arXiv:1101.0383v1 [math.FA] 2 Jan 2011 oteMce oooyon topology Mackey the to pc fShat itiuin for distributions Schwartz of space aua usin rs:()Is (i) arise: questions natural in Let endon defined called upre etoso the of sections supported ebgnwt imninmanifold Riemannian a with begin We i eiaie r one pt order to up bounded are derivatives Lie coe 1 2018 31, October ′ ′ F F U ? suiul eemndi erqiethat require we if determined uniquely is E Then . k pointed etesaeof space the be uu ndffrnilfrs[0 9.Tesaehsgo prope good has space The currents [9]. by [10] forms differential on culus hti sasprbeutaonlgcl( ultrabornological separable a is it that eas iesvrlnwbif(huhnncntutv)de non-constructive) (though brief new several b give difference also We fundamental a identifying reflexive, generally Abstract. ′ ′ B B ( ( U ieeta hisaecoe ne ulvrin ffundam of versions dual under closed are chains Differential U U D efidapeulto predual a find We . o l open all for ) endo noe subset open an on defined ) EATETO UEMTEAISADMTEAIA STATISTIC MATHEMATICAL AND MATHEMATICS PURE OF DEPARTMENT k OOOIA SET FDFEETA CHAINS DIFFERENTIAL OF ASPECTS TOPOLOGICAL k sa ( an is can in -chains B nti ae eivsiaetetplgclpoete ft of properties topological the investigate we paper this In ′ ( LF U C or ) ∞ U -pc,a nutv ii fF´ce pcs h space The Fr´echet of spaces. limit inductive an )-space, k U D ⊂ t xeirpwro h agn udeΛ bundle tangent the of power exterior -th k P frsdfie on defined -forms Let . ′ M ( NVRIYO AIONA BERKELEY CALIFORNIA, OF UNIVERSITY k U ′ h ns oal ovxtplg on topology convex locally finest the , u o eeal ntesrn ultplg of topology dual strong the in generally not but , F .Freape hisspotdi ntl aypit r d are points many finitely in supported chains example, For ). EATETO MATHEMATICS OF DEPARTMENT eeie i)I hr osrciedfiiino h oooyon topology the of definition constructive a there Is (ii) reflexive? F k NVRIYO CAMBRIDGE OF UNIVERSITY F = and 0 = hti,acmlt l.c.s. complete a is, that , U F M ( faReana manifold Riemannian a of 1. r U Let . and , DF eacmlt oal ovxsaeo ieeta forms differential of space convex locally complete a be ) .HARRISON J. Introduction P U .PUGH H. k )-space. U U ednein dense be B = and , 1 k ⊂ Ê lim = n M Let . D ←− eoe and open be k r h pc of space the tencret n ieeta chains. differential and currents etween B B ntoso h space the of finitions ′ k r te oeo hc r o exhibited not are which of some rties F k r . n httetplg on topology the that and , eteFr´echet of the space be na prtr fteCra cal- Cartan the of operators ental ′ M F eso that show We . uhta ( that such esaeo ieeta chains differential of space he P k k k P ( frswt opc support compact with -forms = U T k uhta ( that such P B .Eeet of Elements ). k ′ ( ( U ′ U ′ B ). F h pc ffinitely of space the ) ( U ) D ′ ′ ,adprove and ), B k ′ = ( U stecelebrated the is F ′ snot is ) k nein ense F S h predual The . frswhose -forms ) ′ ′ F = P k restricts F ( U Two . are ) 2 J. HARRISON & H. PUGH

′ Most of this paper concerns the space Bk which is now well developed ( [5–10]). First appearing 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 polyhedral chains can be found in [4].) We use an equivalent definition below using differential forms

Bk and the space of pointed k-chains Pk which has the advantage of of brevity.

P s We can write an element A ∈ k(U) as a formal sum A = Pi=1(pi; αi) where pi ∈ U, and αi ∈ Λk(TpU). Define a family of norms on Pk,

R-A ω kAkBr = sup r r

06=ω∈Bk kωkC - s Bˆ r for r ≥ 0, where RA ω := Pi=1 ω(pi; αi). Let k denote the Banach space on completion, and Bˆ = lim Bˆ r 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 unknown whether Bˆ k is complete. Since Bˆ k is a locally convex space, we may take its completion ′ ′ (see Schaefer [13], p. 17) in any case, and denote the resulting space by Bk(U). Elements of Bk(U) are called “differential k-chains1 in U.”

′B B′ The reader might ask how k(U) relates to the space k(U) of currents, endowed with the strong ′ dual topology. We prove below that Bk(U) is not generally reflexive. However, under the canonical ′B B′ inclusion u : k(U) → k(U), this subspace of currents is closed under the primitive and fun- 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 P B′ P ′B chains. That is, while k is dense in k in the weak topology, k is in fact dense in k in the B′ ′B strong topology. More specifically, when k(U) is given the strong topology, the space u( k(U)) ′ equipped with the subspace topology is topologically isomorphic to Bk(U). Thus in the case of n differential forms Bk(Ê ) question (i) has a negative, and (ii) has an affirmative answer.

2. Topological Properties

Proposition 2.0.1. Bˆ k is an ultrabornological, bornological, barreled, (DF ), Mackey, Hausdorff, ′ and locally convex space. Bk is barreled, (DF ), Mackey, Hausdorff and locally convex.

Proof. By definition, the topology on Bˆ k is locally convex. We showed that Bˆ k is Hausdorff in [10]. According to K othe [11], p. 403, a locally convex space is a bornological (DF ) space if and only if it is the inductive limit of an increasing sequence of normed spaces. It is ultrabornological if it is the inductive limit of Banach spaces. Therefore, Bˆ k is an ultrabornological (DF )-space.

1 previously known as “k-chainlets” TOPOLOGY OF CHAINS 3

Every inductive limit of metrizable convex spaces is a Mackey space (Robertson [1] p. 82). Therefore,

Bˆ k is a Mackey space. It is barreled according to Bourbaki [2] III 45, 19(a).

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 exercise 24(d). But the completion of a bornological space may not be bornological (Valdivia [15]) 

′ Theorem 2.0.2 (Characterization 1). The space of differential chains Bk is the completion of pointed chains Pk given the Mackey topology τ(Pk, Bk).

Proof. Since (Pk, Bk) is a dual pair, the Mackey topology τ(Pk, Bk) is well-defined (Robertson [1]). P P′ B This is the finest locally convex topology on k such that the continuous dual k is equal to k.

P P Bˆ Let τ|Pk be the subspace topology on pointed chains k given the inclusion of k into k. By

(2.0.1), the topology on Bˆ k is Mackey; explicitly it is the topology of uniform convergence on relatively σ(Bk, Bˆk)-compact sets where σ(Bk, Bˆk) is the weak topology on the dual pair (Bk, Bˆk). B P But then τ|Pk is the topology of uniform convergence on relatively σ( k, k)-compact sets, which P P B is the same as the Mackey topology τ on k. Therefore, τ|Pk = τ( k, k).



3. Relation of Differential Chains to Currents

B B r The Fr´echet topology F on = k is determined by the seminorms kωkC = supJ∈Qr ω(J), where r Bˆ r 2 Bˆ r Bˆ .Q is the image of the unit ball in k via the inclusion k ֒→ k

Lemma 3.0.3. (B,β(B, ′B)) = (B, F ).

Proof. First, we show F is coarser than 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 r0 r other words, Q , the polar of Q in Br, is absorbent and hence k·kCr is a seminorm for β.

On the other hand, β(B, ′B), as the strong dual topology of a (DF ) space, is also Fr´echet, and hence by [1] the two topologies are equal. 

′ n ′ n Ê Theorem 3.0.4. The vector space B(Ê ) is a proper subspace of the vector space B ( ).

2 The authors show this inclusion is compact in a sequel. 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 n bounded on Ê , and endows it with the Fr´echet space topology, just as we have done. He writes

1 ˙ ′ n on page 56, “(DL ), (B), (B), ne sont pas r´eflexifs.” Suppose Bk(Ê ) is reflexive. By Lemma 3.0.3

′ n n n

Ê Ê the strong dual of Bk(Ê ) is (B( ), F ). This implies that (B( ), F ) is reflexive, contradicting Schwartz. 

We immediately deduce:

′B B′ B′ Theorem 3.0.5. The space k carries the subspace topology of k, where k is given the strong B′ B topology β( k, k).

Corollary 3.0.6 (Characterization 2). The topology τ(Pk, Bk) on Pk is the subspace topology on P B′ B′ B k considered as a subspace of ( k,β( k, k)). P B′ Remarks 3.0.7. We immediately see that k is not dense in k. Compare this to the Banach- P B′ P ′B Alaoglu theorem, which implies k is weakly dense in k, whereas k is strongly dense in k.

In fact, Corollaries 2.0.2 and 3.0.6 suggest a more general statement: let E be an arbitrary locally convex topological vector space. Elements of E will be our “generalized forms.” Let P be the vector space generated by extremal points of open neighborhoods of the origin in E′ given the strong topology. These will be our “generalized pointed 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 as a subspace of E′ with the strong topology. We ask the following questions: under what conditions 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?

′ n Theorem 3.0.8. The space Bk(Ê ) is not nuclear, normable, metrizable, Montel, or reflexive.

ˆ n Proof. The fact that Bk(Ê ) is not reflexive follows from Theorem 3.0.4. It is well known that

n ′ n Ê Bk(Ê ) is not a normable space. Therefore, Bk( ) is not normable. If E is metrizable and (DF ), ′ n then E is normable (see p. 169 of Grothendieck [3]). Since Bk(Ê ) is a (DF ) space, it is not 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.

′ n ˆ n Ê Therefore, Bk(Ê ) is not nuclear. Since all Montel spaces are reflexive, we know that Bk( ) is not Montel.  TOPOLOGY OF CHAINS 5

4. Independent Characterization

We can describe our topology τ(Pk, Bk) on Pk in another non-constructive manner for U open in n Ê , this time without reference to the space B.

Theorem 4.0.9 (Characterization 3). The topology τ(Pk, Bk) is the finest locally convex topology

µ on Pk such that

(1) bounded mappings (Pk,µ) → F are continuous whenever F is locally convex; 0 (2) K = {(p; α) ∈ Pk : kαk = 1} is bounded in (Pk,µ), where kαk is the mass norm of n α ∈ Λk(Ê );

(3) Pv : (Pk,µ) → (Pk,µ) given by Pv(p; α) := limt→0(p + v; α/t) − (p; α/t) is well-defined and n bounded for all vectors v ∈ Ê .

Proof. A l.c.s. E is bornological if and only if bounded mappings S : E → F are continuous whenever 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 ∈ (Pk,µ) . Then

ωPv(p; α)= ω(lim(p + tv; α/t) − (p; α/t)) = lim ω((p + tv; α/t) − (p; α/t)) t→0 t→0

= lim ω(p + tv; α/t) − ω(p; α/t)= Lvω(p; α), t→0 0 0 where Lv is the Lie derivative of ω. Since ω is continuous and K is bounded, it follows that ω(K ) r r−1 is bounded in Ê. (see [2] III.11 Proposition 1(iii)). Similarly, K = Pv(K ) is bounded implies r ′ ω(K ) is bounded. It follows that ω ∈ Bk. Hence (Pk,µ) ⊂ Bk. Now µ is Mackey since it is ′ ′ bornological. Since (Pk,µ) ⊂ (Pk, τ) and τ is also Mackey, we know that τ is finer than µ by the Mackey-Arens theorem.



Example 4.0.10. Let kAk = lim kAk r . This is a norm on pointed chains (Harrison [4]). The ♮ −→r B 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.

References

[1] Alexander and Wendy Robertson. Topological Vector Spaces. Cambridge University Press, Cambridge, 1964. [2] Nicolas Bourbaki. Elements of Mathematics: Topological Vector Spaces. Springer-Verlag, Berlin, 1981. [3] Alexander Grothendieck. Topological Vector Spaces. Gordon Breach, 1973. 6 J. HARRISON & H. PUGH

[4] J. Harrison. Differential complexes and exterior calculus. [5] Jenny Harrison. Stokes’ theorem on nonsmooth chains. Bulletin of the American Mathematical Society, 29:235– 242, 1993. [6] Jenny Harrison. Continuity of the integral as a function of the domain. Journal of Geometric Analysis, 8(5):769– 795, 1998. [7] Jenny Harrison. Isomorphisms of differential forms and cochains. Journal of Geometric Analysis, 8(5):797–807, 1998. [8] Jenny Harrison. Geometric Hodge star operator with applications to the theorems of Gauss and Green. Mathe- matical Proceedings of the Cambridge Philosophical Society, 140(1):135–155, 2006. [9] Jenny Harrison. Geometric Poincare Lemma. submitted, March 2010. [10] Jenny Harrison. Operator calculus – the exterior differential complex. submitted, March 2010. [11] Gottfried K¨othe. Topological Vector Spaces, volume I. Springer-Verlag, Berlin, 1966. [12] Harrison Pugh. Applications of differential chains to complex analysis and dynamics. Harvard senior thesis, 2009. [13] Helmut Schaefer. Topological Vector Spaces. McMillan Company, 1999. [14] Laurent Schwartz. Th´eorie des Distributions, Tome II. Hermann, Paris, 1954. [15] Manuel Valdivia. On the completion of a bornological space. Arch. Math. Basel, 29:608–613, 1977.