn⊂U; choosexn∈Un. Choosezn∈Exnsuch thatzn =1/3, and choose
σn∈Γ(π )such thatσn(xn)=znandσn =1/3. Choose continuous functionsan: X→[0,1]such thatan(xn)=1 and such thatan(X\Un)=0. Letτn=σ+anσn. Thenτn(y) = σ (y)+an(y)σn(y) = σ (y) ≤1, ify∈Un. On the other hand, ify∈Un, thenτn(y) ≤ σ (y)+an(y)∗1/3≤1, so thatτn∈B(Γ(π )). Evidently, τn−σ = anσn =1/3.
Now, letφ∈Γ(π )∗. By [5], there existsµ≥0∈M(X)and a selectionη:X→•{E∗x: x∈X}, the disjoint union of theEx∗, such that (a) for allσ∈Γ(π ),xσ (x), η(x) is Borel measurable; (b)η(x) =1µ-a.e.; and (c)σ , φ =Xσ (x), η(x)dµ. Then
τn−σ , φ= X
τn(x)−σ (x), η(x)
dµ
=
X
an(x)σn(x), η(x)
dµ
≤
Un
an(x)σn(x)dµ
≤13µUn →0.
(1)
Thus,σ is not a sequential point of continuity.
σ (xn) = σ (xn)>1/3. We again have
τn−σ , φ=
Xσn(x)−σ (x), η(x)
dµ
=
Vn
anσn(x), η(x)
dµ
≤
Vn
σn(x)dµ
≤µVn →0.
(2)
In this case also,σ is not a sequential point of continuity.
Recall that a pointz∈B(E)is called a strongly extreme point for eachε >0 there existsδ >0 such that ifz±y ≤1+δ, theny ≤ε. It is shown in [3] that a function
σ ∈B(C(X, E))is a strongly extreme point if and only ifσ (x)is a strongly extreme point ofB(E)for eachx∈X. We can do somewhat better in the bundle case in one direction.
Recall (see [4] or [10]) that ifC⊂Xis a closed set, then there is a bundleπC:ᏱC→C, called the restriction ofπtoC, whereᏱC=Ᏹ∩π−1(C). We haveΓ(πC)Γ(π )/ICΓ(π ), whereIC= {a∈C(X):a(x)=0 for allx∈C}andICΓ(π )is the closed span inΓ(π ) of{a·σ:a∈IC, σ∈Γ(π )}. WriteIxforI{x}. An elementτC∈Γ(πC)if and only ifτC is the restriction toCof someτ∈Γ(π ). In particular,ExΓ(π{x})Γ(π )/IxΓ(π ).
Proposition3. Letπ:Ᏹ→Xbe a bundle. Ifσ ∈B(Γ(π )) is a strongly extreme point, thenσC=the restriction ofσ toCis a strongly extreme point inΓ(πC)for each closedC⊂X. In particular, for eachx∈X, σ (x)is a strongly extreme point in the unit ball ofEx(and, by the above discussion,ExΓ(π{x})).
Proof. We use the Tietze extension theorem for bundles; see [9]. Letσ∈B(Γ(π ))
be a strongly extreme point, let C ⊂X be closed, and let ε >0 be given. Choose
δ >0 such that ifσ±ω ≤1+δ thenω ≤ε. Suppose thatτC ∈Γ(πC)is such thatσC±τC<1+δ/2. There existsτ∈Γ(π )such that the restriction ofτ toC isτC and such thatτ = τC. Clearly, then,σ±τ extendsσC±τC. By the upper semicontinuity of the norm, there exists a neighborhoodVofCsuch that ify∈Vthen (σ±τ)(y)<1+δ. Choose a continuous functioniV:X→[0,1]such thatiV(C)=0 andiV(X\V )=1. Thenσ±(1−iV)τ<1+δ, and(1−iV)τ is an extension ofτC. By the choice ofδ, we haveτC ≤ (1−iV)τ< ε.
The converse to this result is demonstrated in [3] for the case ofC(X, E). With the next two propositions, we provide a condition which will give us a converse for the case of section spaces, and show that the condition actually obtains in the case of
Proposition4. Suppose thatσ∈Γ(π )and thatσ (x)is a strongly extreme point of
B(Ex)for eachx∈X. Suppose also that the functionxδ0(x, ε, σ )is bounded away from0for eachε >0. Thenσ is a strongly extreme point ofB(Γ(π )). In particular, ifxδ0(x, ε, σ )is lower semicontinuous onX, thenσ is a strongly extreme point of
B(Γ(π )).
Proof. Fixε >0. Letδ0=δ0(ε, σ )=inf{δ0(x, ε, σ ):x∈X}>0. Suppose that
τ∈Γ(π )is such thatσ±τ ≤1+δ0. Then, givenx∈X, we haveσ (x)±τ(x) ≤ 1+δ0≤1+δ0(x, ε, σ ), so thatτ(x) ≤ε. Henceτ ≤ε.
Proposition5. Letσ∈C(X, E), and suppose thatσ (x)is a strongly extreme point ofB(Ex)for eachx∈X. Thenxδ0(x, ε, σ )is lower semicontinuous for eachε >0.
Proof. Fixx∈X. By the continuity ofσ, for eachk∈N,k≥2, there exists a neigh-borhoodVk ofx such that ify∈Vk, thenσ (x)−σ (y)< δ0(x, ε, σ )/k. Suppose thaty∈Vk, and thatσ (y)±z ≤1+((k−1)/k)δ0(x, ε, σ ). Then
σ (x)±z≤σ (x)−σ (y)+σ (y)±z
<δ0(x, ε, σ )
k +1+
k−1
k δ0(x, ε, σ ) =1+δ0(x, ε, σ ),
(3)
so thatz< ε. Thus, wheny∈Vk,δ0(y, ε, σ ) > ((k−1)/k)δ0(x, ε, σ ), so thatx
δ0(x, ε, σ )is lower semicontinuous.
We now consider extreme point theorems inB(Γ(π )∗). Recall that for a bundleπ: Ᏹ→X, there is a canonical isometric injectionjx:E∗x→Γ(π )∗, given byσ , jx(f ) = σ (x), f; that is,jx(f )=f◦evx. Conversely, it can be shown that there is a surjective L-projectionPx:Γ(π )∗→jx(Ex∗)given byPx(φ)=φx, whereφx=weak-∗limV((1− iV)φ), whereV∈ᐂ, the system of open neighborhoods ofx∈X, and, for eachV∈ᐂ, the continuous functioniV:X→[0,1],iV(x)=0 andiV(X\V )=0 (see [4,11]). Then forf∈Ex∗, we havePx(jx(f ))=jx(f ), andPxis an isometry onjx(E∗x). Using the notation precedingProposition 3, the range ofPxcan also be expressed as(IxΓ(π ))⊥; this is anL-summand inΓ(π )∗.
Proposition6. Letπ:Ᏹ→Xbe a bundle, and suppose thatf∈B(Ex∗)is a strongly extreme point. Thenjx(f )=f◦evxis a strongly extreme point inB(Γ(π )∗). Conversely, ifφ∈B(Γ(π )∗) is a strongly extreme point, then there existx∈X and a strongly extreme pointf∈B(Ex∗)such thatφ=jx(f )=f◦evx.
applying Rao’s lemma, we see that jx(f )=f◦evx is a strongly extreme point in B(Γ(π )∗).
Recall that a pointf∈E∗,f =1, is a point of weak-∗continuity ofB(E∗)if and only if whenever we have a netfα→f weak-∗, withfα ≤1, thenfα−f →0. If π:Ᏹ→Xis a bundle, then forφ∈Γ(π )∗andx∈X, writeφx=Px(φ), wherePxis theL-projection described earlier. We can find some of the weak-∗points of continuity
B(Γ(π )∗)by using techniques of [6].
We first make the following observation: letx∈Xbe isolated, and letz∈Ex. Then the selectionσ onXdefined by
σ (y)=
z, ifx=y,
0, ifx≠y (4)
is an element ofΓ(π ). Moreover, ifφ∈Γ(π )∗, we have
σ , φ−φx
=lim V
σ , φ−1−iV φ
=lim V
iVσ , φ
=0, (5)
asVranges over the neighborhood system ofx, since the functioniV which is 1 atx and 0 elsewhere is an identity for the maximal idealIx⊂C(X). Sinceφx∈(IxΓ(π ))⊥ Ex∗, we may writeφx=fx◦evxfor somefx∈E∗x. Thus,σ , φx = σ (x), fx.
Proposition7. Letπ:Ᏹ→Xbe a bundle, and suppose thatφ∈Γ(π )∗ has the form φ=x∈Iφx, whereI= {x ∈X:xis isolated}. Suppose further that for each x ∈I either fx =0 or fx/fx is a weak-∗ point of continuity of B(Ex∗)and that
x∈Ifx =
x∈Iφx =1, where φx=fx◦evx. Thenφis a weak-∗ point of con-tinuity ofB(Γ(π )∗).
Proof. It can be easily shown thatx∈IPx is anL-projection, and so it follows thatφ =x∈Iφx =
x∈I0φx, whereI0= {x∈I:φx≠0}.
Letx∈I0be arbitrary. Ifφα→φweak-∗ inΓ(π )∗, withφα ≤1 and ifz∈Ex, then a selection of the formσ (y)=z, ifx=y, andσ (y)=0 otherwise, is an element ofΓ(π ). From the preceding observation, we have
σ ,φα x
=σ (x),fα x
=σ , φα
→σ , φx
= σ , φ =σ (x), fx
, (6)
and so (fα)x→fx weak-∗ inE∗x. Since the norm is lower semicontinuous with re-spect to the weak-∗ topology, we obtain lim inf(fα)x ≥ fx, and hence find that lim inf(φα)x ≥ φx.
We now have
1= φ = x∈I
φx =
x∈I0
φx
≤
x∈I0
lim infφα x≤lim inf
x∈I0
φα x
=lim infφα≤1.
(7)
Now, by assumption, fx/fx is a weak-∗ point of continuity ofB(E∗x) for each x∈I0. Since(fα)x/(fα)x →fx/fxweak-∗, we then have
fα x fα x
− fx fx
→0, (8)
and thus(fα)x−fx = (φα)x−φx →0 for eachx∈I0. Finally, forε >0, we can choose a finiteI ⊂I0such that
x∈I φx =
x∈I fx> 1−ε. Whenαis sufficiently large,x∈I (fα)x−fx =x∈I (φα)x−φx< ε. Then for all these largeαwe have
φα−φ= x∈I
φα x−φx+
φα−φ −
x∈I
φα x−φx
< ε+φα−
x∈I
φα x + φ−
x∈I φx
=ε+φα−
x∈I
φα x
+φ−
x∈I φx
≤ε+2+ x∈I
φα x−φx−2
x∈I φx
< ε+2+ε+2(1−ε)
=4ε.
(9)
Thus,φα→φin norm, and soφis a weak-∗ point of continuity. Clearly, this proof owes its details to [6, Theorem 6].
It is shown in [6] that the condition analogous to the above forC(X, E)∗ actually characterizes weak-∗points of continuity inB(C(X, E)∗). The proof in this case uses very strongly the description ofC(X, E)∗. However, even in the case of a continuous and separable bundle, where there is at least some description ofΓ(π )∗, it is not clear how to arrive at the whole converse to Proposition 7. We do obtain a rather unsatisfactory partial result for bundles.
Proposition8. Letπ:Ᏹ→Xbe a bundle, and suppose thatφ∈B(Γ(π )∗),φ =1, is a weak-∗point of continuity. Suppose that for a givenx∈X,φx=fx◦evx≠ 0. Then fx/fxis a weak-∗ point of continuity ofB(Ex∗), andφx/φx is a weak-∗ point of continuity ofB(Γ(π )∗).
Proof. Let {gα} be a net in B(Ex∗) such that gα → fx/fx weak-∗. Let φα = φ−φx+jx(fxgα). Thenφa→φweak-∗inΓ(π )∗, and since
φα=φ−φx+jxfxgα ≤φ−φx+jxfxgα ≤φ−φx+fx =φ−φx+φx = φ,
it follows thatφα−φ = φx−jx(fxgα) = fx− fxgα →0. Thus,fx is a weak-∗point of continuity ofB(Ex∗).
Now, suppose that{ψα}is a net inB(Γ(π )∗)such thatψα→φx/φxweak-∗. We then haveφ−φx+φxψα→φweak-∗, and
φ−φx+φxψα≤φ−φx+φxψα≤1, (11)
so thatφ−φx+φxψα−φ = φxψα−φx →0, and thusφx/φxis a weak-∗ point of continuity ofB(Γ(π )∗).
In further analogy with [6], we can identify some points of continuity and sequential points of continuity inB(Γ(π )∗); see [6, Proposition 11]. The difficulties in a complete characterization of points of continuity inB(C(X, E)∗)which are discussed there are even less tractable forB(Γ(π )∗).
Proposition9. Letπ:E→Xbe a bundle, and suppose thatφ∈B(Γ(π )∗)has the formφ=x∈Xfx◦evx=x∈Xφx, where for eachx∈X eitherfx=0orfx/fx is a point of continuity (sequential point of continuity) of B(Ex∗), and φ =1 =
x∈Xfx◦evx. Then φ is a point of continuity (sequential point of continuity) of B(Γ(π )∗).
Proof. Ifx∈Xandφ∗∗x ∈Ex∗∗, defineΦ∗∗x ∈Γ(π )∗∗byφ,Φx∗∗=jx−1(φx), φ∗∗x for eachφ∈Γ(π )∗, whereφx andjx are as described earlier. Ifφ∈Γ(π )∗, and if φα→φweakly inΓ(π )∗, then for eachx∈Xwe have
φα,Φx∗∗
=j−x1
φα x , φ∗∗x
→φ,Φ∗∗ x
=j−x1
φx , φ∗∗x
, (12)
so thatjx−1((φα)x)→jx−1(φx)weakly inEx∗. Hence, ifφis of the form described in the statement of the proposition andφα→φweakly inΓ(π )∗, we havejx−1((φα)x)− jx−1(φx)=(φα)x−φx →0. By repeating the argument at the end ofProposition 7, we obtain thatφα−φ →0. The case of sequential continuity is clearly similar.
Example10. It is well known that for a Banach spaceE, we haveC(X, E)∗M(X, E∗), the space ofE∗-valued countably additive Borel measures of bounded variation onX, equipped with the variation norm. In [6] it is shown that ifµ∈B(C(X, E)∗)is a point of weak-∗continuity, thenµ({x})=0 wheneverxis non-isolated. The analogous re-sult for B(Γ(π )∗)then takes this form: wheneverφ∈B(Γ(π )∗)and φis a weak-∗ point of continuity, andx∈X is isolated, thenφx=0. In general, this is not true. LetX=[0,1], and letXdbeXwith its discrete topology. We can regardc0(Xd), the closure under the sup-norm of the set of (real-valued) functions onXwith finite sup-port, as the spaceΓ(π )of sections of a bundleπ :Ᏹ→X; this is referred to in [10] as the spiky bundle. ThenΓ(π )∗can be identified withl1(X), the space of summable functions onX. A point evaluationevx∈Γ(π )∗can be interpreted as both a strongly extreme point ofEx∗R, and a strongly extreme point ofΓ(π )∗, butxis not isolated inX. The norm-continuity condition on a bundle is thus necessary.
Proposition11. Letπ:Ᏹ→Xbe a bundle, and suppose thatExis strictly convex for eachx∈X.Ifσ∈Γ(π )andσ (x) =1for eachx∈X, thenσ∈extr(B(Γ(π ))).
Proof. Suppose thatσ =1/2∗(τ+τ ), whereτ≠τ ∈B(Γ(π )). Then for some
x∈X, we haveσ (x) =1= 1/2∗(τ+τ )(x)andτ(x)≠τ (x). But 1/2∗(τ+ τ )(x)is a strict convex combination from the unit ball ofEx, and so by hypothesis 1/2∗(τ+τ )(x)<1,a contradiction.
Thus, in the case where eachExis strictly convex, we obtain the characterizationσ∈ B(Γ(π ))is an extreme point if and only ifσ (x) =1 for allx∈X. (The converse to the above follows fromProposition 1.) Note that this is the same proof as forC(X, E).
Example12. There is a continuous normed real bundleπ:Ᏹ→X and a section
σ∈extr(B(Γ(π )))such that for somep∈X,σ (p)∈extrB(Ep). SetE=R2with norm given byα1= |α1| + |α2|, Ep= R2with αp=(|α1|p+ |α2|p)1/p, 1≤p≤2. Let
X=[1,2]. Forα∈R2, note thatα
1≥ αp≥ αq≥ α2when 1≤p≤q≤2. Thus, forp≥1 andα∈R2fixed, there existsc
p≥1 such thatcpαp= α1, and pcpis continuous (becausepαp=(|α1|p+|α2|p)1/p is continuous).
Suppose now thatσ∈C(X, E). Thenpσ (p)pis continuous. For, ifε >0, we can chooseqso close topthat both|σ (p)p−σ (p)q|< εandσ (p)−σ (q)1< ε. Then
σ (p)p−σ (q)q≤σ (p)p−σ (p)q+σ (p)q−σ (q)q ≤ε+σ (p)−σ (q)q
≤ε+σ (p)−σ (q)1
<2ε.
(13)
Thus the space of selections{σ :X→•{Ep: 1≤p≤2}, σ ∈C(X, E)} is a dense
subspace of continuous-normed selections in a bundle π : Ᏹ=•p∈XEp → X. For σ ∈C(X, E), with normσ =sup{σ (p)1:p∈X}, we can renormσ byσ = sup{σ (p)p:p∈X}. Then the mapI:C(X, E)→Γ(π ),(σ ,·)(σ ,·)is easily checked to be norm-decreasing, with dense range. But since forσ∈C(X, E)we also haveσ ≤√2σ, the range ofI is therefore closed, and so there is a topological linear isomorphism ofC(X, E)andΓ(π ).
Consider the sectionσ ∈C(X, E)given byσ (p)≡(1/2,1/2)/(1/2,1/2)p. Then σ (p)p=1 for allp; for 1< p≤2,σ (p)is an extreme point ofB(Ep); andσ (1)is not an extreme point ofB(E1). However, byProposition 1,σ is an extreme point of
B(Γ(π )).
assumption of norm continuity in the bundle binds fibers together more tightly than norm upper semicontinuity. The separability condition allows for a description of the dual of the section space. Thus, the addition of the conditions of norm continuity and separability of the bundle may eventually allow us to get around some of these difficulties, but it is not immediately apparent how this ought to be done. In general, there appears to the author to be a loose sort of “zero-one” law in effect: when an extremal result forΓ(π )orΓ(π )∗is proved here, it is arrived at through a tweaking of similar results for spacesC(X, E). On the other hand, many results, which in the
C(X, E)case use very strongly either a description ofC(X, E)∗or a metric notion of nearness, appear in the bundle situation to be nonobvious.
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D. A. Robbins: Department of Mathematics, Trinity College, Hartford, CT06106, USA
