7.5 Higher-Order models with Distributions
7.5.1 Higher-order distributions and Kernel
In all this section the term reflexive refers toβ-reflexive spaces. Without further indication, the notationE1will refer to the lcsEβ1, the strong dual of the lcsE.
Definition 7.5.2. ConsiderEa lcs and two continuous linear injective functionsf :Rn (Eandg :Rm(E such thatněm. We say thatf ěgwhen
f “g|Rn
or said otherwise, whenf “g˝ιn,mwhereιn,m:Rn //Rmis the canonical injection. This defines indeed an order on the set of all linear continuous injective functions from an euclidean space toE. Then we consider the
family of spacespE1pRnq:“C8pRnq1, f :Rn (Eqf, directed by the order above mentioned. Iff ďgthen we have an inclusion seen as a linear continuous injective mapSf,g:Eg1pRnq //Ef1pRmq, phiÞÑ phÞÑφph˝ιn,mqq.
Definition 7.5.3. ConsiderEany lcs. We define then the space of distributions onEas the inductive limit4in the category TOPVEC:
E1pEq:“ lim
f:RÝÑn(E
Ef1pRnq
wheref : Rn ( E islinear continuous injective, andEf1pRnqdenotes the copy ofE1pRnqcorresponding to f :Rn(E.
Remark7.5.4. With this definition we are basically saying that distributions with compact support onEare the distributions with a finite dimensional compact supportKĂRn.
Proposition 7.5.5. For any lcsE,E1pEqis reflexive lcs.
Proof. As the inductive limit of reflexive and thus barrelled spaces,E1pEqis barrelled (see [44, 11.3.1]). Thus we just need to prove thatE1pEqis semi-reflexive, that is that every bounded closed subset ofE1pEqis weakly compact (then we will have that the strong dualEβ1 coincides to the Mackey dual, and thus that we have the linear isomorphismpEβ1q1„E, see section3.5.2).
Let use a well known method and show that this inductive limit is regular, that is any bounded setBis contained in one of theEf1pRnq. Suppose that it is not. Then we have some bounded setBsuch that for everynthere isfn
such that we havebn PBzEf1pRnq. The lcsF “lim
ÝÑnEf1npRnqis a strict (i.e. indexed byN) inductive limit of spaces such thatEf1npRn`1qis closed inEf1npRn`1q, and as such it is regular ([44, 4.6.2]). But the topology onF is the one induced its inclusion inEpEq, and thus the imageBF ofBinF is bounded. So asFis a regular limit BF should be included in one of theEf1npRnq, and we have a contradiction.
Thus any bounded setBofE1pEqis bounded in oneEf1pRnq. As these spaces are reflexive (as duals of nuclear (F)-spaces, see section7.2.2),B is in particularEf1pRnq-weakly compact. But the dual of an inductive limit is contained in the product of the duals ([44, 8.8]) and as suchBis also weakly compact inEpEq. ThusE1pEqis semi-reflexive, and as it is barrelled it is reflexive.
In fact, E1pEqis can be characterized as an inductive limit of nuclear DF spaces. What we showed in the previous proof is that any inductive limit of nuclear DF space is reflexive. By analogy with thestrictinductive limit of Fréchet spaces which are denotedLF-spacesin the literature, we introduce in section7.5.4the characterization of LNDFspaces. Now we show that our notation is indeed coherent with the one used in the previous sections.
Proposition 7.5.6. IfE »Rmfor somem, thenE1pEq »E1pRmq, whereE1pRmqdenotes here the usual space of distributions onRmwith compact support.
Proof. IfE » Rm, the linear continuous injective mapid : Rm ( Rm, results, by definition of the inductive limit, in a linear continuous injective mapEi1dpRmq » E1pRmq //E1pEq. Now considerφP E1pEq. Suppose that we have an indexf :Rn //Eon whichφis non-null. Then asf is an injection one hasnďmand thus a linear injectionE1pRnqãÑE1pRmq. This injection is continuous: indeed any compact inRnis compact inRm, and thus the topology induced byE1pRmqonE1pRnqis exactly the strong topology onE1pRnq. As we have in particular a linear mapf :Rm //Rm, we have the desired equality.
Definition 7.5.7. For any reflexive spaceEwe denote byEpEqthe strong dualpE1pEqq1βofE1pEq. As the strong dual of a reflexive space it is reflexive.
We recall the duality between projective and inductive limits: the dual of a projective limit islinearly isomor-phicto the inductive limit of the duals, however the topology may not coincide by [44, 8.8.12]. WhenEis endowed with its Mackey topology (which is the case in particular whenEis reflexive), then the topologies coincides.
Notation 7.5.8. We consider the projective systempEpRnq :“ C8pRnq, f : Rn (Eqf, directed by the order above mentioned. We denote byEfpRnqthe lcs copy ofEpRnq.
Proposition 7.5.9. WhenEis reflexive, the lcsEpEqis linearly homeomorphic to the projective limit limÐÝ
f:Rn(E
EfpRnq,
4see section3.1.4
indexed by
Tg,f “S1f,g:EgpRmq //EfpRnq,gÞÑg˝ιn,m
forf ďg, that is forf “g˝ιn,m.
Proof. According to [44, 8.8.7], asF “lim
ÐÝf:Rn(ECf8pRnqis reduced, then its dual is linearly isomorphic to F1“ lim
f:RÝÑn(E
Cf8pRnq
by reflexivity of the spacesCf8pRnq. Let us prove that this is a linear homeomorphism. As a reflexive space, EfpRnqfis endowed with its Mackey topology. As the Mackey-topology is preserved by inductive limits, we have that the topology on the lcsF1 »E1pEq »lim
ÐÝf:Rn(ECf8pRnq1 is also the Mackey-topologyµpF, F2q. But as we know thatEpEqis reflexive, this topology is exactly the strong dual topology.
Remark7.5.10. Thus elementsf PEpEqare familiespffqf:Rn(E such that iff :Rn (E,g :Rm (F and f “g˝ιn,m, we have :
ff “fg˝ιn,m.
Remark7.5.11. Thus we are saying that distributions a lcsEare in fact distributions with compact support in an euclidean space, or equivalently that smooth functions a lcsE are those with are smooth when restricted to one Rn. This makes it possible to define multinomials on lcsE:
PpxPRkq “ ÿ
IĂr|1,n|s
aαxα11. . . xαnIn
where we embedRk inRn fork ă n, and project it fork ą n. This definition, which as we said is inspired by work on differential algebras [46]. One could thought of another setting restricted specifically to higher order spaces of distributions and not to every reflexive space. Indeed, we would like to describe smooth scalar functions on a lcsE:“EpRnqas :
hPEpRnq ÞÑfp0q2,
thus taking into account that we have as input non-linear functions. This seems to indicate that we would like to construct smooth functions indexed by dirac functions, that is by functionsδ:Rn(E1“E1pRnq.
The Kernel Theorem As what was argued for negative interpretations ofLLpolin2.3.2, here we need to prove the Kernel theorem on negative spaces, that is on spaces of functions asEpEq. Indeed, the spaces of functions are the one which can be described as projective limits, and projective limits are the one which commutes with a completed projective tensor product. Let us highlight the fact that this theorem strongly depend of the fact that the considered spaces of functions are nuclear.
Theorem 7.5.12. For every lcsEandFwe have a linear homeomorphism:
EpEqbˆπEpFq »EpE‘Fq.
Proof. By section3.6or [44, 15.4.2] we have that the completed projective tensor product of two projective limits indexed respectively byIandJis the projective limit, indexed byIˆJwith the pointwise order, of the completed tensor product. Thus:
E1pEqbˆπE1pFq » p lim
f:RÐÝn(E
Ef1pRnqqbˆπp lim
g:RÐÝm(F
Eg1pRmqq
» lim
pf,gqÐÝ
Ef1pRnqpbˆπEg1pRmqq
» lim
pf,gqÐÝ
E1pRn‘Rmqthrough the Kernel theorem7.3.9
However, the direct sumRnˆRmis isomorphic toRn`m as we deal here with finite indices. Thus any linear continuous injective functionh:Rk //E‘Fis by definition of the biproduct topology a sumf`g:Rn‘Rm(
E‘F5. Indeed, ashis linear we have thath´1pE,0qis a sub-linear space ofRk, that is an euclidean spaceRn fornďk. Thus the indexing by pairspf, gq: Rn`m //EˆF coincide with the indexation by linear smooth functionsh:Rk //EˆF1, with corresponding pre-order, and thus:
E1pEqbˆπE1pFq » lim
f:Rk ÐÝ//EˆF
E1pRkq
»E1pEˆFq
Now we proceed to the definition of a functor?, which agrees to the previous characterization?E»C8pE1,Rq whenE»Rn.
Notation 7.5.13. We denote byREFLisoand linear homeomorphism between them.
Proposition 7.5.14. We extend the definition ofEpEqto a functor? :REFLiso //REFLiso, by:
?`with the projectionsπf is continuous, and injective as`1is injective.
Proof. We check immediately that indeed?Id“ Id. Let us see that?is covariant: consider`1 : E //F and
`2 : F //Glinear continuous surjective functions between the lcsE,F andG. Considerh : Rk //G1, and
Notation 7.5.15. We denote byREFLisothe category of reflexive lcs and linear continuous isomorphisms between them.
Definition 7.5.16. We define the functor! :REFLiso //REFLisodefined as:
! :
5Notice here the link between direct sum, biproduct and sums of smooth functionsf:RK //Eandg:Rk //F, as exposed in section 2.4.2.
Remark7.5.17. We showed previously that whenE »Rn, then?E »EpRnq. The notation!E:“E1pEqis still coherent with the euclidean case, aspRnq1 »Rnand thusEpE1q1»E1pRnqwhenE»Rn.
Remark7.5.18. Let us point out that constructingEpEqandE1pEqas projective and inductive limits over linear functionsf :Rn (E1(thus embedding into the dual of the codomainEand not intoEitself) would have made the definition of the codereliction and the co-multiplication much easier. Indeed, for example, the codereliction would have been defined as :
xPEÞÑfPEpEq ÞÑD0pfqpf1pxqq,
wheref1 :E2»E //Rn denotes the transpose off. However, we must indexE1pEqbyf :Rn (E in order to makeE a covariant functor (proposition7.5.14).
Outlook14. In order to avoid the use of isomorphisms in REFL, and to have the functoriality of!with respect to any linear continuous functionE ( F, we could try to define the following. ConsiderE andF two lcs and
` : E ( F a linear continuous map. Letφ P E1pEqand we denote by f : Rn ( E the linear continuous injections indexingE1pEq.
Then one defines a linear continuous injection as :
`z˝F: Rn
Kerp`˝fq //E.
The quotient vector space Kerp`˝fqRn is finite dimensional and thus isomorphic to someRp, forpďn. We denote byπn,pthe projection ofRnontoRp. Then one could define:
Ep`qpφqpgq “ pg
`˝fy ˝πn,pqf. The linear continuous mapz`˝f is indeed injective, whileg
`˝fy ˝πn,p PC8pRn,Rq. However, one still needs to check that the dereliction, co-multiplication and co-dereliction stated below are still natural with that definition of!on maps.