$M, P ofLLpolas a morphism in COMPL:
JπKPLpP1, M1b...ˆ bMˆ nq.
Through the strong monoidal closure between the negations, this is equivalent to interpreting a proof of the sequent in COMPLpP1b...bMˆ n1, Pq. A proof of the sequent$Mis interpreted as:
JπKPLpR, M1b...ˆ bMˆ nq
The interpretation of connectives follows the one of section2.3.2.3. The only difference is that the cut-rule coin-cides to the composition of linear continuous functions in COMPL. Theˆis interpreted as the completion making a space in NDFcomplete.
7.3 Kernel theorems and Distributions
Traditionally, one tackles the search for a denotational model ofLLby looking for a cartesian closed category, which will allow for an interpretation of the Seely isomorphism. Here, we take another point of view by looking for strong monoidal functors. Let us denote informally spaces of distributions are the duals of spaces of smooth functions: E1pRnq:“ pC8pRn,Rqq1. In particular, iff is a function with compact support, then for any smooth functionhthe integral
ż
fpxqhpxqdx is well defined, and defines a distributionTf :hÞÑş
fpxqhpxqdx. Now if you consider natural numbersnandm, then any distributionkinC8pRn`m,Rq1defines a linear operator, calledKernel,K:C8pRn,Rq //C8pRm,Rq1:
Kphqph1q “kppx1, ..., xn, y1, ..., ymq ÞÑhpxqh1pyqq.
Schwartz’s theorem says that the converse operation always possible: from a kernelKone can constructs a distri-butionk. In linear logic, if we denote!Rn:“ pC8pRn,Rqq1this amounts to say that
!pRnˆRmq „!Rn`!Rm.
In fact, because we will work with nuclear spaces, this`is a completed tensor product, which is associative and commutative on (F)-spaces.
In this Chapter we will try to look for denotational models of LL and DiLL in which the exponential is inter-preted by spaces of distributions. This will lead us to some intermediate syntax ( Smooth DiLL), of DiLL without higher order, and we will then exploit a categorical extension of distributions to interpret higher order terms.
Notation 7.3.1. If X is an open subset ofRn, we denote byCC8pXqthe vector space of all smooth functions f :X //Rwith compact support (i.e. there isKcompact inXsuch that ifxPXzK,fpxq “0). It is a lcs when endowed with the topology of uniform convergence of all derivatives of finite order on every compact ofX. This topology has been described in Definition3.2.6. This space is sometimes called the space of test functions, and denoted byDpRnqorDpXq.
Notation 7.3.2. If X is an open subset ofRn, we denote byC8pXqthe vector space of all smooth functions f :X //RIt is a lcs when endowed with the topology of uniform convergence of all derivatives of finite order on every compact ofX. This topology has been described Definition3.2.6. Following the literature it is also denoted byEpXqorEpRnq.
7.3.1 Kernel theorems for functions
Kernel theorems come from the fact that one can approximate functions with compact support by polynomials (their Taylor sums in fact, see Chapter 15 of [76]).
Proposition 7.3.3. [76, 39.2] ConsiderX(resp. Y) an open subset ofRn(resp.Rm). Then the algebraic tensor productCc8pXq bCc8pYqis sequentially dense inCc8pXˆYq.
The proof of the previous theorem consists in approximating a function inCc8pXˆYqby polynomialspPkqk, and then using partition of the unitygandhonthe projection on X or Y of the support off, to define a sequence of polynomialspgbhqPkwhich converge tof inCc8pXˆYq. This theorem is also true under a refined statement:
Cckp8, Xq bCclpYqis sequentially dense inCck`lpXˆYq2. From this theorem we deduce:
Proposition 7.3.4. ConsiderX (resp. Y) an open subset ofRn (resp. Rm). Then the algebraic tensor product C8pXq bC8pYqis sequentially dense inC8pXˆYq.
The following kernel theorem for functions is a direct consequence of the fact that the considered spaces of functions are nuclear.
Theorem 7.3.5. [76, Theorems 39.2 and 51.6] ConsiderX (resp.Y) an open subset ofRn (resp.Rm). Then we have the linear homeomorphism:
C8pXqbCˆ 8pYq »C8pXˆYq.
Proof. In the previous theorem, the connectorbˆ equivalently denotes the completed projective tensor product or the completed injective tensor product, as it involves nuclear spaces. Let us show that the lcsC8pXˆYqinduces on the vector spaceC8pXq bC8pYqthe topologybπ “ b. As the second is dense in the first, and the first is complete, the completion of the second will be linearly homeomorphic to the first.
The topology induced byC8pX ˆYqis coarser than the projective topology: indeed the bilinear mapping C8pXq ˆC8pYq //C8pXˆYqis separately continuous and thus continuous as we are dealing with (F)-spaces.
The topology induced byC8pXˆYqis finer than the injective topology: tensor productAbBof equicon-tinuous sets inC8pXq1andC8pYq1respectively are equicontinuous inC8pXˆYq1.
Thus the topology induced byC8pXˆYqonC8pXq bC8pYqis exactly the projective and injective topology.
Kernel theorems for other spaces of functions Various forms of kernel theorems exists for formal power series, holomorphic functions (spaces of holomorphic functions are also nuclear), spacesCc8pRnqof smooth functions with compact support, and measurable functions.
Let us emphasize what happens for measurable functions. From the same techniques one deduces that, for 1 ď p ă 8,LppXq bLppXq is dense inLppX ˆYqwhereX andY are measurable spaces (showing that measurable maps forms a good basis to model probabilistic programs). However this density result on measurable maps could not lead to an isomorphism as introduced in the introduction of section7.3, by detailing the particular case ofL2pXq. The dual ofL2pXqis alsoL2pXq, and any linear mapkPL2pXˆYq1induces indeed a "kernel"
K:L2pXq //L2pYq1»L2pXqthrough the computation exposed in the introduction of the section:
Kpf PL2pXqq:gPL2pYq ÞÑkppx, yq ÞÑfpxqgpyqq.
2that is, every point ofCck`lpXˆYqis the limit of a sequence, and not only a filter, inCckpXq bCclpYq
This embedding of L2pX ˆYq1 intoLpL2pXq, L2pYq1qis however not surjective. Indeed whenX “ Y the identityK“IdL2pXqis an element ofLpL2pXq, L2pXq1q “LpL2pXq, L2pXqq, while it would coincide to
k:hPL2pX, Yq ÞÑδx´yp‰0iffx“yq which is not a measurablefunctiononXˆY.
This coincides to the necessity to consider spaces of distributions in generality, spaces which includes in par-ticular Dirac maps.
7.3.2 Distributions and distributions with compact support
Interpreting the exponential by spaces of distributions with compact support is recurrent in models of Linear Logic.
Following a remark by Frölicher and Kriegl, the authors of [6] point out that their exponential coincides with the space of distributions with compact support when the exponential is defined on a euclidean spaceRn[26, 5.1.8].
In Köthe spaces, Ehrhard notes that!1contains the distributions with compact support onR.
Notation 7.3.6. The strong dual ofDpXqis called the space of distributions and denotedD1pXq. The strong dual ofEpXqis called the space of distributions with compact support and denotedE1pXq.
The idea is that distributions are generalized smooth functions and distributions with compact support are generalized functions with compact support.
Indeed, any smooth scalar functionfdefines a distribution Tf:gPDpRnq ÞÑ
ż f g
while smooth scalar function with compact support defines a distribution with compact support:
Tf :gPEpRnq ÞÑ ż
f g.
The integral is well-defined as in each case,f orghas compact support.
Proposition 7.3.7. For anynPN,EpRnqis anpFq ´spaceandE1pRnqis a completepDFq ´space.
Example7.3.8. A distribution must be considered as a generalized function, and acts as such. The key idea is that, iff PCc8pRnqthen on defines a compact distribution by
gPC8pRnq ÞÑ ż
fpxqgpxqdx.
Typical examples of distributions which do not follow this pattern are the dirac distribution or its iterated deriva-tives: forxPRnone defines the dirac atxas:δx:f PEpRnq ÞÑfpxq.Thenδpkq0 :f ÞÑ p´1qkfpkqp0q.
Taking the strong dual of the kernel theorem for functions7.3.5gives:
E1pRnˆRmq » ppEpRnqbˆEpRmqq1.
Let us recall that asEpRnqis a nuclear space, the operatorbˆ equivalently denotes the projective tensor product bˆπor the injective tensor productbˆε. However asEpRnqis a (F)-space we have through proposition7.2.30that pEpRnqbˆEpRmqq1»E1pRnqbˆE1pRmqand thus:
Theorem 7.3.9([76] 51.6). For anyn, mPNwe have:
E1pRn`mq »E1pRnqbˆπE1pRmq »LpE1pRmq,EpRnqq
Definition 7.3.10. The support of a distributionφ P D1pRnqis the set of points x P Rn such that there is no neighbourhoodU ofx, with a non-null functionf PC8c pRnqwhose support is included inUsuch thatφpfq “0.
Then the terminology is coherent: the distributions inD1pRnqwhich have compact support are exactly the distributions which extends toEpRnq. This is proved for example by Hormander in [41, 1.5.2].
7.3.3 Distributions and convolution product
Definition 7.3.11. Considern P NandΩan open set inRn. We consider the space of smooth functions with compact supportDpΩq:“Cc8pΩ,Rqendowed with the topology described in3.2.2, and its strong dualD1pΩq:“
Cc8pΩ,Rq1β.
Then we have in particularDpΩq ĂEpΩq, the first being dense in the second, and thusE1pΩq ĂD1pΩq.
Definition 7.3.12. Considerf andgtwo continuous functions onRn, one of which has compact support. Then we have a continuous functionf˚gwith compact support:
f˚g:xÞÑ ż
fpx´yqgpyqdy.
This operation called the convolution of functions, and is commutative and associative on functions when at leat two of the functions considered have compact suupport[41, Thm 1.6.2].
Definition 7.3.13. We define the convolution between a distributionφPD1pRnqand a functionf PCc8pRnqas the functionφ˚f :xÞÑφpyÞÑfpx´yqq. Thenφ˚f PC8pRnq. In particular, beware that the function resulting from the convolution does not necessarily have a compact support.
Remark7.3.14. For anyf PCc8pRnqwe haveδ0˚f “f.
Convolution is now extended to a convolution product between distributions by the following unicity results.
Proposition 7.3.15. [41, Thm 1.6.3]Cc8pRnqis sequentially dense inC8pRnq, and thusC8pRnqis weakly se-quentially dense inD1pRnq.
Proof. The first assertion follows by definition of the topology onC8pRnq, and by multiplying a functionf by partition of the unity. Thus by Hahn-Banach C8pRnqis sequentially dense inD1pRnqendowed with its weak topology.
Proposition 7.3.16. [41, Thm 1.6.4] Consider a continuous linear mappingU :Cc8pRnq //C8pRqwhich com-mutes with translations. Then there is a uniqueφPD1pRnqsuch thatUpfq “φ˚f for allf PCc8pRnq.
Proof. The functionφ:f ÞÑUpfqp0qis linear and continuous and thusφPD1pRnq. We have indeed forxPRn: φ˚upxq “φpy ÞÑfpx´yqq “UpyÞÑfpx´yqqp0q, and this equalsUpfqpxqasU is invariant by translation.
The unicity ofφfollows from the previous theorem.
Definition 7.3.17. Considerψ PE1pRnqandφPD1pRnq. Thenφ˚ψis the unique distribution inD1pRnqsuch that:
@f PDpRnq,pφ˚ψq ˚f “φ˚ pψ˚fq.
This definition is made possible by the fact thatf ÞÑ φ˚ pψ˚fqis invariant by translation. Thusφ˚ψis defined asφ˚ψ:f ÞÑφ˚ pψ˚fqp0q “φpyÞÑψ˚fp0´yqq “φpyÞÑψpzÞÑfpy´zqq.
Proposition 7.3.18. [41, Thm 1.6.5] The convolution product defines a commutative operation onE1pRnq, and the support ofφ˚ψis included in the sum of the support ofφandψrespectively. Ifφ, φ1PE1pRnqandψPD1pRnq, then we have associativity:
φ˚ pψ˚φ1q “ pφ˚ψq ˚φ1
Remark7.3.19. By definition and proposition7.3.16, the convolution product is a bilinear continuous operation
˚:D1pRnq ˆE1pRnq //E1pRnq. The previous theorem implies that it is a commutative and associative operation, and moreover that it preservesE1pRnq: ifφandψhave compact support, then so hasφ˚ψ.
As a corollary, we have the following domains of definition for the convolution product. This classification will be important for the model considered below in section7.4, and later in Chapter8.
Proposition 7.3.20. The convolution product˚defines a bilinear continuous function:
• fromE1pRnq ˆD1pRnqtoD1pRnq,
• fromE1pRnq ˆE1pRnqtoE1pRnq.
7.3.4 Other spaces of distributions and Fourier transforms
We refer to [76] for this section, and in particular to Chapter 25.
Definition 7.3.21. The space of tempered functions is defined as SpRnq:“ tf PC8pRnq,@P, QPRrX1, ..., Xns,sup
xPRn
|PpxqQpB{Bxqfpxq| ă 8u and is endowed with the topology generated by the semi-normsk_kP :f ÞÑsupx|PpxqQpB{Bxqfpxq|.
We define on tempered functions aFourier transformFand aninverse Fourier transformF¯: F:f ÞÑ pfˆ:ζPRnÞÑ
ż
expp´2iπxx, ζyqfpxqdxq, F¯ :f ÞÑ pf˘:xPRnÞÑ
ż
expp2iπxx, ζyqfpζqdζq.
Theorem 7.3.22. FandF¯defines linear homeomorphisms ofSpRnqinto itself.
Moreover, we have continuous linear injections: D ãÑ S ãÑE, and thus if we defineS1pRnqas the strong dual ofSpRnq, we have the continuous linear injections
E1ãÑS1ãÑD1.
The lcs S1pRnqis called the space of tempered distributionsonRn. The Fourier transform of a tempered distributionφis thenφˆ:f ÞÑφpfˆq. Likewise, Fourier transform is a linear homeomorphism ofS1pRnq. Moreover, this operation behaves remarkably with respect to convolution and partial differential operators:
Proposition 7.3.23. [76, Chapter 30]
• Considerφ, ψ PE1pRnq. Then the Fourier transform of the convolution is the scalar multiplication of the Fourier transformsφz˚ψ“φˆψˆ3.
• Considerf PEpRnqandjP t1, ...., nu. ThenFpBxBf
jqpζq “iζjFpfqpζq.
Generalized to polynomials, the second points thus gives for any polynomialP PRrx1, ..., Xns:FpPpBxBqpfqq “ PpiqFpfq.