Topological Aspects of Differential Chains

0.11 MB·English

by  Jenny Harrison

January 4, 2011
TOPOLOGICAL ASPECTS OF DIFFERENTIAL CHAINS
J.HARRISON
DEPARTMENTOFMATHEMATICS
UNIVERSITYOFCALIFORNIA,BERKELEY
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1
H.PUGH
0
2 DEPARTMENTOFPUREMATHEMATICSANDMATHEMATICALSTATISTICS
UNIVERSITYOFCAMBRIDGE
n
a
J

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Abstract. Inthispaperweinvestigatethetopologicalpropertiesofthespaceofdifferentialchains

] ′B(U) defined on an open subset U of a Riemannian manifold M. We show that ′B(U) is not
A
generally reflexive, identifying a fundamental difference between currents and differential chains.
F Wealsogiveseveralnewbrief(thoughnon-constructive)definitionsofthespace ′B(U),andprove
h. thatitisaseparableultrabornological(DF)-space.
t Differential chains areclosed under dual versions of fundamental operators of the Cartancal-
a
m culusondifferentialforms[10][9]. Thespacehasgoodpropertiessomeofwhicharenotexhibited
by currents B′(U)orD′(U). For example, chains supported in finitely many points are dense in
[
′B(U)forallopenU ⊂M,butnotgenerallyinthestrongdualtopologyofB′(U).

1
v
3
8
3
0 1. Introduction
.
1
0
1 WebeginwithaRiemannianmanifoldM. LetU ⊂M beopenandP =P (U)thespaceoffinitely
k k
1 supportedsectionsofthe k-thexteriorpowerofthe tangentbundle Λ (TU). ElementsofP (U)are
: k k
v called pointed k-chains in U. Let F =F(U) be a complete locally convex space of differential forms
i
X definedonU. WefindapredualtoF,thatis,acompletel.c.s. ′F suchthat(′F)′ =F. The predual
ar ′F isuniquelydeterminedifwerequirethatPk bedensein ′F,andthatthetopologyon ′F restricts
to the Mackey topology on P , the finest locally convex topology on P such that (′F)′ = F. Two
k k
natural questions arise: (i) Is ′F reflexive? (ii) Is there a constructive definition of the topology on
′F?
LetE bethespaceofC∞ k-formsdefinedonU,andD thespaceofk-formswithcompactsupport
k k
inU. ThenD isan(LF)-space,aninductivelimitofFr´echetspaces. ThespaceD′ isthecelebrated
k k
spaceofSchwartzdistributionsfork =0andU = n. LetBr betheFr´echetspaceofk-formswhose
R k
Lie derivatives are bounded up to order r, and B =lim Br.
k ←−r k
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2 J.HARRISON&H.PUGH
Most of this paper concerns the space ′B which is now well developed ( [5–10]). First appearing
k
in [5] is a constructive, geometric definition using “difference chains,” which does not rely on a
space of differential forms. (See also [12] for an elegant exposition. An earlier approach based on
polyhedralchainscanbefoundin[4].) Weuseanequivalentdefinitionbelowusingdifferentialforms
B and the space of pointed k-chains P which has the advantage of of brevity.
k k
We can write an element A ∈ Pk(U) as a formal sum A = Psi=1(pi;αi) where pi ∈ U, and αi ∈
Λ (T U). Define a family of norms on P ,
k p k
R- ω
kAkBr = sup A
06=ω∈Brk kωkCr
for r ≥ 0, where R-Aω := Psi=1ω(pi;αi). Let Bˆrk denote the Banach space on completion, and
Bˆ = lim Bˆr the inductive limit. We endow Bˆ with the inductive limit topology τ. Since the
k −→r k k
norms are decreasing, the Banach spaces form an increasing nested sequence. As of this writing, it
is unknownwhether Bˆ is complete. Since Bˆ is a locally convexspace,we maytake its completion
k k
(seeSchaefer[13],p. 17)inanycase,anddenotetheresultingspaceby ′B (U). Elementsof ′B (U)
k k
1
are called “differential k-chains in U.”
The reader might ask how ′B (U) relates to the space B′(U) of currents, endowed with the strong
k k
dualtopology. Weprovebelowthat ′B (U)isnotgenerallyreflexive. However,underthecanonical
k
inclusion u : ′B (U) → B′(U), this subspace of currents is closed under the primitive and fun-
k k
damental operators used in the Cartan calculus (see Harrison [10]). Thus, differential chains form
a distinguished subspace of currents that is constructively defined and approximable by pointed
chains. That is, while P is dense in B′ in the weak topology, P is in fact dense in ′B in the
k k k k
strong topology. More specifically, when B′(U) is given the strong topology, the space u(′B (U))
k k
equipped with the subspace topology is topologically isomorphic to ′B (U). Thus in the case of
k
differential forms B ( n) question (i) has a negative, and (ii) has an affirmative answer.
k
R
2. Topological Properties
Proposition 2.0.1. Bˆ is an ultrabornological, bornological, barreled, (DF), Mackey, Hausdorff,
k
and locally convex space. ′B is barreled, (DF), Mackey, Hausdorff and locally convex.
k
Proof. By definition, the topology on Bˆ is locally convex. We showed that Bˆ is Hausdorff in [10].
k k
According to Kothe [11], p. 403, a locally convex space is a bornological(DF) space if and only if
itis the inductive limitofanincreasingsequence ofnormedspaces. Itis ultrabornological if itis the
inductive limit of Banach spaces. Therefore, Bˆ is an ultrabornological(DF)-space.
k
1previouslyknownas“k-chainlets”
TOPOLOGY OF CHAINS 3
EveryinductivelimitofmetrizableconvexspacesisaMackeyspace(Robertson[1]p. 82). Therefore,
Bˆ is a Mackey space. It is barreled according to Bourbaki [2] III 45, 19(a).
k
The completion of any locally convex Hausdorff space is also locally convex and Hausdorff. The
completion of a barreled space is barreled by Schaefer [13], p. 70 exercise 15 and the completion of
a (DF)-space is (DF) by [13] p.196 exercise24(d). But the completion of a bornologicalspace may
not be bornological(Valdivia [15]) (cid:3)
Theorem 2.0.2 (Characterization 1). The space of differential chains ′B is the completion of
k
pointed chains P given the Mackey topology τ(P ,B ).
k k k
Proof. Since (P ,B ) is a dual pair, the Mackey topology τ(P ,B ) is well-defined (Robertson [1]).
k k k k
This is the finest locally convex topology on P such that the continuous dual P′ is equal to B .
k k k
Let τ|Pk be the subspace topology on pointed chains Pk given the inclusion of Pk into Bˆk. By
(2.0.1), the topology on Bˆ is Mackey; explicitly it is the topology of uniform convergence on
k
relatively σ(B ,Bˆ )-compact sets where σ(B ,Bˆ ) is the weak topology on the dual pair (B ,Bˆ ).
k k k k k k
But then τ|Pk is the topology of uniform convergence on relatively σ(Bk,Pk)-compact sets, which
is the same as the Mackey topology τ on Pk. Therefore, τ|Pk =τ(Pk,Bk).
(cid:3)
3. Relation of Differential Chains to Currents
The Fr´echet topology F on B=Bk is determined by the seminorms kωkCr =supJ∈Qrω(J), where
Qr is the image of the unit ball in Bˆr via the inclusion2 Bˆr ֒→Bˆ .
k k k
Lemma 3.0.3. (B,β(B, ′B))=(B,F).
Proof. First, we show F is coarserthan β(B, ′B). It is enough to show that the Fr´echet seminorms
are seminorms for β. For every form ω ∈Bk there exists a scalar λω,r such that kλω,rωkCr ≤1. In
other words, Qr0, the polar of Qr in Br, is absorbent and hence k·kCr is a seminorm for β.
Ontheotherhand,β(B, ′B),asthestrongdualtopologyofa(DF)space,isalsoFr´echet,andhence
by [1] the two topologies are equal. (cid:3)
Theorem 3.0.4. The vector space ′B( n) is a proper subspace of the vector space B′( n).
R R
2
Theauthorsshowthisinclusioniscompact inasequel.
4 J.HARRISON&H.PUGH
Remark: ′B and B are barreled. Thus semi-reflexive and reflexive are identical for both spaces.
Proof. Schwartz defines (B) in §8 on page 55 of [14] as the space of functions with all derivatives
bounded on n, and endows it with the Fr´echet space topology, just as we have done. He writes
R
on page 56, “(DL1),(B˙),(B), ne sont pas r´eflexifs.” Suppose ′Bk( n) is reflexive. By Lemma 3.0.3
R
the strong dual of ′B ( n) is (B( n),F). This implies that (B( n),F) is reflexive, contradicting
k
R R R
Schwartz. (cid:3)
We immediately deduce:
Theorem 3.0.5. The space ′B carries the subspace topology of B′, where B′ is given the strong
k k k
topology β(B′,B ).
k k
Corollary 3.0.6 (Characterization 2). The topology τ(P ,B ) on P is the subspace topology on
k k k
P considered as a subspace of (B′,β(B′,B )).
k k k k
Remarks 3.0.7. We immediately see that P is not dense in B′. Compare this to the Banach-
k k
Alaoglu theorem, which implies P is weakly dense in B′, whereas P is strongly dense in ′B .
k k k k
In fact, Corollaries 2.0.2 and 3.0.6 suggest a more general statement: let E be an arbitrary locally
convextopologicalvectorspace. ElementsofE willbe our“generalizedforms.” LetP bethevector
space generated by extremal points of open neighborhoods of the origin in E′ given the strong
topology. These will be our “generalizedpointed chains.” Then (P,E) forms a dual pair and so we
may put the Mackey topology τ on P. We may also put the subspace topology σ on P, considered
asasubspaceofE′ withthestrongtopology. Weaskthefollowingquestions: underwhatconditions
on E will τ = σ? Under what conditions will P be strongly dense in E′? What happens when we
replace B with D, E or S, the Schwartz space of forms rapidly decreasing at infinity?
Theorem 3.0.8. The space ′B ( n) is not nuclear, normable, metrizable, Montel, or reflexive.
k
R
Proof. The fact that Bˆ ( n) is not reflexive follows from Theorem 3.0.4. It is well known that
k
R
B ( n) is not a normable space. Therefore, ′B ( n) is not normable. If E is metrizable and(DF),
k k
R R
then E is normable (see p. 169 of Grothendieck [3]). Since ′B ( n) is a (DF) space, it is not
k
R
metrizable. If a nuclear space is complete, then it is semi-reflexive,that is, the space coincides with
its second dual as a set of elements.
Therefore, ′B ( n)isnotnuclear. SinceallMontelspacesarereflexive,weknowthatBˆ ( n)isnot
k k
R R
Montel. (cid:3)
TOPOLOGY OF CHAINS 5
4. Independent Characterization
We can describe our topology τ(P ,B ) on P in another non-constructive manner for U open in
k k k
n, this time without reference to the space B.
R
Theorem 4.0.9 (Characterization 3). The topology τ(P ,B ) is the finest locally convex topology
k k
µ on P such that
k
(1) bounded mappings (P ,µ)→F are continuous whenever F is locally convex;
k
(2) K0 = {(p;α) ∈ P : kαk = 1} is bounded in (P ,µ), where kαk is the mass norm of
k k
α∈Λ ( n);
k
R
(3) Pv :(Pk,µ)→(Pk,µ) given by Pv(p;α):= limt→0(p+v;α/t)−(p;α/t) is well-defined and
bounded for all vectors v ∈ n.
R
Proof. Al.c.s. EisbornologicalifandonlyifboundedmappingsS :E →F arecontinuouswhenever
F is locally convex. Any subspace of a bornological space is bornological. So by proposition 2.0.1,
τ satisfies (1). Properties (2) and (3) are established in Harrison [4,10].
Now suppose µ satisfies (1)-(3). Suppose ω ∈(P ,µ)′. Then
k
ωP (p;α)=ω(lim(p+tv;α/t)−(p;α/t))= limω((p+tv;α/t)−(p;α/t))
v
t→0 t→0
= limω(p+tv;α/t)−ω(p;α/t)=L ω(p;α),
v
t→0
where L is the Lie derivative of ω. Since ω is continuousand K0 is bounded, it follows that ω(K0)
v
is bounded in . (see [2] III.11 Proposition 1(iii)). Similarly, Kr = P (Kr−1) is bounded implies
v
R
ω(Kr) is bounded. It follows that ω ∈ B . Hence (P ,µ)′ ⊂ B . Now µ is Mackey since it is
k k k
bornological. Since (P ,µ)′ ⊂ (P ,τ)′ and τ is also Mackey, we know that τ is finer than µ by the
k k
Mackey-Arens theorem.
(cid:3)
Example 4.0.10. Let kAk♮ = l−i→mrkAkBr. This is a norm on pointed chains (Harrison [4]). The
Banach space (P,♮) satisfies (1)-(3). The topology ♮ is strictly coarser than τ since (P,t)′ =B and
(P,♮)′ is the space of differential forms with a uniform bound on all directional derivatives.
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