The Convenient Setting for Denjoy--Carleman Differentiable Mappings of Beurling and Roumieu Type
aa r X i v : . [ m a t h . F A ] J a n THE CONVENIENT SETTING FORDENJOY–CARLEMAN DIFFERENTIABLE MAPPINGSOF BEURLING AND ROUMIEU TYPE
ANDREAS KRIEGL, PETER W. MICHOR, AND ARMIN RAINER
Abstract.
We prove in a uniform way that all Denjoy–Carleman differen-tiable function classes of Beurling type C ( M ) and of Roumieu type C { M } ,admit a convenient setting if the weight sequence M = ( M k ) is log-convex andof moderate growth: For C denoting either C ( M ) or C { M } , the category of C -mappings is cartesian closed in the sense that C ( E, C ( F, G )) ∼ = C ( E × F, G )for convenient vector spaces. Applications to manifolds of mappings are given:The group of C -diffeomorphisms is a regular C -Lie group if C ⊇ C ω , but notbetter. Introduction
Denjoy–Carleman differentiable functions form classes of smooth functions thatare described by growth conditions on the Taylor expansion. The growth is pre-scribed in terms of a sequence M = ( M k ) of positive real numbers which serves asa weight for the iterated derivatives: for compact K the sets n f ( k ) ( x ) ρ k k ! M k : x ∈ K, k ∈ N o are required to be bounded. The positive real number ρ is subject to either auniversal or an existential quantifier, thereby dividing the Denjoy–Carleman classesinto those of Beurling type, denoted by C ( M ) , and those of Roumieu type, denotedby C { M } , respectively. For the constant sequence M = ( M k ) = (1), as Beurlingtype we recover the real and imaginary parts of all entire functions on the onehand, and as Roumieu type the real analytic functions on the other hand, where1 /ρ plays the role of a radius of convergence. Moreover, Denjoy–Carleman classesare divided into quasianalytic and non-quasianalytic classes, depending on whetherthe mapping to infinite Taylor expansions is injective on the class or not.That a class of mappings C admits a convenient setting means essentially thatwe can extend the class to mappings between admissible infinite dimensional spaces E, F, . . . so that C ( E, F ) is again admissible and we have C ( E × F, G ) canonically C -diffeomorphic to C ( E, C ( F, G )). This property is called the exponential law ; itincludes the basic assumption of variational calculus. Usually the exponential law
Mathematics Subject Classification.
Key words and phrases.
Convenient setting, Denjoy–Carleman classes of Roumieu and Beurl-ing type, quasianalytic and non-quasianalytic mappings of moderate growth, Whitney jets onBanach spaces.AK was supported by FWF-Project P 23028-N13; PM by FWF-Project P 21030-N13; AR byFWF-Project P 22218-N13. comes hand in hand with (partially nonlinear) uniform boundedness theorems whichare easy C -detection principles.The class C ∞ of smooth mappings admits a convenient setting. This is dueoriginally to [9], [10], and [20], [21]. For the C ∞ convenient setting one can testsmoothness along smooth curves. Also real analytic ( C ω ) mappings admit a conve-nient setting, due to [22]: A mapping is C ω if and only if it is C ∞ and in additionis weakly C ω along weakly C ω -curves (i.e., curves whose compositions with anybounded linear functional are C ω ); indeed, it suffices to test along affine lines in-stead of weakly C ω -curves. See the book [23] for a comprehensive treatment, orthe three appendices in [25] for a short overview of the C ∞ and C ω cases. We shalluse convenient calculus of C ∞ -mappings in this paper, and we shall reprove that C ω admits a convenient calculus.We now describe what was known about convenient settings for Denjoy-Carlemanclasses before: In [25] we developed the convenient setting for non-quasianalytic log-convex Denjoy–Carleman classes of Roumieu type C { M } having moderate growth,and we showed that moderate growth and a condition that guarantees stabilityunder composition (like log-convexity) are necessary. There a mapping is C { M } ifand only if it is weakly C { M } along all weakly C { M } -curves. The method of proofrelies on the existence of C { M } partitions of unity.We succeeded in [26] to prove that some quasianalytic log-convex Denjoy–Carleman classes of Roumieu type C { M } having moderate growth admit a con-venient setting. The method consisted of representing C { M } as the intersection ofall larger non-quasianalytic log-convex classes C { L } . A mapping is C { M } if andonly if it is weakly C { L } along each weakly C { L } -curve for each non-quasianalyticlog-convex L ≥ M . We constructed countably many classes which satisfy all theserequirements, but many reasonable quasianalytic classes C { M } , like the real ana-lytic class, are not covered by this approach.In this paper we prove that all log-convex Denjoy–Carleman classes of moderategrowth admit a convenient setting. This is achieved through a change of philosophy:instead of testing along curves as in our previous approaches [25] and [26] we testalong Banach plots , i.e., mappings of the respective weak class defined in opensubsets of Banach spaces. By ‘weak’ we mean: the mapping is in the class aftercomposing it with any bounded linear functional. In this way we are able to treat allDenjoy–Carleman classes uniformly, no matter if quasianalytic, non-quasianalytic,of Beurling, or of Roumieu type, including C ω and real and imaginary parts ofentire functions. Furthermore, it makes the proofs shorter and more transparent.Smooth mappings between Banach spaces are C ( M ) or C { M } if their derivativessatisfy the boundedness conditions alluded to above. A smooth mapping betweenadmissible locally convex vector spaces is C ( M ) or C { M } if and only if it mapsBanach plots of the respective class to Banach plots of the same class. This impliesstability under composition, see Theorem 4.9.We equip the spaces of C ( M ) or C { M } mappings between Banach spaces withnatural locally convex topologies which are just the usual ones if the involved Ba-nach spaces are finite dimensional, see Section 4.1. In order to show completenesswe need to work with Whitney jets on compact subsets of Banach spaces satisfyinggrowth conditions of Denjoy–Carleman type, see Proposition 4.1. Having foundnothing in the literature we introduce Whitney jets on Banach spaces in Section 3. ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 3
In Theorem 7.1 we show that the Roumieu type classes of Denjoy–Carleman dif-ferentiable mappings studied in the present paper coincide bornologically with theclasses considered previously in [25] and [26] and, most notably, with the structure C ω of real analytic mappings introduced in [22] (see also [23]). We want to stressthat thereby we provide a considerably simpler proof for the real analytic conve-nient setting. But for the results that testing along curves suffices one still has torely on [22], [25], and [26].For a class of mappings C that admits a convenient setting one can hope thatthe space C ( A, B ) of all C -mappings between finite dimensional C -manifolds (with A compact for simplicity) is again a C -manifold, that composition is C , and thatthe group Diff C ( A ) of all C -diffeomorphisms of A is a regular infinite dimensional C -Lie group. In Section 9 this is proved for all log-convex Denjoy–Carleman classesof moderate growth C { M } and for the classes C ( M ) containing C ω .A further area of application is the perturbation theory for linear unboundedoperators; see [27] and [30].This paper is organized as follows. In Section 2 we recall basic facts aboutDenjoy–Carleman classes C [ M ] (which stands for C { M } or C ( M ) ) in finite dimen-sions and discuss corresponding sequence spaces. In Section 3 we introduce Whitneyjets on Banach spaces. In Section 4 we define C [ M ] -mappings in infinite dimensions,first between Banach spaces with the aid of jets and then between convenient vec-tor spaces, and we show that they form a category, if M = ( M k ) is log-convex. InSection 5 we prove that this category is cartesian closed, if M = ( M k ) has moderategrowth. In Section 6 we show the C [ M ] uniform boundedness principle. In Section7 we prove that the structures studied in this paper coincide bornologically withthe structures considered in our previous work [25], [26], [22], and [23]. In Section8 we further study the spaces of C [ M ] -mappings. In Section 9 we apply this theoryto prove that the space of C [ M ] -mappings between finite dimensional (compact)manifolds is naturally an infinite dimensional C [ M ] -manifold, and that the group of C [ M ] -diffeomorphisms of a compact manifold is a C [ M ] -regular Lie group. Notation.
We use N = N > ∪ { } . For each multi-index α = ( α , . . . , α n ) ∈ N n ,we write α ! = α ! · · · α n !, | α | = α + · · · + α n , and ∂ α = ∂ | α | /∂x α · · · ∂x α n n .A sequence r = ( r k ) of reals is called increasing if r k ≤ r k +1 for all k .We write f ( k ) ( x ) = d k f ( x ) for the k -th order Fr´echet derivative of f at x ; by d kv we mean k times iterated directional derivatives in direction v .For a convenient vector space E and a closed absolutely convex bounded subset B ⊆ E , we denote by E B the linear span of B equipped with the Minkowskifunctional k x k B = inf { λ > x ∈ λB } . Then E B is a Banach space. If U ⊆ E then U B := i − B ( U ), where i B : E B → E is the inclusion of E B in E .We denote by E ∗ (resp. E ′ ) the dual space of continuous (resp. bounded) linearfunctionals. L ( E , . . . , E k ; F ) is the space of k -linear bounded mappings E × · · · × E k → F ; if E i = E for all i , we also write L k ( E, F ). If E and F are Banach spaces,then k k L k ( E,F ) denotes the operator norm on L k ( E, F ). By L k sym ( E, F ) we denotethe subspace of symmetric k -linear bounded mappings. We write oE for the openunit ball in a Banach space E .The notation C [ M ] stands locally constantly for either C ( M ) or C { M } ; this means:Statements that involve more than one C [ M ] symbol must not be interpreted bymixing C ( M ) and C { M } . A. KRIEGL, P.W. MICHOR, A. RAINER
From Section 2.3 on, if not specified otherwise, a positive sequence M = ( M k )is assumed to satisfy M = 1 ≤ M . In Section 9 we also assume that M = ( M k )is log-convex and has moderate growth, and in the Beurling case C [ M ] = C ( M ) weadditionally require C ω ⊆ C ( M ) .2. Denjoy–Carleman differentiable functions in finite dimensions
Denjoy–Carleman differentiable functions of Beurling and Roumieutype in finite dimensions.
Let M = ( M k ) k ∈ N be a sequence of positive realnumbers. Let U ⊆ R n be open, K ⊆ U compact, and ρ >
0. Consider the set(1) n ∂ α f ( x ) ρ | α | | α | ! M | α | : x ∈ K, α ∈ N n o . We define the
Denjoy–Carleman classes C ( M ) ( U ) := { f ∈ C ∞ ( U ) : ∀ compact K ⊆ U ∀ ρ > } ,C { M } ( U ) := { f ∈ C ∞ ( U ) : ∀ compact K ⊆ U ∃ ρ > } . The elements of C ( M ) ( U ) are said to be of Beurling type ; those of C { M } ( U ) of Roumieu type . If M k = 1, for all k , then C ( M ) ( U ) consists of the restrictions to U of the real and imaginary parts of all entire functions, while C { M } ( U ) coincideswith the ring C ω ( U ) of real analytic functions on U . We shall also write C [ M ] andthereby mean that C [ M ] stands for either C ( M ) or C { M } .A sequence M = ( M k ) is log-convex if k log( M k ) is convex, i.e.,(2) M k ≤ M k − M k +1 for all k. If M = ( M k ) is log-convex, then k ( M k /M ) /k is increasing and(3) M l M k ≤ M M l + k for all l, k ∈ N . Let us assume M = 1 from now on. Furthermore, we have that k k ! M k is log-convex (since Euler’s Γ-function is so); if M = ( M k ) satisfies this weaker conditionwe say that it is weakly log-convex . If M = ( M k ) is weakly log-convex, then C [ M ] ( U )is a ring, for all open subsets U ⊆ R n .If M = ( M k ) is log-convex, then (see the proof of [25, 2.9]) we have(4) M j M k ≥ M j M α · · · M α j for all α i ∈ N > with α + · · · + α j = k. Condition (4) implies that the class of C [ M ] -mappings is stable under composition.This is due to [34] in the Roumieu case, see also [7] or [1, 4.7]; the same proof worksin the Beurling case. We reproof it in Theorem 4.9; compare also with Lemma 2.3.For a partial converse, see [31].If M = ( M k ) is log-convex, then the inverse function theorem for C { M } holds([17]; see also [1, 4.10]), and C { M } is closed under solving ODEs (due to [18]). Ifadditionally we have M k +1 /M k → ∞ , then also C ( M ) is closed under taking theinverse and solving ODEs (again by [17] and [18]). See [39], [40], [32], and Section9.2 for Banach space versions of these results.Suppose that M = ( M k ) and N = ( N k ) are such that sup k ( M k /N k ) /k < ∞ ,i.e.(5) ∃ C, ρ > ∀ k ∈ N : M k ≤ Cρ k N k . Then C ( M ) ( U ) ⊆ C ( N ) ( U ) and C { M } ( U ) ⊆ C { N } ( U ). The converse is true if M = ( M k ) is weakly log-convex: In the Roumieu case the inclusion C { M } ( U ) ⊆ ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 5 C { N } ( U ) implies (5) thanks to the existence of a function f ∈ C { M } ( R ) such that | f ( k ) (0) | ≥ k ! M k for all k (see [38, Thm. 1]; and also Section 2.2). In the Beurlingcase the equivalence of C ( M ) ( U ) ⊆ C ( N ) ( U ) and (5) follows from the closed graphtheorem; see Bruna [2]. As a consequence we see that the following three conditionsare equivalent: C ω ( U ) is contained in C { M } ( U ), the restrictions of entire functionsare contained in C ( M ) ( U ), and lim M /kk > C [ M ] is stable under derivations (alias derivation closed ) if(6) sup k ∈ N > (cid:16) M k +1 M k (cid:17) k < ∞ . The converse is true if M = ( M k ) is weakly log-convex: C { M } is stable underderivations if and only if (6) holds.A sequence M = ( M k ) is said to have moderate growth if(7) sup j,k ∈ N > (cid:16) M j + k M j M k (cid:17) j + k < ∞ . Moderate growth implies (6) and thus stability under derivations. If M = ( M k ) isweakly log-convex and has moderate growth, then C [ M ] ( U ) is stable under ultra-differential operators , see [15, 2.11 and 2.12].For sequences M = ( M k ) and N = ( N k ) of positive real numbers we define M ✁ N : ⇔ ∀ ρ > ∃ C > M k ≤ Cρ k N k ∀ k ∈ N ⇔ lim k →∞ (cid:16) M k N k (cid:17) k = 0 . If M ✁ N , then we have C { M } ( U ) ⊆ C ( N ) ( U ). If M = ( M k ) is weakly log-convex,also the converse is true: C { M } ( U ) ⊆ C ( N ) ( U ) implies M ✁ N . This follows from theexistence of a function f ∈ C { M } ( R ) with | f ( k ) (0) | ≥ k ! M k for all k (see [38, Thm.1]). As a consequence C ω ( U ) is contained in C ( M ) ( U ) if and only if M /kk → ∞ . Theorem 2.1 (Denjoy–Carleman [6], [3]) . For a sequence M = ( M k ) of positivereal numbers the following statements are equivalent: (1) C [ M ] is quasianalytic , i.e., for open connected U ⊆ R n and each x ∈ U , theTaylor series homomorphism centered at x from C [ M ] ( U, R ) into the spaceof formal power series is injective. (2) P ∞ k =1 1 m ♭ ( i ) k = ∞ where m ♭ ( i ) k := inf { ( j ! M j ) /j : j ≥ k } is the increasingminorant of ( k ! M k ) /k . (3) P ∞ k =1 ( M ♭ ( lc ) k ) /k = ∞ where M ♭ ( lc ) k is the log-convex minorant of k ! M k ,given by M ♭ ( lc ) k := inf { ( j ! M j ) l − kl − j ( l ! M l ) k − jl − j : j ≤ k ≤ l, j < l } . (4) P ∞ k =0 M ♭ ( lc ) k M ♭ ( lc ) k +1 = ∞ . For contemporary proofs of the equivalence of (2), (3), (4) and quasianalyticityof C { M } , see for instance [14, 1.3.8] or [35, 19.11]. For the equivalence of theseconditions to the quasianalyticity of C ( M ) , see [15, 4.2]. A. KRIEGL, P.W. MICHOR, A. RAINER
Sequence spaces.
Let M = ( M k ) k ∈ N be a sequence of positive real numbers,and ρ >
0. We consider (where F stands for ‘formal power series’) F Mρ := n ( f k ) k ∈ N ∈ R N : ∃ C > ∀ k ∈ N : | f k | ≤ Cρ k k ! M k o , F ( M ) := \ ρ> F Mρ , and F { M } := [ ρ> F Mρ . Lemma.
Consider the following conditions for two positive sequences M i = ( M ik ) , i = 1 , , and < σ < ∞ : (1) sup k ( M k /M k ) /k = σ . (2) For all ρ > we have F M ρ ⊆ F M ρσ . (3) F { M } ⊆ F { M } . (4) F ( M ) ⊆ F ( M ) . (5) M ✁ M . (6) F { M } ⊆ F ( M ) .Then we have (1) ⇔ (2) ⇔ (3) ⇒ (4) and (5) ⇔ (6) . Proof. (1) ⇒ (2) Let f = ( f k ) ∈ F M ρ , i.e., there is a C > | f k | ≤ Cρ k k ! M k ≤ C ( ρσ ) k k ! M k , for all k . So f ∈ F M ρσ .(2) ⇒ (3) and (2) ⇒ (4) follow by definition.(3) ⇒ (1) Let f k := k ! M k . Then f = ( f k ) ∈ F { M } ⊆ F { M } , so there exists ρ > k ! M k ≤ ρ k +1 k ! M k for all k .(5) ⇒ (6) Let f = ( f k ) ∈ F M ρ . As M ✁ M , for each σ > C > | f k | ≤ C ( σρ ) k k ! M k for all k . So f ∈ F M σρ for all σ .(6) ⇒ (5) Since ( k ! M k ) ∈ F { M } ⊆ F ( M ) , for each ρ > C > k ! M k ≤ Cρ k k ! M k for all k , i.e., M ✁ M . (cid:3) Theorem 2.2.
Let M = ( M k ) be a (weakly) log-convex sequence of positive realnumbers. Then we have (8) F { M } = \ L F ( L ) = \ L F { L } , where the intersections are taken over all (weakly) log-convex L = ( L k ) with M ✁ L . Proof.
The inclusions F { M } ⊆ T L F ( L ) ⊆ T L F { L } follow from Lemma 2.2. So itremains to prove that F { M } ⊇ T L F { L } . Let f = ( f k ) / ∈ F { M } , i.e.,(9) lim (cid:16) | f k | k ! M k (cid:17) k = ∞ . We must show that there exists a (weakly) log-convex L = ( L k ) with M ✁ L suchthat f / ∈ F { L } .Choose a j , b j > a j ր ∞ , b j ց
0, and a j b j → ∞ . Now (9) implies thatthere exists a strictly increasing sequence k j ∈ N such that (cid:16) | f k j | ( k j )! M k j (cid:17) kj ≥ a j . ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 7
Passing to a subsequence we may assume that k > < β j := b j (cid:16) | f k j | ( k j )! M k j (cid:17) kj ր ∞ . Passing to a subsequence again we may also get(10) β j +1 ≥ ( β j ) k j . We define a piecewise affine function φ by setting φ ( k ) := k = 0 ,k j log β j if k = k j ,c j + d j k for the minimal j with k ≤ k j , where c j and d j are chosen such that φ is well defined and φ ( k j − ) = c j + d j k j − ,i.e., for j ≥ c j + d j k j = k j log β j ,c j + d j k j − = k j − log β j − , and c = 0 ,d = log β . This implies first that c j ≤ β j ≤ d j = k j log β j − k j − log β j − k j − k j − ≤ k j k j − k j − log β j (10) ≤ log β j +1 k j − k j − ≤ log β j +1 . Thus j d j is increasing and so φ is convex. The fact that all c j ≤ φ ( k ) /k is increasing.Now let L k := e φ ( k ) · M k . Then L = ( L k ) is (weakly) log-convex, since so is M = ( M k ). As φ ( k ) /k isincreasing and e φ ( k j ) /k j = β j → ∞ , we find M ✁ L . Finally, f / ∈ F { L } , since wehave (cid:16) | f k j | ( k j )! L k j (cid:17) kj = (cid:16) | f k j | ( k j )! M k j (cid:17) kj · e − φ ( k j ) /k j = (cid:16) | f k j | ( k j )! M k j (cid:17) kj · β − j = b − j → ∞ . The proof is complete. (cid:3)
Remark. (1) If M = 1 ≤ M we also have L = 1 ≤ L .(2) The proof also shows that, if M = ( M k ) is just any positive sequence, then(8) still holds if the intersections are taken over all positive sequences L = ( L k )with M ✁ L . Lemma 2.3.
Let M = ( M k ) and L = ( L k ) be sequences of positive real numbers.Then for the composition of formal power series we have (11) F [ M ] ◦ F [ L ] > ⊆ F [ M ◦ L ] , where ( M ◦ L ) k := max { M j L α . . . L α j : α i ∈ N > , α + · · · + α j = k } . A. KRIEGL, P.W. MICHOR, A. RAINER
Here F [ L ] > is the space of formal power series in F [ L ] with vanishing constantterm. Proof.
Let f ∈ F ( M ) and g ∈ F ( L ) (resp. f ∈ F { M } and g ∈ F { L } ). For k > f ◦ g ) k k ! : = k X j =1 f j j ! X α ∈ N j> α + ··· + α j = k g α α ! . . . g α j α j ! , | ( f ◦ g ) k | k !( M ◦ L ) k ≤ k X j =1 | f j | j ! M j X α ∈ N j> α + ··· + α j = k | g α | α ! L α . . . | g α j | α j ! L α j ≤ k X j =1 ρ jf C f X α ∈ N j> α + ··· + α j = k ρ kg C jg ≤ k X j =1 ρ jf C f (cid:18) k − j − (cid:19) ρ kg C jg = ρ kg ρ f C f C g k X j =1 ( ρ f C g ) j − (cid:18) k − j − (cid:19) = ρ kg ρ f C f C g (1 + ρ f C g ) k − = ( ρ g (1 + ρ f C g )) k ρ f C f C g ρ f C g . This implies (11) in the Roumieu case. For the Beurling case, let τ > σ > τ = √ σ + σ . If we set ρ g = √ σ and ρ f = √ σ/C g , then f ◦ g ∈ F M ◦ Lτ . (cid:3) Convention.
For a positive sequence M = ( M k ) ∈ ( R > ) N consider the fol-lowing properties:(0) M = 1 ≤ M .(1) M = ( M k ) is weakly log-convex, i.e., k log( k ! M k ) is convex.(2) M = ( M k ) is log-convex, i.e., k log( M k ) is convex.(3) M = ( M k ) is derivation closed, i.e., k ( M k +1 M k ) k is bounded.(4) M = ( M k ) has moderate growth, i.e., ( j, k ) ( M j + k M j M k ) j + k is bounded.(5) M k +1 M k → ∞ .(6) M /kk → ∞ , or equivalently, C ω ⊆ C ( M ) .Henceforth, if not specified otherwise, we assume that M = ( M k ), N = ( N k ), L = ( L k ), etc., satisfy condition (0). It will be explicitly stated when some of theother properties (1)–(6) are assumed. Remarks.
Let M = ( M k ) be a positive sequence. We may replace ( M k ) k by( Cρ k M k ) k with C, ρ > F [ M ] (see Section 2.2). In particular, itis no loss of generality to assume that M > Cρ > /M ) and M = 1 (put C := 1 /M ). Each one of the properties (1)–(6) is preserved by this modification.Furthermore M = ( M k ) is quasianalytic if and only if the modified sequence is so,since ( M ♭ ( lc ) k ) k (see Theorem 2.1) is modified in the same way.Conditions (0) and (1) together imply that k k ! M k is monotone increasing,while (0) and (2) together imply that k M k is monotone increasing. ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 9 Whitney jets on Banach spaces
Whitney jets.
Let E and F be Banach spaces. For open U ⊆ E consider thespace C ∞ ( U, F ) of arbitrarily often Fr´echet differentiable mappings f : U → F . Forsuch f we have the derivatives f ( k ) : U → L k sym ( E, F ), where L k sym ( E, F ) denotesthe space of symmetric k -linear bounded mappings E × · · · × E → F . We also havethe iterated uni-directional derivatives d kv f ( x ) ∈ F defined by d kv f ( x ) := (cid:18) ddt (cid:19) k f ( x + t v ) | t =0 . Let j ∞ : C ∞ ( U, F ) → J ∞ ( U, F ) := Q k ∈ N C ( U, L k sym ( E, F )) be the jet mapping f ( f ( k ) ) k ∈ N . On L k sym ( E, F ) we consider the operator norm k ℓ k L k sym ( E,F ) := sup n k ℓ ( v , . . . , v k ) k F : k v j k E ≤ j ∈ { , . . . , k } o . Note that by the polarization equality (see [23, 7.13.1])sup {k ℓ ( v, . . . , v ) k F : k v k E ≤ } ≤ k ℓ k L k sym ( E,F ) ≤ (2 e ) k sup {k ℓ ( v, . . . , v ) k F : k v k E ≤ } For an infinite jet f = ( f k ) k ∈ N ∈ Q k ∈ N L k sym ( E, F ) X on a subset X ⊆ E let theTaylor polynomial ( T ny f ) k : X → L k sym ( E, F ) of order n at y be( T ny f ) k ( x )( v , . . . , v k ) := n X j =0 j ! f j + k ( y )( x − y, . . . , x − y, v , . . . v k )and the remainder( R ny f ) k ( x ) := f k ( x ) − ( T ny f ) k ( x ) = ( T nx f ) k ( x ) − ( T ny f ) k ( x ) ∈ L k sym ( E, F ) . Let k f k k := sup {k f k ( x ) k L k sym ( E,F ) : x ∈ X } ∈ [0 , + ∞ ] and ||| f ||| n,k := sup n ( n + 1)! k ( R ny f ) k ( x ) k L k sym ( E,F ) k x − y k n +1 : x, y ∈ X, x = y o ∈ [0 , + ∞ ] . By Taylor’s theorem, for f ∈ C ∞ ( U, F ) and [ x, y ] ⊆ U we have( R ny f ) k ( x ) = f ( k ) ( x ) − X j ≤ n f ( k + j ) ( y )( x − y ) j j != Z (1 − t ) n n ! f ( k + n +1) ( y + t ( x − y ))( x − y ) n +1 dt and hence for convex X ⊆ U : ||| j ∞ f | X ||| n,k :== sup n ( n + 1)! k ( R ny f ) k ( x )( v , . . . , v k ) k F k x − y k n +1 : k v j k E ≤ , x, y ∈ X, x = y o ≤ sup n k f ( k + n +1) ( x )( v , . . . , v k , x − y, . . . , x − y ) k F k x − y k n +1 : k v j k E ≤ , x = y o ≤ k j ∞ f | X k n + k +1 . (12) We supply C ∞ ( U, F ) with the semi-norms f
7→ k j ∞ f | K k n for all compact K ⊆ U and all n ∈ N . For compact convex K ⊆ E the space C ∞ ( E ⊇ K, F ) of Whitney jets on K isdefined by C ∞ ( E ⊇ K, F ) :== n f = ( f k ) k ∈ N ∈ Y k ∈ N C ( K, L k sym ( E, F )) : ||| f ||| n,k < ∞ for all n, k ∈ N o and is supplied with the seminorms k k n for n ∈ N together with ||| ||| n,k for n, k ∈ N . Lemma 3.1.
For Banach spaces E and F and compact convex K ⊆ E the space C ∞ ( E ⊇ K, F ) is a Fr´echet space. Proof.
The injection of C ∞ ( E ⊇ K, F ) into Q k ∈ N C ( K, L k sym ( E, F )) is continuousby definition and C ( K, L k sym ( E, F )) is a Banach space, so a Cauchy sequence ( f p ) p in C ∞ ( E ⊇ K, F ) has an infinite jet f ∞ = ( f k ∞ ) k as component-wise limit in Q k ∈ N C ( K, L k sym ( E, F )) with respect to the seminorms k k n . This is the limit alsowith respect to the finer structure of C ∞ ( E ⊇ K, F ) with the additional seminorms ||| ||| n,k as follows: For given n, k ∈ N and ǫ > p such that ||| f p − f q ||| n,k < ǫ/ p, q ≥ p . By the convergence f q → f ∞ in Q k ∈ N C ( K, L k sym ( E, F )) there exists for given x, y ∈ K with x = y a q ≥ p suchthat for all m ≤ k + n k f q − f ∞ k m ≤ k x − y k n +1 ( n + 1)! ǫ { , e −k x − y k } and hence k ( T ny f q ) k ( x ) − ( T ny f ∞ ) k ( x ) k L k sym ( E,F ) ≤≤ n X j =0 k f k + jq ( y ) − f k + j ∞ ( y ) k L k + j sym ( E,F ) k x − y k j j ! ≤ n X j =0 k f q − f ∞ k k + j k x − y k j j ! ≤ k x − y k n +1 ( n + 1)! ǫ e −k x − y k n X j =0 k x − y k j j ! ≤ k x − y k n +1 ( n + 1)! ǫ . So ( n + 1)! k ( R ny f p ) k ( x ) − ( R ny f ∞ ) k ( x ) k L k sym ( E,F ) k x − y k n +1 ≤≤ ||| f p − f q ||| n,k + ( n + 1)! k ( R ny f q ) k ( x ) − ( R ny f ∞ ) k ( x ) k L k sym ( E,F ) k x − y k n +1 ≤ ǫ n + 1)! k f kq ( x ) − f k ∞ ( x ) k L k sym ( E,F ) k x − y k n +1 + ( n + 1)! k ( T ny f q ) k ( x ) − ( T ny f ∞ ) k ( x ) k L k sym ( E,F ) k x − y k n +1ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 11 ≤ ǫ ǫ and finally ||| f p − f ∞ ||| n,k ≤ ǫ for all p ≥ p . Consequently, ||| f ∞ ||| n,k ≤ ||| f ∞ − f p ||| n,k + ||| f p ||| n,k < ∞ , i.e., f ∞ ∈ C ∞ ( E ⊇ K, F ). (cid:3) The category of Denjoy–Carleman differentiable mappings
Spaces of Denjoy–Carleman jets and mappings between Banachspaces.
Let E and F be Banach spaces, K ⊆ E compact, and ρ >
0. Let C Mρ ( E ⊇ K, F ) : = n ( f m ) m ∈ Y m ∈ N C ( K, L m sym ( E, F )) : k f k ρ < ∞ o , where k f k ρ : = max (cid:26) sup n k f k m m ! ρ m M m : m ∈ N o , sup n ||| f ||| n,k ( n + k + 1)! ρ n + k +1 M n + k +1 : k, n ∈ N o(cid:27) , cf. [4, 11], [5, 11] and [37, 3], and, for an open neighborhood U of K in E , let C MK,ρ ( U, F ) : = n f ∈ C ∞ ( U, F ) : j ∞ f | K ∈ C Mρ ( E ⊇ K, F ) o supplied with the semi-norm f
7→ k j ∞ f | K k ρ . This space is not Hausdorff and forinfinite dimensional E it(s Hausdorff quotient) will not always be complete. Thisis the reason for considering the jet spaces C Mρ ( E ⊇ K, F ) instead. Note that forconvex K we have ||| j ∞ f | K ||| n,k ≤ k j ∞ f | K k n + k +1 by (12) and hence the seminorm f
7→ k j ∞ f | K k ρ on C MK,ρ ( U, F ) coincides with f sup n k f ( n ) ( x ) k L n sym ( E,F ) n ! ρ n M n : x ∈ K, n ∈ N o =: k f k K,ρ . Thus C MK,ρ ( U, F ) = n f ∈ C ∞ ( U, F ) : ( k j ∞ f | K k m ) m ∈ F Mρ o and the bounded subsets B ⊆ C MK,ρ ( U, F ) are exactly those
B ⊆ C ∞ ( U, F ) for which( b m ) m ∈ F Mρ , where b m := sup {k j ∞ f | K k m : f ∈ B} .For open convex U ⊆ E and compact convex K ⊆ U let C ( M ) ( E ⊇ K, F ) : = \ ρ> C Mρ ( E ⊇ K, F ) ,C { M } ( E ⊇ K, F ) : = [ ρ> C Mρ ( E ⊇ K, F ) , and C [ M ] ( U, F ) : = n f ∈ C ∞ ( U, F ) : ∀ K : ( f ( k ) | K ) ∈ C [ M ] ( E ⊇ K, F ) o . That means, we consider the projective limit C ( M ) ( E ⊇ K, F ) := lim ←− ρ> C Mρ ( E ⊇ K, F ) , the inductive limit C { M } ( E ⊇ K, F ) := lim −→ ρ> C Mρ ( E ⊇ K, F ) , and the projective limits C [ M ] ( U, F ) := lim ←− K ⊆ U C [ M ] ( E ⊇ K, F ) , where K runs through all compact convex subsets of U .Furthermore, we consider the projective limit C ( M ) K ( U, F ) := lim ←− ρ> C MK,ρ ( U, F ) , and the inductive limit C { M } K ( U, F ) := lim −→ ρ> C MK,ρ ( U, F ) . Thus C [ M ] K ( U, F ) = n f ∈ C ∞ ( U, F ) : ( k j ∞ f | K k m ) m ∈ F [ M ] o . Furthermore, the bounded subsets
B ⊆ C [ M ] K ( U, F ) are exactly those
B ⊆ C ∞ ( U, F )for which ( b m ) m ∈ F [ M ] , where b m := sup {k j ∞ f | K k m : f ∈ B} .Finally, the projective limitslim ←− K ⊆ U C [ M ] K ( U, F ) = n f ∈ C ∞ ( U, F ) : ∀ K : ( k j ∞ f | K k m ) m ∈ F [ M ] o , where K runs through all compact convex subsets of U , are for E = R n and F = R exactly the vector spaces of Section 2.1 and the topology is the usual one.For the inductive limits with respect to ρ > ρ ∈ N only. Proposition 4.1.
We have the following completeness properties: (1)
The spaces C Mρ ( E ⊇ K, F ) are Banach spaces. (2) The spaces C ( M ) ( E ⊇ K, F ) are Fr´echet spaces. (3) The spaces C { M } ( E ⊇ K, F ) are compactly regular (i.e., compact subsetsare contained and compact in some step) (LB)-spaces hence ( c ∞ -)complete,webbed and (ultra-)bornological. (4) The spaces C [ M ] ( U, F ) are complete spaces. (5) As locally convex spaces C [ M ] ( U, F ) := lim ←− K ⊆ U C [ M ] ( E ⊇ K, F ) = lim ←− K ⊆ U C [ M ] K ( U, F ) . ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 13
Proof. (1) The injection C Mρ ( E ⊇ K, F ) → Q k ∈ N C ( K, L k sym ( E, F )) is by defi-nition continuous and C ( K, L k sym ( E, F )) is a Banach space, so a Cauchy sequence( f p ) p in C Mρ ( E ⊇ K, F ) has an infinite jet f ∞ = ( f k ∞ ) k as component-wise limit in Q k ∈ N C ( K, L k sym ( E, F )). This is the limit also with respect to the finer structure of C Mρ ( E ⊇ K, F ) as follows: For fixed n, k and x = y we have that ( R ny f p ) k ( x ) con-verges to ( R ny f ∞ ) k ( x ). So we choose for ǫ > p ∈ N such that k f p − f q k ρ < ǫ/ p, q ≥ p and given x, y, n , and k we can choose q > p such that( n + 1)! k ( R ny f q ) k ( x ) − ( R ny f ∞ ) k ( x ) k L k sym ( E,F ) ( n + k + 1)! ρ n + k +1 M n + k +1 k x − y k n +1 < ǫ k f nq ( x ) − f n ∞ ( x ) k L n sym ( E,F ) n ! ρ n M n < ǫ . Thus( n + 1)! k ( R ny f p ) k ( x ) − ( R ny f ∞ ) k ( x ) k L k sym ( E,F ) ( n + k + 1)! ρ n + k +1 M n + k +1 k x − y k n +1 < k f p − f q k ρ + ( n + 1)! k ( R ny f q ) k ( x ) − ( R ny f ∞ ) k ( x ) k L k sym ( E,F ) ( n + k + 1)! ρ n + k +1 M n + k +1 k x − y k n +1 < ǫ and hence ||| f p − f ∞ ||| n,k ( n + k + 1)! ρ n + k +1 M n + k +1 ≤ ǫ and similarly for k f p − f ∞ k n n ! ρ n M n . Thus k f p − f ∞ k ρ ≤ ǫ for all p ≥ p .(2) This is obvious; they are countable projective limits of Banach spaces.(3) For finite dimensional E and F it is shown in [37] that the connectingmappings are nuclear. For infinite dimensional E the connecting mappings in C { M } ( E ⊇ K, F ) = lim −→ ρ> C Mρ ( E ⊇ K, F ) cannot be compact, since the set { ℓ ∈ E ′ : k ℓ k ≤ } is bounded in C Mρ ( E ⊇ K, R ) for each ρ ≥
1. In fact, k ℓ k = sup {| ℓ ( x ) | : x ∈ K } ≤ sup {k x k : x ∈ K } , k ℓ k = k ℓ k ≤ k ℓ k m = 0 for m ≥
2, moreover, ( R ny ℓ ) k = 0 for n + k ≥ R y ℓ ) ( x ) = ℓ ( x − y ). It is notrelatively compact in any of the spaces C Mρ ( E ⊇ K, R ), ρ ≥
1, since it is not evenpointwise relatively compact in C ( K, L ( E, R )).In order to show that the (LB)-space in (3) is compactly regular it suffices by[29, Satz 1] to verify condition (M) of [33]: There exists a sequence of increasing0-neighborhoods U n ⊆ C Mn ( E ⊇ K, F ), such that for each n there exists an m ≥ n for which the topologies of C Mk ( E ⊇ K, F ) and of C Mm ( E ⊇ K, F ) coincide on U n for all k ≥ m .For ρ ′ ≥ ρ we have k f k ρ ′ ≤ k f k ρ . So consider the ǫ -balls U ρǫ ( f ) := { g : k g − f k ρ ≤ ǫ } in C Mρ ( E ⊇ K, F ). It suffices to show that for ρ > ρ := 2 ρ , ρ > ρ , ǫ > f ∈ U ρ := U ρ (0) there exists a δ > U ρ δ ( f ) ∩ U ρ ⊆ U ρ ǫ ( f ). Since f ∈ U ρ we have k f k n ≤ n ! ρ n M n and ||| f ||| n,k ≤ ( n + k + 1)! ρ n + k +1 M n + k +1 for all n, k. Let N < ǫ and δ := ǫ (cid:16) ρ ρ (cid:17) N − . Let g ∈ U ρ δ ( f ) ∩ U ρ , i.e., k g k n ≤ n ! ρ n M n for all n, k g − f k n ≤ δ n ! ρ n M n for all n, ||| g ||| n,k ≤ ( n + k + 1)! ρ n + k +1 M n + k +1 for all n, k, ||| g − f ||| n,k ≤ δ ( n + k + 1)! ρ n + k +12 M n + k +1 for all n, k. Then k g − f k n ≤ k g k n + k f k n ≤ n ! ρ n M n = 2 n ! ρ n M n n < ǫ n ! ρ n M n for n ≥ N and k g − f k n ≤ δ n ! ρ n M n ≤ ǫ n ! ρ n M n for n < N. Moreover, ||| g − f ||| n,k ≤ ||| g ||| n,k + ||| f ||| n,k ≤ n + k + 1)! ρ n + k +1 M n + k +1 = 2 ( n + k + 1)! ρ n + k +11 M n + k +1 n + k +1 < ǫ ( n + k + 1)! ρ n + k +11 M n + k +1 for n + k + 1 ≥ N and ||| g − f ||| n,k ≤ δ ( n + k + 1)! ρ n + k +12 M n + k +1 ≤ ǫ ( n + k + 1)! ρ n + k +11 M n + k +1 for n + k + 1 < N. (4) This is obvious; they are projective limits of complete spaces.(5) Since j ∞ | K : C MK,ρ ( U, F ) → C Mρ ( E ⊇ K, F ) is by definition a well-definedcontinuous linear mapping, it induces such mappings C [ M ] K ( U, F ) → C [ M ] ( E ⊇ K, F ) and lim ←− K C [ M ] K ( U, F ) → lim ←− K C [ M ] ( E ⊇ K, F ). The last mapping is obvi-ously injective (use K := { x } for the points x ∈ U ).Conversely, let f kK ∈ C ( K, L k sym ( E, F )) be given, such that for each K thereexists ρ > ρ >
0) we have ( f kK ) k ∈ N ∈ C Mρ ( E ⊇ K, F ) and suchthat f kK | K ′ = f kK ′ . They define an infinite jet ( f k ) k ∈ N ∈ J ∞ ( U, F ) by setting f k ( x ) := f k { x } ( x ) which satisfies f k | K = f kK for all k ∈ N and all K .We claim that f ∈ C ∞ ( U, F ) and ( f ) ( k ) = f k for all k , i.e., j ∞ f | K = ( f kK ) k for all k ∈ N and all K .By [23, 5.20] it is enough to show by induction that d kv f ( x ) = f k ( x )( v, . . . , v ). For k = 0 this is obvious, so let k >
0. Then d kv f ( x ) : = lim t → d k − v f ( x + t v ) − d k − v f ( x ) t = lim t → f k − ( x + t v )( v k − ) − f k − ( x )( v k − ) t = lim t → ( R x f ) k − ( x + t v )( v k − ) t + f k ( x )( v k ) = f k ( x )( v k ) . Finally, f defines an element in lim ←− K C [ M ] K ( U, F ), since ∀ K we have f ∈ C MK,ρ ( U, F ) = { g ∈ C ∞ ( U, F ) : j ∞ g | K ∈ C Mρ ( E ⊇ K, F ) } for some (resp. all) ρ > ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 15
That this bijection is an isomorphism follows, since the seminorm k k
K,ρ on C MK,ρ ( U, F ) is the pull-back of the seminorm k k ρ on C Mρ ( E ⊇ K, F ). (cid:3) Spaces of Denjoy–Carleman differentiable mappings between conve-nient vector spaces.
For convenient vector spaces E and F , and c ∞ -open U ⊆ E ,we define: C ( M ) b ( U, F ) := n f ∈ C ∞ ( U, F ) : ∀ B ∀ compact K ⊆ U B ∀ ρ > (cid:8) f ( k ) ( x )( v , . . . , v k ) k ! ρ k M k : k ∈ N , x ∈ K, k v i k B ≤ (cid:9) is bounded in F o = n f ∈ C ∞ ( U, F ) : ∀ B ∀ compact K ⊆ U B ∀ ρ > (cid:8) d kv f ( x ) k ! ρ k M k : k ∈ N , x ∈ K, k v k B ≤ (cid:9) is bounded in F o , and C { M } b ( U, F ) := n f ∈ C ∞ ( U, F ) : ∀ B ∀ compact K ⊆ U B ∃ ρ > (cid:8) f ( k ) ( x )( v , . . . , v k ) k ! ρ k M k : k ∈ N , x ∈ K, k v i k B ≤ (cid:9) is bounded in F o = n f ∈ C ∞ ( U, F ) : ∀ B ∀ compact K ⊆ U B ∃ ρ > (cid:8) d kv f ( x ) k ! ρ k M k : k ∈ N , x ∈ K, k v k B ≤ (cid:9) is bounded in F o . Here B runs through all closed absolutely convex bounded subsets in E , E B is thevector space generated by B with the Minkowski functional k v k B = inf { λ ≥ v ∈ λB } as complete norm, and U B = U ∩ E B . For Banach spaces E and F obviously C [ M ] b ( U, F ) = C [ M ] ( U, F ) . Now we define the spaces of main interest in this paper: C [ M ] ( U, F ) := n f ∈ C ∞ ( U, F ) : ∀ ℓ ∈ F ∗ ∀ B : ℓ ◦ f ◦ i B ∈ C [ M ] ( U B , R ) o , where B again runs through all closed absolutely convex bounded subsets in E ,the mapping i B : E B → E denotes the inclusion of E B in E , and U B = i − B ( U ) = U ∩ E B . It will follow from Lemmas 4.2 and 4.3 that for Banach spaces E and F this definition coincides with the one given earlier in Section 4.1.We equip C [ M ] ( U, F ) with the initial locally convex structure induced by alllinear mappings C [ M ] ( U, F ) − C [ M ] ( i B ,ℓ ) → C [ M ] ( U B , R ) , f ℓ ◦ f ◦ i B . Then C [ M ] ( U, F ) is a convenient vector space as c ∞ -closed subspace in the product Q ℓ,B C [ M ] ( U B , R ), since smoothness can be tested by composing with the inclusions E B → E and with the ℓ ∈ F ∗ , see [23, 2.14.4 and 1.8]. This shows at the sametime, that C [ M ] ( U, F ) = n f ∈ F U : ∀ ℓ ∈ F ∗ ∀ B : ℓ ◦ f ◦ i B ∈ C [ M ] ( U B , R ) o . Lemma 4.2 ( C ( M ) = C ( M ) b ) . Let
E, F be convenient vector spaces, and let U ⊆ E be c ∞ -open. Then a mapping f : U → F is C ( M ) (i.e., is in C ( M ) ( U, F ) ) if andonly if f is C ( M ) b . Proof.
Let f : U → F be C ∞ . We have the following equivalences, where B runsthrough all closed absolutely convex bounded subsets in E : f ∈ C ( M ) ( U, F ) ⇐⇒ ∀ ℓ ∈ F ∗ ∀ B ∀ K ⊆ U B compact ∀ ρ > n ( ℓ ◦ f ) ( k ) ( x )( v , . . . , v k ) ρ k k ! M k : x ∈ K, k ∈ N , k v i k B ≤ o is bounded in R ⇐⇒ ∀ B ∀ K ⊆ U B compact ∀ ρ > ∀ ℓ ∈ F ∗ : ℓ (cid:16)n f ( k ) ( x )( v , . . . , v k ) ρ k k ! M k : x ∈ K, k ∈ N , k v i k B ≤ o(cid:17) is bounded in R ⇐⇒ ∀ B ∀ K ⊆ U B compact ∀ ρ > n f ( k ) ( x )( v , . . . , v k ) ρ k k ! M k : x ∈ K, k ∈ N , k v i k B ≤ o is bounded in F ⇐⇒ f ∈ C ( M ) b ( U, F ) (cid:3) In the Roumieu case C { M } the corresponding equality holds only under addi-tional assumptions: Lemma 4.3 ( C { M } = C { M } b ) . Let
E, F be convenient vector spaces, and let U ⊆ E be c ∞ -open. Assume that there exists a Baire vector space topology on the dual F ∗ for which the point evaluations ev x are continuous for all x ∈ F . Then a mapping f : U → F is C { M } if and only if f is C { M } b . Proof. ( ⇒ ) Let B be a closed absolutely convex bounded subset of E . Let K becompact in U B . We consider the sets A ρ,C := n ℓ ∈ F ∗ : | ( ℓ ◦ f ) ( k ) ( x )( v , . . . , v k ) | ρ k k ! M k ≤ C for all x ∈ K, k ∈ N , k v i k B ≤ o which are closed subsets in F ∗ for the given Baire topology. We have S ρ,C A ρ,C = F ∗ . By the Baire property there exist ρ and C such that the interior int( A ρ,C ) of A ρ,C is non-empty. If ℓ ∈ int( A ρ,C ), then for each ℓ ∈ F ∗ there is a δ > δℓ ∈ int( A ρ,C ) − ℓ , and, hence, for all k ∈ N , x ∈ K , and k v i k B ≤
1, we have | ( ℓ ◦ f ) ( k ) ( x )( v , . . . ) | ≤ δ (cid:16) | (( δ ℓ + ℓ ) ◦ f ) ( k ) ( x )( v , . . . ) | + | ( ℓ ◦ f ) ( k ) ( x )( v , . . . ) | (cid:17) ≤ Cδ ρ k k ! M k . So the set n f ( k ) ( x )( v , . . . , v k ) ρ k k ! M k : x ∈ K, k ∈ N , k v i k B ≤ o is weakly bounded in F and hence bounded. Since B and K were arbitrary, weobtain f ∈ C { M } b ( U, F ).( ⇐ ) is obvious. (cid:3) The following example shows that the additional assumption in Lemma 4.3 can-not be dropped.
ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 17
Example 4.4.
By [38, Thm. 1], for each weakly log-convex sequence M = ( M k )there exists f ∈ C { M } ( R , R ) such that | f ( k ) (0) | ≥ k ! M k for all k ∈ N . Then g : R → R given by g ( s, t ) = f ( st ) is C { M } , whereas there is no reasonabletopology on C { M } ( R , R ) such that the associated mapping g ∨ : R → C { M } ( R , R ) is C { M } b . For a topology on C { M } ( R , R ) to be reasonable we require only that allevaluations ev t : C { M } ( R , R ) → R are bounded linear functionals. Proof.
The mapping g is obviously C { M } . If g ∨ were C { M } b , for s = 0 there existed ρ such that n ( g ∨ ) ( k ) (0) k ! ρ k M k : k ∈ N o was bounded in C { M } ( R , R ). We apply the bounded linear functional ev t for t = 2 ρ and then get | ( g ∨ ) ( k ) (0)(2 ρ ) | k ! ρ k M k = (2 ρ ) k | f ( k ) (0) | k ! ρ k M k ≥ k , a contradiction. (cid:3) This example shows that for C { M } b one cannot expect cartesian closedness. Usingcartesian closedness, i.e., Theorem 5.2, and Lemma 5.1 this also shows (for F = C { M } ( R , R ) and U = R = E ) that C { M } ( U, F ) ) \ B,V C { M } b ( U B , F V ) , where F V is the completion of F/p − V (0) with respect to the seminorm p V inducedby the absolutely convex closed 0-neighborhood V .If we compose g ∨ with the restriction mapping (incl N ) ∗ : C { M } ( R , R ) → R N := Q t ∈ N R , then we get a C { M } -curve, since the continuous linear functionals on R N are linear combinations of coordinate projections ev t with t ∈ N . However, thiscurve cannot be C { M } b as the argument above for t > ρ shows.In the following Lemmas 4.5 and 4.6 we find projective descriptions for C ( M ) ( U, F ) and C { M } ( U, F ), if E , F are Banach spaces, and U ⊆ E is open. Thisis of vital importance for the development of the convenient setting of C { M } ( U, F ).The spaces C ( M ) ( U, F ), however, already are projective by definition, and thusLemma 4.5 just gives a further projective description; see also Theorem 8.5. Weinclude and use Lemma 4.5 in order to treat the Beurling and Roumieu case in auniform and efficient way.
Lemma 4.5.
Let
E, F be Banach spaces, U ⊆ E open, and f : U → F a C ∞ -mapping. The following are equivalent: (1) f is C ( M ) = C ( M ) b . (2) For each sequence ( r k ) with r k ρ k → for some ρ > and each compact K ⊆ U , the set n f ( k ) ( a )( v , . . . , v k ) k ! M k r k : a ∈ K, k ∈ N , k v i k ≤ o is bounded in F . (3) For each sequence ( r k ) satisfying r k > , r k r ℓ ≥ r k + ℓ , and r k ρ k → forsome ρ > , each compact K ⊆ U , and each δ > , the set n f ( k ) ( a )( v , . . . , v k ) k ! M k r k δ k : a ∈ K, k ∈ N , k v i k ≤ o is bounded in F . Proof. (1) ⇒ (2) For ( r k ) and K , and ρ > r k ρ k → (cid:13)(cid:13)(cid:13)(cid:13) f ( k ) ( a ) k ! M k r k (cid:13)(cid:13)(cid:13)(cid:13) L k ( E,F ) = (cid:13)(cid:13)(cid:13)(cid:13) f ( k ) ( a ) k ! ρ k M k (cid:13)(cid:13)(cid:13)(cid:13) L k ( E,F ) · | r k ρ k | is bounded uniformly in k ∈ N and a ∈ K (by Lemma 4.2).(2) ⇒ (3) Apply (2) to the sequence ( r k δ k ).(3) ⇒ (1) Let a k := sup a ∈ K k f ( k ) ( a ) k ! M k k L k ( E,F ) . By the following lemma, the a k arethe coefficients of a power series with infinite radius of convergence. Thus a k /ρ k isbounded for every ρ > (cid:3) Lemma.
For a formal power series P k ≥ a k t k with real coefficients the followingare equivalent: (4) The radius of convergence is infinite. (5)
For each sequence ( r k ) satisfying r k > , r k r ℓ ≥ r k + ℓ , and r k ρ k → forsome ρ > , and each δ > , the sequence ( a k r k δ k ) is bounded. Proof. (4) ⇒ (5) The series P a k r k δ k = P ( a k ( δρ ) k ) r k ρ k converges absolutely foreach δ . Hence ( a k r k δ k ) is bounded.(5) ⇒ (4) Suppose that the radius of convergence ρ is finite. So P k | a k | n k = ∞ for n > ρ . Set r k = 1 /n k . Then, by (5), a k n k k = a k r k n k k = a k r k (2 n ) k < C, for some C > k . Consequently, P k | a k | n k ≤ C P k k , a contradiction. (cid:3) Lemma 4.6.
Let
E, F be Banach spaces, U ⊆ E open, and f : U → F a C ∞ -mapping. The following are equivalent: (1) f is C { M } = C { M } b . (2) For each sequence ( r k ) with r k ρ k → for all ρ > , and each compact K ⊆ U , the set n f ( k ) ( a )( v , . . . , v k ) k ! M k r k : a ∈ K, k ∈ N , k v i k ≤ o is bounded in F . (3) For each sequence ( r k ) satisfying r k > , r k r ℓ ≥ r k + ℓ , and r k ρ k → forall ρ > , and each compact K ⊆ U , there exists δ > such that n f ( k ) ( a )( v , . . . , v k ) k ! M k r k δ k : a ∈ K, k ∈ N , k v i k ≤ o is bounded in F . Proof. (1) ⇒ (2) For K , there exists ρ > (cid:13)(cid:13)(cid:13) f ( k ) ( a ) k ! M k r k (cid:13)(cid:13)(cid:13) L k ( E,F ) = (cid:13)(cid:13)(cid:13) f ( k ) ( a ) k ! ρ k M k (cid:13)(cid:13)(cid:13) L k ( E,F ) · | r k ρ k | ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 19 is bounded uniformly in k ∈ N and a ∈ K (by Lemma 4.3).(2) ⇒ (3) Use δ = 1.(3) ⇒ (1) Let a k := sup a ∈ K k f ( k ) ( a ) k ! M k k L k ( E,F ) . Using [23, 9.2(4 ⇒ a k /ρ k isbounded for some ρ > (cid:3) Definition 4.7 (Banach plots) . Let E be a convenient vector space. A C [ M ] (Banach) plot in E is a mapping c : D → E of class C [ M ] , where D is an open setin some Banach space F . It suffices to only consider the open unit ball D = oF . Theorem 4.8.
Let M = ( M k ) be log-convex. Let U ⊆ E be c ∞ -open in a conve-nient vector space E , let F be a Banach space, and let f : U → F be a mapping.Then: f ∈ C [ M ] ( U, F ) = ⇒ f ◦ c ∈ C [ M ] , for all C [ M ] -plots c. Note that the converse ( ⇐ ) holds by Section 4.2. Proof.
We treat first the Beurling case C ( M ) : We have to show that f ◦ c is C ( M ) for each C ( M ) -plot c : G ⊇ D → E . By Lemma 4.5(3), it suffices to show that, foreach sequence ( r k ) satisfying r k > r k r ℓ ≥ r k + ℓ , and r k t k → t > K ⊆ D , and each δ >
0, the set(13) n ( f ◦ c ) ( k ) ( a )( v , . . . , v k ) k ! M k r k δ k : a ∈ K, k ∈ N , k v i k G ≤ o is bounded in F .So let δ , the sequence ( r k ), and a compact (and without loss of generality convex)subset K ⊆ D be fixed. For each ℓ ∈ E ∗ the set(14) n ( ℓ ◦ c ) ( k ) ( a )( v , . . . , v k ) k ! M k r k (2 δ ) k : a ∈ K, k ∈ N , k v i k G ≤ o is bounded in R , by Lemma 4.5(2) applied to the sequence ( r k (2 δ ) k ). Thus, the set(15) n c ( k ) ( a )( v , . . . , v k ) k ! M k r k (2 δ ) k : a ∈ K, k ∈ N , k v i k G ≤ o is contained in some closed absolutely convex bounded subset B of E and hence k c ( k ) ( a ) k L k ( G,E B ) r k δ k k ! M k ≤ k . Furthermore c : K → E B is Lipschitzian, since c ( x ) − c ( y ) = Z c ′ ( y + t ( x − y )) ( x − y ) dt ∈ M k x − y k G r δ B, and hence c ( K ) is compact in E B . By Fa`a di Bruno’s formula for Banach spaces(see [8] for the 1-dimensional version), for k ≥ f ◦ c ) ( k ) ( a ) k ! = sym (cid:16) X j ≥ X α ∈ N j> α + ··· + α j = k j ! f ( j ) ( c ( a )) (cid:16) c ( α ) ( a ) α ! , . . . , c ( α j ) ( a ) α j ! (cid:17)(cid:17) , where sym denotes symmetrization. Using (4) and f ∈ C ( M ) ( U, F ), we find thatfor each ρ >
C > a ∈ K and k ∈ N > , (cid:13)(cid:13)(cid:13) ( f ◦ c ) ( k ) ( a ) k ! M k r k δ k (cid:13)(cid:13)(cid:13) L k ( G,F ) ≤ X j ≥ M j X α ∈ N j> α + ··· + α j = k k f ( j ) ( c ( a )) k L j ( E B ,F ) j ! M j | {z } ≤ C ρ j j Y i =1 k c ( α i ) ( a ) k L αi ( G,E B ) r α i δ α i α i ! M α i | {z } ≤ / αi ≤ X j ≥ M j (cid:18) k − j − (cid:19) C ρ j k = M ρ (1 + M ρ ) k − C k ≤ C , (16)as required, where in the last inequality we set ρ := 1 /M .Let us now consider the Roumieu case C { M } : Let now c : G ⊇ D → E be a C { M } -plot. We have to show that f ◦ c is C { M } . By Lemma 4.6(3), it suffices toshow that for each sequence ( r k ) satisfying r k > r k r ℓ ≥ r k + ℓ , and r k t k → t >
0, and each compact K ⊆ D , there exists δ > F .By Lemma 4.6(2) (applied to ( r k k ) instead of ( r k )), for each ℓ ∈ E ∗ , eachsequence ( r k ) with r k t k → t >
0, and each compact K ⊆ D , the set (14)with δ = 1 is bounded in R , and, thus, the set (15) with δ = 1 is contained in someclosed absolutely convex bounded subset B of E . Using that f ∈ C { M } ( U, F ) andcomputing as above we find that, for some ρ >
C > δ := M ρ , theleft-hand side of (16) is bounded by C M ρ M ρ . (cid:3) Theorem 4.9 ( C [ M ] is a category) . Let M = ( M k ) be log-convex. Let E, F, G beconvenient vector spaces, U ⊆ E , V ⊆ F be c ∞ -open, and f : U → F , g : V → G ,and f ( U ) ⊆ V . Then: f, g ∈ C [ M ] = ⇒ g ◦ f ∈ C [ M ] . Proof.
By Section 4.2, we must show that for all closed absolutely convex bounded B ⊆ E and for all ℓ ∈ G ∗ the composite ℓ ◦ g ◦ f ◦ i B : U B → R belongs to C [ M ] . U f / / V g / / ℓ ◦ g & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ G ℓ (cid:15) (cid:15) U Bi B O O f ◦ i B ♣♣♣♣♣♣♣♣♣♣♣♣♣ R By assumption, f ◦ i B and ℓ ◦ g are C [ M ] . So the assertion follows from Theorem4.8. (cid:3) The exponential law
Lemma 5.1.
Let E be a Banach space, and U ⊆ E be open. Let F be a convenientvector space, and let S be a family of bounded linear functionals on F which togetherdetect bounded sets (i.e., B ⊆ F is bounded if and only if ℓ ( B ) is bounded for all ℓ ∈ S ). Then: f ∈ C [ M ] ( U, F ) ⇐⇒ ℓ ◦ f ∈ C [ M ] ( U, R ) , for all ℓ ∈ S . ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 21
Proof.
For C ∞ -curves this follows from [23, 2.1 and 2.11], and, by composing withsuch, it follows for C ∞ -mappings f : U → F .In the Beurling case C ( M ) : By Lemma 4.5, for ℓ ∈ F ∗ , the function ℓ ◦ f is C ( M ) if and only if the set(17) n ( ℓ ◦ f ) ( k ) ( a )( v , . . . , v k ) k ! M k r k : a ∈ K, k ∈ N , k v i k ≤ o is bounded, for each sequence ( r k ) with r k ρ k → ρ > K ⊆ U . So the smooth mapping f : U → F is C ( M ) if and only if the set(18) n f ( k ) ( a )( v , . . . , v k ) k ! M k r k : a ∈ K, k ∈ N , k v i k ≤ o is bounded in F , for each such ( r k ) and K . This is in turn equivalent to ℓ ◦ f ∈ C ( M ) for all ℓ ∈ S , since S detects bounded sets.The same proof works in the Roumieu case C { M } if we use Lemma 4.6 anddemand that r k ρ k → ρ > (cid:3) Theorem 5.2 (Cartesian closedness) . We have: (1)
Let M = ( M k ) be weakly log-convex and have moderate growth. Then, forconvenient vector spaces E , E , and F and c ∞ -open sets U ⊆ E and U ⊆ E , we have the exponential law: f ∈ C [ M ] ( U × U , F ) ⇐⇒ f ∨ ∈ C [ M ] ( U , C [ M ] ( U , F )) . The direction ( ⇐ ) holds without the assumption that M = ( M k ) has moder-ate growth. The direction ( ⇒ ) holds without the assumption that M = ( M k ) is weakly log-convex. (2) Let M = ( M k ) be log-convex and have moderate growth. Then the categoryof C [ M ] -mappings between convenient real vector spaces is cartesian closed,i.e., satisfies the exponential law. Note that C [ M ] is not necessarily a category if M = ( M k ) is just weakly log-convex. Proof. (2) is a direct consequence of (1) and Theorem 4.9. Let us prove (1). Wehave C ∞ ( U × U , F ) ∼ = C ∞ ( U , C ∞ ( U , F )), by [23, 3.12]; thus, in the followingall mappings are assumed to be smooth. We have the following equivalences, where B ⊆ E × E and B i ⊆ E i run through all closed absolutely convex bounded subsets,respectively: f ∈ C [ M ] ( U × U , F ) ⇐⇒ ∀ ℓ ∈ F ∗ ∀ B : ℓ ◦ f ◦ i B ∈ C [ M ] (( U × U ) B , R ) ⇐⇒ ∀ ℓ ∈ F ∗ ∀ B , B : ℓ ◦ f ◦ ( i B × i B ) ∈ C [ M ] (( U ) B × ( U ) B , R )For the second equivalence we use that every bounded B ⊆ E × E is containedin B × B for some bounded B i ⊆ E i , and, thus, the inclusion ( E × E ) B → ( E ) B × ( E ) B is bounded.On the other hand, we have: f ∨ ∈ C [ M ] ( U , C [ M ] ( U , F )) ⇐⇒ ∀ B : f ∨ ◦ i B ∈ C [ M ] (( U ) B , C [ M ] ( U , F )) ⇐⇒ ∀ ℓ ∈ F ∗ ∀ B , B : C [ M ] ( i B , ℓ ) ◦ f ∨ ◦ i B ∈ C [ M ] (( U ) B , C [ M ] (( U ) B , R )) For the second equivalence we use Lemma 5.1 and the fact that the linear mappings C [ M ] ( i B , ℓ ) generate the bornology.These considerations imply that in order to prove cartesian closedness we mayrestrict to the case that U i ⊆ E i are open in Banach spaces E i and F = R .( Direction ⇒ ) We assume that M = ( M k ) has moderate growth. Let f ∈ C [ M ] ( U × U , R ). It is clear that f ∨ takes values in C [ M ] ( U , R ). Claim. f ∨ : U → C [ M ] ( U , R ) is C ∞ with d j f ∨ = ( ∂ j f ) ∨ . Since C [ M ] ( U , R ) is a convenient vector space, by [23, 5.20] it is enough to showthat the iterated unidirectional derivatives d jv f ∨ ( x ) exist, equal ∂ j f ( x, )( v j ), andare separately bounded for x , resp. v , in compact subsets. For j = 1 and fixed x , v ,and y consider the smooth curve c : t f ( x + tv, y ). By the fundamental theorem f ∨ ( x + tv ) − f ∨ ( x ) t ( y ) − ( ∂ f ) ∨ ( x )( y )( v ) = c ( t ) − c (0) t − c ′ (0)= t Z s Z c ′′ ( tsr ) dr ds = t Z s Z ∂ f ( x + tsrv, y )( v, v ) dr ds. Since ( ∂ f ) ∨ ( K )( oE ) is obviously bounded in C [ M ] ( U , R ) for each compact sub-set K ⊆ U , this expression is Mackey convergent to 0 in C [ M ] ( U , R ) as t → d v f ∨ ( x ) exists and equals ∂ f ( x, )( v ).Now we proceed by induction, applying the same arguments as before to( d jv f ∨ ) ∧ : ( x, y ) ∂ j f ( x, y )( v j ) instead of f . Again ( ∂ ( d jv f ∨ ) ∧ ) ∨ ( K )( oE ) =( ∂ j +21 f ) ∨ ( K )( oE , oE , v, . . . , v ) is bounded, and also the separated boundednessof d jv f ∨ ( x ) follows. So the claim is proved.We have to show that f ∨ : U → C [ M ] ( U , R ) is C [ M ] .In the Beurling case C ( M ) : U f ∨ / / C ( M ) ( U , R ) lim ←− K lim ←− ρ C Mρ ( E ⊇ K , R ) ℓ / / (cid:15) (cid:15) R lim ←− ρ C Mρ ( E ⊇ K , R ) (cid:15) (cid:15) K ?(cid:31) O O / / C MK ,ρ ( U , R ) / / C Mρ ( E ⊇ K , R ) K K (19)By Lemma 5.1, it suffices to show that f ∨ : U → C Mρ ( E ⊇ K , R ) is C ( M ) b = C ( M ) (see Lemma 4.2) for each compact K ⊆ U and each ρ >
0, since every ℓ ∈ C ( M ) ( U , R ) ∗ factors over some C Mρ ( E ⊇ K , R ). Thus it suffices to provethat, for all compact K ⊆ U , K ⊆ U and all ρ , ρ >
0, the set(20) n d k f ∨ ( x )( v , . . . , v k ) k ! ρ k M k : x ∈ K , k ∈ N , k v j k E ≤ o ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 23 is bounded in C MK ,ρ ( U , R ), or, equivalently, for all K , K , ρ , ρ the set(21) n ∂ k ∂ k f ( x , x )( v , . . . , v k ; v , . . . , v k ) k ! k ! ρ k ρ k M k M k : x i ∈ K i , k i ∈ N , k v ij k E i ≤ o is bounded in R .Since M = ( M k ) has moderate growth, i.e., M k + k ≤ σ k + k M k M k for some σ >
0, we obtain, for x ∈ K , k ∈ N , and k v j k E ≤ (cid:13)(cid:13)(cid:13) d k f ∨ ( x )( v , . . . , v k ) k ! ρ k M k (cid:13)(cid:13)(cid:13) K ,ρ = sup n | ∂ k ∂ k f ( x , x )( v , . . . , v k ; v , . . . , v k ) | k ! k ! ρ k ρ k M k M k : x ∈ K , k ∈ N , k v j k E ≤ o ≤ sup n (2 σ ) k + k | ∂ k ∂ k f ( x , x )( v , . . . ; v , . . . ) | ( k + k )! ρ k ρ k M k + k : x ∈ K , k ∈ N , k v j k E ≤ o . (22)If for given ρ , ρ > ρ := σ min { ρ , ρ } , then (22) is bounded by(23) sup n | ∂ k ∂ k f ( x , x )( v , . . . ; v , . . . ) | ( k + k )! ρ k + k M k + k : x i ∈ K i , k i ∈ N , k v ij k E i ≤ o which is finite, since f is C ( M ) . Thus, f ∨ is C ( M ) .In the Roumieu case C { M } : U f ∨ / / C { M } ( U , R ) lim ←− K lim −→ ρ C Mρ ( E ⊇ K , R ) ℓ / / (cid:15) (cid:15) R lim −→ ρ C Mρ ( E ⊇ K , R ) K ?(cid:31) O O / / C MK ,ρ ( U , R ) / / C Mρ ( E ⊇ K , R ) O O (24)By Lemma 5.1, it suffices to show that f ∨ : U → lim −→ ρ C Mρ ( E ⊇ K , R ) is C { M } b ⊆ C { M } for each compact K ⊆ U , since every ℓ ∈ C { M } ( U , R ) ∗ factorsover some lim −→ ρ C Mρ ( E ⊇ K , R ). Thus it suffices to prove that, for all compact K ⊆ U and K ⊆ U there exists ρ >
0, such that the set (20) is bounded inlim −→ ρ C Mρ ( E ⊇ K , R ). For that it suffices to show that for all K , K there are ρ , ρ so that the set (21) is bounded in R .Since f is C { M } , there exists ρ > ρ i := 2 σρ , then (22) is bounded by (23). It follows that f ∨ is C { M } .( Direction ⇐ ) Let f ∨ : U → C [ M ] ( U , R ) be C [ M ] . Clearly, f ∨ : U → C [ M ] ( U , R ) → C ∞ ( U , R ) is C ∞ , see Proposition 8.1, and so it remains to showthat f ∈ C [ M ] ( U × U , R ).In the Beurling case C ( M ) : Consider diagram (19). For each compact K ⊆ U and each ρ >
0, the mapping f ∨ : U → C Mρ ( E ⊇ K , R ) is C ( M ) = C ( M ) b . That means that, for all compact K ⊆ U , K ⊆ U and all ρ , ρ >
0, the set (20)is bounded in C Mρ ( E ⊇ K , R ). Since it is contained in C MK ,ρ ( U , R ) = { g ∈ C ∞ ( U , R ) : j ∞ g | K ∈ C Mρ ( E ⊇ K , R ) } and k g k K ,ρ = k j ∞ g | K k ρ , it is alsobounded in this space, and hence the set (21) is bounded.Since M = ( M k ) is weakly log-convex, thus, k ! k ! M k M k ≤ ( k + k )! M k + k ,we have, for x ∈ K , k ∈ N , and k v j k E ≤ (cid:13)(cid:13)(cid:13) d k f ∨ ( x )( v , . . . , v k ) k ! ρ k M k (cid:13)(cid:13)(cid:13) K ,ρ = sup n | ∂ k ∂ k f ( x , x )( v , . . . , v k ; v , . . . , v k ) | k ! k ! ρ k ρ k M k M k : x ∈ K , k ∈ N , k v j k E ≤ o ≥ sup n | ∂ k ∂ k f ( x , x )( v , . . . ; v , . . . ) | ( k + k )! ρ k ρ k M k + k : x ∈ K , k ∈ N , k v j k E ≤ o . (25)This implies that f is C ( M ) .In the Roumieu case C { M } : Consider diagram (24). For each compact K ⊆ U ,the mapping f ∨ : U → lim −→ ρ C Mρ ( E ⊇ K , R ) is C { M } . The inductive limit isregular, by Proposition 4.1(3). So the dual space (lim −→ ρ C Mρ ( E ⊇ K , R )) ∗ can beequipped with the Baire topology of the countable limit lim ←− ρ C Mρ ( E ⊇ K , R ) ∗ ofBanach spaces. Thus, the mapping f ∨ : U → lim −→ ρ C Mρ ( E ⊇ K , R ) is C { M } b , byLemma 4.3. By regularity, for each compact K ⊆ U there exists ρ > C Mρ ( E ⊇ K , R ) for some ρ >
0. Since thisset is contained in C MK ,ρ ( U , R ) = { g ∈ C ∞ ( U , R ) : j ∞ g | K ∈ C Mρ ( E ⊇ K , R ) } and k g k K ,ρ = k j ∞ g | K k ρ , it is also bounded in this space, and hence the set (21)is bounded. Then (25) implies that f is C { M } . The proof is complete. (cid:3) Remarks 5.3.
Theorem 8.2 below states that, if M = ( M k ) is (weakly) log-convex, E, F are convenient vector spaces, and U ⊆ E is c ∞ -open, then(26) C { M } ( U, F ) = lim ←− L C ( L ) ( U, F )as vector spaces with bornology, where the projective limits are taken over all(weakly) log-convex L = ( L k ) with M ✁ L . Using this equality we can give analternative proof of the direction f ∨ ∈ C { M } ( U , C { M } ( U , F )) ⇒ f ∈ C { M } ( U × U , F )in Theorem 5.2 as follows: If f ∨ ∈ C { M } ( U , C { M } ( U , F )) then f ∨ ∈ C ( L ) ( U , C ( L ) ( U , F )) for all L = ( L k ) with M ✁ L , by (26). By cartesian closednessof C ( L ) (i.e., Theorem 5.2, the implication which holds without moderate growth),we have f ∈ C ( L ) ( U × U , F ) for all L , and, by (26) again, f ∈ C { M } ( U × U , F ).The proof of (26) in Theorem 8.2 uses the C { M } uniform boundedness principle,i.e., Theorem 6.1, and the proof of the latter uses completeness of the inductivelimit lim −→ ρ C Mρ ( E ⊇ K, F ), where
E, F are Banach spaces and K ⊆ E is compact,see Proposition 4.1. Here is a direct proof of (26), where we only assume that M = ( M k ) is positive: ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 25
The spaces coincide as vector spaces by Sections 4.1, 4.2, and by Theorem 2.2.For K compact in a Banach space E and ρ >
0, the inclusion C Mρ ( E ⊇ K, R ) → C Lσ ( E ⊇ K, R ) is continuous for all σ > M ✁ L . It follows that the inclusionlim −→ ρ C Mρ ( E ⊇ K, R ) → lim ←− σ C Lσ ( E ⊇ K, R ) is continuous. This implies that theinclusion C { M } ( U, F ) → C ( L ) ( U, F ) is continuous (by definition of the structure inSection 4.2).Conversely, let B be a bounded set in lim ←− L C ( L ) ( U, F ), i.e., bounded in each C ( L ) ( U, F ). We claim that B is bounded in C { M } ( U, F ). We may assume with-out loss of generality that E is a Banach space and F = R (by composing with C { M } ( i B , ℓ )). Let K ⊆ U be compact and b k := sup {k j ∞ f | K k k : f ∈ B} . Forall L = ( L k ) with M ✁ L the set B is bounded in C ( L ) ( U, F ) by assumption, i.e.,( b k ) k ∈ T L F ( L ) = F { M } by Theorem 2.2. From this follows that B is bounded in C { M } K ( U, F ) and by Proposition 4.1(5) also in C { M } ( U, F ).Note that this independently proves that C { M } ( U, F ) is c ∞ -complete since so islim ←− L C ( L ) ( U, F ). Moreover, it provides an independent proof of the regularity ofthe inductive limit involved in the definition of C { M } ( U, F ) if E and F are Banachspaces (cf. Proposition 4.1 and the remark after Theorem 8.6). Example 5.4 (Cartesian closedness fails without moderate growth) . Let us assumethat M = ( M k ) is weakly log-convex and has non-moderate growth (for instance, M k = q k , q > , see [38, 2.1.3] ). Then: (1) There exists f ∈ C { M } ( R , C ) such that f ∨ : R → C { M } ( R , C ) is not C { M } . (2) There exists a weakly log-convex N = ( N k ) with M ✁ N and an f ∈ C ( N ) ( R , C ) such that f ∨ : R → C ( N ) ( R , C ) is not C ( N ) . Proof. (1) There is a function g ∈ C { M } ( R , C ) such that g ( k ) (0) = i k h k and h k ≥ k ! M k for all k ; see [38, Thm. 1]. Defining f ( s, t ) := g ( s + t ), we obtain afunction f ∈ C { M } ( R , C ) with ∂ α f (0 ,
0) = i | α | h | α | , h | α | ≥ | α | ! M | α | for all α ∈ N . Since M = ( M k ) has non-moderate growth, there exist j n ր ∞ and k n > (cid:16) M k n + j n M k n M j n (cid:17) kn + jn ≥ n. Consider the linear functional ℓ : C { M } ( R , C ) → C given by ℓ ( g ) = X n i j n g ( j n ) (0) j n ! M j n n j n . This functional is continuous, since (cid:12)(cid:12)(cid:12) X n i j n g ( j n ) (0) j n ! M j n n j n (cid:12)(cid:12)(cid:12) ≤ X n | g ( j n ) (0) | j n ! ρ j n M j n ρ j n n j n ≤ C ( ρ ) k g k [ − , ,ρ < ∞ , for suitable ρ , where C ( ρ ) := X n (cid:16) ρn (cid:17) j n < ∞ , for all ρ . But ℓ ◦ f ∨ is not C { M } , since k ℓ ◦ f ∨ k [ − , ,ρ = sup k ∈ N ,t ∈ [ − , | ( ℓ ◦ f ∨ ) ( k ) ( t ) | ρ k k ! M k ≥ sup k ρ k k ! M k (cid:12)(cid:12)(cid:12) X n i j n f ( j n ,k ) (0 , j n ! M j n n j n (cid:12)(cid:12)(cid:12) = sup k ρ k k ! M k (cid:12)(cid:12)(cid:12) X n i j n + k h ( j n ,k ) j n ! M j n n j n (cid:12)(cid:12)(cid:12) = sup k ρ k k ! M k X n h ( j n ,k ) j n ! M j n n j n ≥ sup n ρ k n k n ! M k n h ( j n ,k n ) j n ! M j n n j n ≥ sup n ( j n + k n )! M j n + k n ρ k n k n ! j n ! M k n M j n n j n ≥ sup n n j n + k n ρ k n n j n = ∞ , for all ρ > E, F and c ∞ -opensubsets U ⊆ E , C { M } ( U, F ) = \ N C ( N ) ( U, F ) , where the intersection is taken over all weakly log-convex N = ( N k ) with M ✁ N .Let f be the function in (1). Then there exist weakly log-convex sequences N i =( N ik ), i = 1 ,
2, with M ✁ N i such that f ∨ : R → C ( N ) ( R , C ) is not C ( N ) . Bythe lemma below there exists a weakly log-convex sequence N = ( N k ) such that M ✁ N ≤ N i for i = 1 ,
2. Since f ∈ C { M } ( R , C ) ⊆ C ( N ) ( R , C ), the mapping f ∨ has values in C ( N ) ( R , C ) and thus factors over the inclusion C ( N ) ( R , C ) → C ( N ) ( R , C ) which is obviously continuous. It follows that f ∨ : R → C ( N ) ( R , C ) isnot C ( N ) and consequently not C ( N ) . R f ∨ C ( N / / f ∨ C ( N ⊇ C ( N ) ( ( PPPPPPPPPPPPPP C ( N ) ( R , C ) C ( N ) ( R , C ) ?(cid:31) O O By Theorem 5.2, N = ( N k ) has non-moderate growth. (cid:3) Lemma.
Let M = ( M k ) , N i = ( N ik ) , i = 1 , , be weakly log-convex with M ✁ N i for i = 1 , . Then there exists a weakly log-convex sequence N = ( N k ) such that M ✁ N ≤ N i for i = 1 , . Proof.
Set ¯ N = ( ¯ N k ) := (min { N k , N k } ) and N = ( N k ), where ( k ! N k ) is thelog-convex minorant of ( k ! ¯ N k ). Note that N = 1 ≤ N . We have N ≤ ¯ N ≤ N i ,and M ✁ N i implies M ✁ ¯ N . It remains to show that M ✁ N .We claim that C ( N )global ( R , R ) = C ( ¯ N )global ( R , R ), where for a sequence L = ( L k ) ∈ ( R > ) N we set C [ L ]global ( R , R ) := n f ∈ C ∞ ( R , R ) : (sup x ∈ R | f ( k ) ( x ) | ) k ∈ F [ L ] o . ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 27
In the Roumieu case this a theorem due to Cartan and Gorny, see [19, IV E]; thesame proof with obvious modifications yields the Beurling version, i.e., the claim.Now M ✁ ¯ N implies C { M } global ( R , R ) ⊆ C ( ¯ N )global ( R , R ) = C ( N )global ( R , R ). The function˜ g := Re g + Im g , where g is the function from the proof of (1), is actually anelement of C { M } global ( R , R ) and satisfies | ˜ g ( k ) (0) | ≥ k ! M k for all k ; see [38, Thm. 1].Thus ˜ g ∈ C ( N )global ( R , R ) and therefore M ✁ N . (cid:3) Corollary 5.5 (Canonical mappings) . Let M = ( M k ) be log-convex and havemoderate growth. Let E , F , etc., be convenient vector spaces and let U and V be c ∞ -open subsets of such. Then we have: (1) The exponential law holds: C [ M ] ( U, C [ M ] ( V, G )) ∼ = C [ M ] ( U × V, G ) is a linear C [ M ] -diffeomorphism of convenient vector spaces.The following canonical mappings are C [ M ] . ev : C [ M ] ( U, F ) × U → F, ev( f, x ) = f ( x )(2) ins : E → C [ M ] ( F, E × F ) , ins( x )( y ) = ( x, y )(3) ( ) ∧ : C [ M ] ( U, C [ M ] ( V, G )) → C [ M ] ( U × V, G )(4) ( ) ∨ : C [ M ] ( U × V, G ) → C [ M ] ( U, C [ M ] ( V, G ))(5) comp : C [ M ] ( F, G ) × C [ M ] ( U, F ) → C [ M ] ( U, G )(6) C [ M ] ( , ) : C [ M ] ( F, F ) × C [ M ] ( E , E ) → C [ M ] (cid:16) C [ M ] ( E, F ) , C [ M ] ( E , F ) (cid:17) (7) ( f, g ) ( h f ◦ h ◦ g ) Y : Y C [ M ] ( E i , F i ) → C [ M ] (cid:16)Y E i , Y F i (cid:17) (8) Proof.
This is a direct consequence of cartesian closedness, i.e., Theorem 5.2. See[25, 5.5] or [23, 3.13] for the detailed arguments. (cid:3) Uniform boundedness principles
Theorem 6.1 ( C [ M ] uniform boundedness principle) . Let E , F , G be convenientvector spaces and let U ⊆ F be c ∞ -open. A linear mapping T : E → C [ M ] ( U, G ) isbounded if and only if ev x ◦ T : E → G is bounded for every x ∈ U . Proof. ( ⇒ ) For x ∈ U and ℓ ∈ G ∗ , the linear mapping ℓ ◦ ev x = C [ M ] ( x, ℓ ) : C [ M ] ( U, G ) → R is continuous, thus ev x is bounded. Therefore, if T is boundedthen so is ev x ◦ T .( ⇐ ) Suppose that ev x ◦ T is bounded for all x ∈ U . By the definition of C [ M ] ( U, G ) in Section 4.2 it is enough to show that T is bounded in the case that E and F are Banach spaces and G = R . By Section 4.1, C [ M ] ( U, R ) = lim ←− K C [ M ] ( F ⊇ K, R ), by Proposition 4.1(2), C ( M ) ( F ⊇ K, R ) is a Fr´echet space, and by Proposi-tion 4.1(3), C { M } ( F ⊇ K, R ) is an (LB)-space, so C [ M ] ( F ⊇ K, R ) is webbed and hence the closed graph theorem [23, 52.10] gives the desired result. E T / / ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆ C [ M ] ( U, R ) ev x / / R lim ←− K C [ M ] ( F ⊇ K, R ) / / C [ M ] ( F ⊇ K, R ) (ev x ) O O (27) (cid:3) Remark 6.2.
Alternatively, the C { M } uniform boundedness principle follows fromthe C ( M ) uniform boundedness principle and from the remark after Theorem 8.2,since the structure of C { M } ( U, F ) = lim ←− L C ( L ) ( U, F ) is initial with respect to theinclusions lim ←− L C ( L ) ( U, F ) → C ( L ) ( U, F ) for all L . This is no circular argument,since the first identity in Theorem 8.2 was proved in Remark 5.3 without using the C { M } uniform boundedness principle, i.e., Theorem 6.1.7. Relation to previously considered structures
In [25] and [26] we have developed the convenient setting for all reasonable non-quasianalytic and some quasianalytic (namely, L -intersectable, see Section 7.1)Denjoy–Carleman classes of Roumieu type. We have worked with a definition whichis based on testing along curves . The resulting structures were denoted by C M in[25] and [26] and will be denoted by C { M } curve in this section; this notation does not ap-pear elsewhere in this paper. We shall now show that they coincide bornologicallywith the structure C { M } studied in the present paper. Furthermore, we prove thatthe bornologies induced by C { } and the structure C ω of real analytic mappingsintroduced in [22] are isomorphic; here 1 denotes the constant sequence (1) k . Notethat C { } is not L -intersectable (see [26, 1.8]).7.1. Testing along curves.
Let M = ( M k ) be log-convex, E and F convenientvector spaces, and U a c ∞ -open subset in E . If M = ( M k ) is non-quasianalytic weset C { M } curve ( U, F ) := n f ∈ F U : ∀ ℓ ∈ F ∗ ∀ c ∈ C { M } ( R , U ) : ℓ ◦ f ◦ c ∈ C { M } ( R , R ) o . If M = ( M k ) is quasianalytic and L -intersectable , i.e., F { M } = T L ∈L ( M ) F { L } ,where L ( M ) := n L = ( L k ) : L ≥ M, L is non-quasianalytic log-convex o , we define C { M } curve ( U, F ) := \ L ∈L ( M ) C { L } curve ( U, F ) . Note that non-quasianalytic log-convex sequences are trivially L -intersectable. Fornon-quasianalytic M = ( M k ) we supply C { M } curve ( U, F ) with the initial locally convexstructure induced by all linear mappings: C { M } curve ( U, F ) − C { M } curve ( c,ℓ ) → C { M } ( R , R ) , f ℓ ◦ f ◦ c, ℓ ∈ F ∗ , c ∈ C { M } ( R , U ) , and for quasianalytic and L -intersectable M = ( M k ) by all inclusions C { M } curve ( U, F ) −→ C { L } curve ( U, F ) , L ∈ L ( M ) . ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 29
In both cases C { M } curve ( U, F ) is a convenient vector space.Let C ω ( R , R ) denote the real analytic functions f : R → R and set C ω ( U, F ) := n f ∈ C ∞ ( U, F ) : ∀ ℓ ∈ F ∗ ∀ c ∈ C ω ( R , U ) : ℓ ◦ f ◦ c ∈ C ω ( R , R ) o , where C ω ( R , U ) is the space of all weakly C ω -curves in U . We equip C ω ( U, R ) withthe initial locally convex structure induced by the family of mappings C ω ( U, R ) − c ∗ → C ω ( R , R ) , f f ◦ c, c ∈ C ω ( R , U ) ,C ω ( U, R ) − c ∗ → C ∞ ( R , R ) , f f ◦ c, c ∈ C ∞ ( R , U ) , where C ∞ ( R , R ) carries the topology of compact convergence in each derivativeseparately, and where C ω ( R , R ) is equipped with the final locally convex topologywith respect to the embeddings (restriction mappings) of all spaces of holomor-phic mappings from a neighborhood V of R in C mapping R to R , and each ofthese spaces carries the topology of compact convergence. The space C ω ( U, F ) isequipped with the initial locally convex structure induced by all mappings C ω ( U, F ) − ℓ ∗ → C ω ( U, R ) , f ℓ ◦ f, ℓ ∈ F ∗ . This is again a convenient vector space.
Theorem 7.1.
Let M = ( M k ) be log-convex, E and F convenient vector spaces,and U a c ∞ -open subset in E . We have: (1) If M = ( M k ) is L -intersectable, then C { M } ( U, F ) = C { M } curve ( U, F ) as vector spaces with bornology. (2) If denotes the constant sequence, then C { } ( U, F ) = C ω ( U, F ) as vector spaces with bornology. Proof. (1) If M = ( M k ) is non-quasianalytic, then C { M } ( U, F ) and C { M } curve ( U, F )coincide as vector spaces, by [26, 2.8]. If M = ( M k ) is quasianalytic and L -intersectable, then the non-quasianalytic case implies that C { M } curve ( U, F ) = \ L ∈L ( M ) C { L } curve ( U, F ) = \ L ∈L ( M ) C { L } ( U, F ) = C { M } ( U, F )as vector spaces, where the last equality is a consequence of the definition of C { M } ( U, F ) (see Section 4.2) and of [26, 1.6] (applied to C { M } ( U B , R )). The factthat both spaces C { M } ( U, F ) and C { M } curve ( U, F ) are convenient and satisfy the uni-form boundedness principle with respect to the set of point evaluations, see Theorem6.1 and [26, 2.9], implies that the identity is a bornological isomorphism.(2) We show first that C { } ( U, F ) = C ω ( U, F ) as vector spaces. By Section 4.2and [23, 10.6], it suffices to consider the case that U is open in a Banach space E and F = R .Let f ∈ C ω ( U, R ). By [22, 2.4 and 2.7] or [23, 10.1 and 10.4], this is equivalent to f being smooth and being locally given by its convergent Taylor series. Let K ⊆ U be compact. Since the Taylor series of f converges locally, there exist constants C, ρ > k f ( k ) ( a ) k L k ( E, R ) k ! ≤ Cρ k , for all a ∈ K, k ∈ N , that is, f ∈ C { } ( U, R ).Conversely, the above estimate for compact subsets K of affine lines in E impliesthat the restriction of f to each affine line is real analytic and hence f ∈ C ω ( U, R )by [23, 10.1].The bornologies coincide, since both spaces are convenient and satisfy the uni-form boundedness principle with respect to the set of point evaluations, see Theorem6.1 and [22, 5.6] or [23, 11.12]. (cid:3) More on function spaces
Proposition 8.1 (Inclusions) . Let M = ( M k ) , N = ( N k ) be positive sequences, E , F convenient vector spaces, and U ⊆ E a c ∞ -open subset. We have: (1) C ( M ) ( U, F ) ⊆ C { M } ( U, F ) ⊆ C ∞ ( U, F ) . (2) If there exist
C, ρ > so that M k ≤ Cρ k N k for all k , then C ( M ) ( U, F ) ⊆ C ( N ) ( U, F ) and C { M } ( U, F ) ⊆ C { N } ( U, F ) . (3) If for each ρ > there exists C > so that M k ≤ Cρ k N k for all k , i.e., M ✁ N , then C { M } ( U, F ) ⊆ C ( N ) ( U, F ) . (4) For U = ∅ and F = { } we have: C ω ( U, F ) ⊆ C ( M ) ( U, F ) ⇐⇒ M /kk → ∞ , and C ω ( U, F ) ⊆ C { M } ( U, F ) ⇐⇒ lim M /kk > . All these inclusions are bounded.
Proof.
The inclusions in (1), (2), and (3) follow immediately from the definitionsin Sections 4.1 and 4.2 and Lemma 2.2. Here we use that C { } ( U, F ) = C ω ( U, F )as vector spaces with bornology, see Theorem 7.1.The directions ( ⇐ ) in (4) are direct consequences of (2) and (3). The directions( ⇒ ) follow, since they have been shown in Section 2.1 for E = F = R .All inclusions are bounded, since all spaces are convenient and satisfy the uniformboundedness principle, cf. Theorem 6.1 and [23, 5.26] for C ∞ . (cid:3) Theorem 8.2.
Let M = ( M k ) be (weakly) log-convex, E and F convenient vectorspaces, and U a c ∞ -open subset in E . We have C { M } ( U, F ) = lim ←− L C ( L ) ( U, F ) = lim ←− L C { L } ( U, F ) as vector spaces with bornology, where the projective limits are taken over all(weakly) log-convex sequences L = ( L k ) with M ✁ L . Proof.
The three spaces coincide as vector spaces: By Section 4.2 it suffices toassume that E and F are Banach spaces, and by Section 4.1 and Proposition 4.1(5)it suffices to apply Theorem 2.2 to the sequence ( k j ∞ f | K k m ). ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 31
Each space is convenient (see Section 4.2; projective limits preserve c ∞ -completeness) and each space satisfies the uniform boundedness principle withrespect to the set of point evaluations (see Theorem 6.1; the structure oflim ←− L C [ L ] ( U, F ) is initial with respect to the inclusions lim ←− L C [ L ] ( U, F ) → C [ L ] ( U, F )for all L ). Thus the identity between any two of the three spaces is a bornologicalisomorphism. (cid:3) Remark.
By the remark after Theorem 2.2 the statement of the theorem stillholds, if M = ( M k ) is just any positive sequence, where the projective limits arenow taken over all positive sequences L = ( L k ) with M ✁ L . Proposition 8.3 (Derivatives) . Let M = ( M k ) be a positive sequence and set M +1 = ( M k +1 ) . Let E and F be convenient vector spaces, and U ⊆ E a c ∞ -opensubset. Then we have: (1) Multilinear mappings between convenient vector spaces are C [ M ] if and onlyif they are bounded. (2) If f : E ⊇ U → F is C [ M ] , then the derivative df : U → L ( E, F ) is C [ M +1 ] ,where the space L ( E, F ) of all bounded linear mappings is considered withthe topology of uniform convergence on bounded sets. If M +1 = ( M k +1 ) is weakly log-convex (which is the case if M = ( M k ) is weakly log-convex),also ( df ) ∧ : U × E → F is C [ M +1 ] , (3) The chain rule holds.
Proof. (1) If f is C [ M ] then it is smooth and hence bounded by [23, 5.5]. Con-versely, if f is multilinear and bounded then it is smooth, again by [23, 5.5]. Further-more, f ◦ i B is multilinear and continuous and all derivatives of high order vanish.Thus f is C [ M ] , by Section 4.2.(2) Since f is smooth, by [23, 3.18] the mapping df : U → L ( E, F ) exists andis smooth. We have to show that ( df ) ◦ i B : U B → L ( E, F ) is C [ M +1 ] , for allclosed absolutely convex bounded subsets B ⊆ E . By the uniform boundednessprinciple [23, 5.18] and by Lemma 5.1 it suffices to show that the mapping U B ∋ x ℓ ( df ( i B ( x ))( v )) ∈ R is C [ M +1 ] for each ℓ ∈ F ∗ and v ∈ E .Since ℓ ◦ f is C ( M ) (resp. C { M } ), for each closed absolutely convex bounded B ⊆ E , each compact K ⊆ U B , and each ρ > ρ >
0) the set n k d k ( ℓ ◦ f ◦ i B )( a ) k L k ( E B , R ) k ! ρ k M k : a ∈ K, k ∈ N o is bounded, say by C >
0. The assertion follows in both cases from the followingcomputation. For v ∈ E and those B containing v we then have: k d k ( L ( ℓ, v ) ◦ df ) ◦ i B )( a ) k L k ( E B , R ) = k d k ( d ( ℓ ◦ f )( )( v )) ◦ i B )( a ) k L k ( E B , R ) = k d k +1 ( ℓ ◦ f ◦ i B )( a )( v, . . . ) k L k ( E B , R ) ≤ k d k +1 ( ℓ ◦ f ◦ i B )( a ) k L k +1 ( E B , R ) k v k B ≤ C ( k + 1)! ρ k +1 M k +1 = Cρ (( k + 1) /k ρ ) k k ! M k +1 ≤ Cρ (2 ρ ) k k ! ( M +1 ) k . By Proposition 8.4 below also ( df ) ∧ is C [ M +1 ] , if M = ( M k ) is weakly log-convex.(3) This is valid even for all smooth f by [23, 3.18]. (cid:3) Proposition 8.4.
We have: (1)
For convenient vector spaces E and F , the following topologies have thesame bounded subsets in L ( E, F ) : • The topology of uniform convergence on bounded subsets of E . • The topology of pointwise convergence. • The trace topology of C ∞ ( E, F ) . • The trace topology of C [ M ] ( E, F ) . (2) Let M = ( M k ) be weakly log-convex, E , F , and G convenient vector spaces,and U ⊆ E a c ∞ -open subset. A mapping f : U × F → G which is linear inthe second variable is C [ M ] if and only if f ∨ : U → L ( F, G ) is well definedand C [ M ] .Analogous results hold for spaces of multilinear mappings. Proof. (1) That the first three topologies on L ( E, F ) have the same bounded setshas been shown in [23, 5.3 and 5.18]. The inclusion C [ M ] ( E, F ) → C ∞ ( E, F ) isbounded by Proposition 8.1. Conversely, the inclusion L ( E, F ) → C [ M ] ( E, F ) isbounded by the uniform boundedness principle, i.e., Theorem 6.1.(2) The assertion for C ∞ is true by [23, 3.12] since L ( E, F ) is closed in C ∞ ( E, F ).Suppose that f is C [ M ] . We have to show that f ∨ ◦ i B is C [ M ] into L ( F, G ), forall closed absolutely convex bounded subsets B ⊆ E . By the uniform boundednessprinciple [23, 5.18] and by Lemma 5.1 it suffices to show that the mapping U B ∋ x ℓ (cid:0) f ∨ ( i B ( x ))( v ) (cid:1) = ℓ (cid:0) f ( i B ( x ) , v ) (cid:1) ∈ R is C [ M ] for each ℓ ∈ G ∗ and v ∈ F ; thisis obviously true.Conversely, let f ∨ : U → L ( F, G ) be C [ M ] . By (1) the inclusion L ( F, G ) → C [ M ] ( F, G ) is bounded linear, and so f ∨ : U → C [ M ] ( F, G ) is C [ M ] . By cartesianclosedness, i.e., Theorem 5.2 (the direction which holds without moderate growth), f : U × F → G is C [ M ] and linearity in the second variable is obvious. (cid:3) Remark.
We may prove f ∨ ∈ C [ M ] ( U, L ( F, G )) ⇒ f ∈ C [ M ] ( U × F, G ) withoutusing cartesian closedness: By composing with ℓ ∈ G ∗ we may assume that G = R .By induction we have: d k f ( x, w ) (cid:0) ( v k , w k ) , . . . , ( v , w ) (cid:1) = d k ( f ∨ )( x )( v k , . . . , v )( w )++ k X i =1 d k − ( f ∨ )( x )( v k , . . . , c v i , . . . , v )( w i )Thus for B , B ′ closed absolutely convex bounded subsets of E , F , respectively, K ⊆ U B compact, and x ∈ K we have: k d k f ( x, w ) k L k ( E B × F B ′ , R ) ≤≤ k d k ( f ∨ )( x )( . . . )( w ) k L k ( E B , R ) + k X i =1 k d k − ( f ∨ )( x ) k L k − ( E B ,L ( F B ′ , R )) ≤ k d k ( f ∨ )( x ) k L k ( E B ,L ( F B ′ , R )) k w k B ′ + k k d k − ( f ∨ )( x ) k L k − ( E B ,L ( F B ′ , R )) ≤ C ρ k k ! M k k w k B ′ + k C ρ k − ( k − M k − = C ρ k k ! M k (cid:16) k w k B ′ + M k − ρ M k (cid:17) , for all ρ > C = C ( ρ ) (resp. for some C, ρ > L ( i B ′ , R ) ◦ f ∨ ◦ i B : U B → L ( F B ′ , R ) is C [ M ] . Since k k ! M k is increasing (seethe remarks in Section 2.3), we have M k − M k ≤ k ≤ k , and we may conclude that f is C [ M ] . ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 33
Let r = ( r k ) be a positive sequence, E and F Banach spaces, and K ⊆ E compact convex. Consider C M ( r k ) ( E ⊇ K, F ) : = n ( f m ) m ∈ Y m ∈ N C ( K, L m sym ( E, F )) : k f k ( r k ) < ∞ o , where k f k ( r k ) : = max (cid:26) sup n k f k m m ! r m M m : m ∈ N o , sup n ||| f ||| n,k ( n + k + 1)! r n + k +1 M n + k +1 : k, n ∈ N o(cid:27) . If ( r k ) = ( ρ k ) for some ρ > ρ instead of ( r k ) as indices and recoverthe spaces introduced in Section 4.1. Similarly as in Proposition 4.1(1) one showsthat the spaces C M ( r k ) ( E ⊇ K, F ) are Banach spaces.
Theorem 8.5.
Let E and F be Banach spaces and let U ⊆ E be open and convex.Then we have C ( M ) ( U, F ) = lim ←− K, ( r k ) C M ( r k ) ( E ⊇ K, F ) as vector spaces with bornology. Here K runs through all compact convex subsets of U ordered by inclusion and ( r k ) runs through all sequences of positive real numbersfor which ρ k /r k → for some ρ > . Proof.
Note first that the elements of the space lim ←− K, ( r k ) C M ( r k ) ( E ⊇ K, F ) aresmooth functions f : U → F which can be seen as in the proof of Proposition4.1(5). By Lemma 4.5 it coincides with C ( M ) ( U, F ) as vector space.Obviously the identity is continuous from left to right. The space on the right-hand side is as a projective limit of Banach spaces convenient and C ( M ) ( U, F )satisfies the uniform boundedness principle, i.e., Theorem 6.1, with respect to theset of point evaluations. Thus the identity from right to left is bounded. (cid:3)
Theorem 8.6.
Let E and F be Banach spaces and let U ⊆ E be open and convex.Then we have C { M } ( U, F ) = lim ←− K, ( r k ) C M ( r k ) ( E ⊇ K, F ) as vector spaces with bornology. Here K runs through all compact convex subsets of U ordered by inclusion and ( r k ) runs through all sequences of positive real numbersfor which ρ k /r k → for all ρ > . Proof.
The proof is literally identical with the proof of Theorem 8.5, where wereplace C ( M ) with C { M } and use Lemma 4.6 instead of Lemma 4.5. (cid:3) Remark.
Let us prove that the identity lim ←− K, ( r k ) C M ( r k ) ( E ⊇ K, F ) → C { M } ( U, F )is bounded without using the C { M } uniform boundedness principle, i.e., Theorem6.1: Let B be a bounded set in lim ←− K, ( r k ) C M ( r k ) ( E ⊇ K, F ), i.e., for each compact K and each ( r k ) with ρ k /r k → ρ > B is bounded in C M ( r k ) ( E ⊇ K, F ),i.e., sup {k f | K k ( r k ) : f ∈ B} < ∞ . Since the elements of lim ←− K, ( r k ) C M ( r k ) ( E ⊇ K, F ) are the infinite jets of smoothfunctions, we may estimate ||| f | K ||| n,k by k f | K k n + k +1 by (12), and so the sequence a k := sup n k f | K k k k ! M k : f ∈ B o < ∞ satisfies sup k a k /r k < ∞ for each ( r k ) as above. By [23, 9.2], these are the coeffi-cients of a power series with positive radius of convergence. Thus a k /ρ k is boundedfor some ρ >
0. That means that B is contained and bounded in C Mρ ( E ⊇ K, F ).This also provides an independent proof of the completeness of C { M } ( U, F ) andof the regularity of the involved inductive limit (cf. Proposition 4.1 and Remark5.3).
Lemma 8.7.
For convenient vector spaces E , F , G , and c ∞ -open V ⊆ F the flip ofvariables induces an isomorphism L ( E, C [ M ] ( V, G )) ∼ = C [ M ] ( V, L ( E, G )) as vectorspaces. Proof.
For f ∈ C [ M ] ( V, L ( E, G )) consider ˜ f ( x ) := ev x ◦ f ∈ C [ M ] ( V, G ) for x ∈ E .By the uniform boundedness principle, i.e., Theorem 6.1, the linear mapping ˜ f isbounded, since ev y ◦ ˜ f = f ( y ) ∈ L ( E, G ) for y ∈ V .If conversely ℓ ∈ L ( E, C [ M ] ( V, G )), we consider ˜ ℓ ( y ) = ev y ◦ ℓ ∈ L ( E, G ) for y ∈ V . Since the bornology of L ( E, G ) (see Proposition 8.4) is generated by S := { ev x : x ∈ E } and since ev x ◦ ˜ ℓ = ℓ ( x ) ∈ C [ M ] ( V, G ), it follows that ˜ ℓ : V → L ( E, G )is C [ M ] , by Lemma 5.1 (and by composing with all i B : V B → V ). (cid:3) Lemma 8.8.
Let E be a convenient vector space and let U ⊆ E be c ∞ -open. By λ [ M ] ( U ) we denote the c ∞ -closure of the linear subspace generated by { ev x : x ∈ U } in C [ M ] ( U, R ) ′ and let δ : U → λ [ M ] ( U ) be given by x ev x . Then λ [ M ] ( U ) is thefree convenient vector space over C [ M ] , i.e., for every convenient vector space G the C [ M ] -mapping δ induces a bornological isomorphism L ( λ [ M ] ( U ) , G ) ∼ = C [ M ] ( U, G ) . Proof.
The proof goes along the same lines as in [11, 5.1.1] and [23, 23.6]. Notefirst that λ [ M ] ( U ) is a convenient vector space, since it is c ∞ -closed in the con-venient vector space C [ M ] ( U, R ) ′ . Moreover, δ is C [ M ] , by Lemma 5.1 (and bycomposing with all i B : U B → U ), since ev h ◦ δ = h for all h ∈ C [ M ] ( U, R ). So δ ∗ : L ( λ [ M ] ( U ) , G ) → C [ M ] ( U, G ) is a well-defined linear mapping. This mapping isinjective, since each bounded linear mapping λ [ M ] ( U ) → G is uniquely determinedon δ ( U ) = { ev x : x ∈ U } . Let now f ∈ C [ M ] ( U, G ). Then ℓ ◦ f ∈ C [ M ] ( U, R ) forevery ℓ ∈ G ∗ , and hence ˜ f : C [ M ] ( U, R ) ′ → Q G ∗ R given by ˜ f ( φ ) = ( φ ( ℓ ◦ f )) ℓ ∈ G ∗ is a well-defined bounded linear mapping. Since it maps ev x to ˜ f (ev x ) = δ ( f ( x )),where δ : G → Q G ∗ R denotes the bornological embedding given by y ( ℓ ( y )) ℓ ∈ G ∗ ,it induces a bounded linear mapping ˜ f : λ [ M ] ( U ) → G satisfying ˜ f ◦ δ = f . Thus δ ∗ is a linear bijection. That it is a bornological isomorphism follows from the uniformboundedness principle, i.e., Theorem 6.1, and from Proposition 8.4. (cid:3) Theorem 8.9 (Canonical isomorphisms) . Let M = ( M k ) and N = ( N k ) be positivesequences. Let E , F be convenient vector spaces and let W i be c ∞ -open subsets insuch. We have the following natural bornological isomorphisms: (1) C ( M ) ( W , C ( N ) ( W , F )) ∼ = C ( N ) ( W , C ( M ) ( W , F )) , ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 35 (2) C { M } ( W , C { N } ( W , F )) ∼ = C { N } ( W , C { M } ( W , F )) , (3) C ( M ) ( W , C { N } ( W , F )) ∼ = C { N } ( W , C ( M ) ( W , F )) , (4) C [ M ] ( W , C ∞ ( W , F )) ∼ = C ∞ ( W , C [ M ] ( W , F )) . (5) C [ M ] ( W , C ω ( W , F )) ∼ = C ω ( W , C [ M ] ( W , F )) . (6) C [ M ] ( W , L ( E, F )) ∼ = L ( E, C [ M ] ( W , F )) . (7) C [ M ] ( W , ℓ ∞ ( X, F )) ∼ = ℓ ∞ ( X, C [ M ] ( W , F )) . (8) C [ M ] ( W , L ip k ( X, F )) ∼ = L ip k ( X, C [ M ] ( W , F )) .In (7) the space X is an ℓ ∞ -space, i.e., a set together with a bornology inducedby a family of real valued functions on X , cf. [11, 1.2.4] . In (8) the space X is a L ip k -space, cf. [11, 1.4.1] . The spaces ℓ ∞ ( X, F ) and L ip k ( X, F ) are defined in [11,3.6.1 and 4.4.1] . Proof.
Let C and C denote any of the functions spaces mentioned above and X and X the corresponding domains. In order to show that the flip of coordinates f ˜ f , C ( X , C ( X , F )) → C ( X , C ( X , F )) is a well-defined bounded linearmapping we have to show: • ˜ f ( x ) ∈ C ( X , F ), which is obvious, since ˜ f ( x ) = ev x ◦ f : X →C ( X , F ) → F . • ˜ f ∈ C ( X , C ( X , F )), which we will show below. • f ˜ f is bounded and linear, which follows by applying the appropriateuniform boundedness theorems for C and C , since f ev x ◦ ev x ◦ ˜ f =ev x ◦ ev x ◦ f is bounded and linear.All occurring function spaces are convenient and satisfy the uniform S -boundednesstheorem, where S is the set of point evaluations: C [ M ] by Section 4.2 and Theorem 6.1. C ∞ by [23, 2.14.3 and 5.26] C ω by [23, 11.11 and 11.12] or by Theorems 6.1 and 7.1, L by [23, 2.14.3 and 5.18] ℓ ∞ by [23, 2.15, 5.24, and 5.25] or [11, 3.6.1 and 3.6.6] L ip k by [11, 4.4.2 and 4.4.7]It remains to check that ˜ f is of the appropriate class:(1)–(4) For α ∈ { ( M ) , { M }} and β ∈ { ( N ) , { N } , ∞} we have C α ( W , C β ( W , F )) ∼ = L ( λ α ( W ) , C β ( W , F )) by Lemma 8.8 ∼ = C β ( W , L ( λ α ( W ) , F )) by Lemma 8.7, [23, 3.13.4 and 5.3] ∼ = C β ( W , C α ( W , F )) by Lemma 8.8.(5) follows from (2), (3), and Theorem 7.1.(6) is exactly Lemma 8.7.(7) follows from (6), using the free convenient vector spaces ℓ ( X ) over the ℓ ∞ -space X , see [11, 5.1.24 or 5.2.3], satisfying ℓ ∞ ( X, F ) ∼ = L ( ℓ ( X ) , F ).(8) follows from (6), using the free convenient vector spaces λ k ( X ) overthe L ip k -space X , see [11, 5.1.24 or 5.2.3], satisfying L ip k ( X, F ) ∼ = L ( λ k ( X ) , F ). (cid:3) Manifolds of C [ M ] -mappings Hypothesis.
In this section we assume that M = ( M k ) is log-convex andhas moderate growth. In the Beurling case C [ M ] = C ( M ) we also require that C ω ⊆ C ( M ) , equivalently, M /kk → ∞ or M k +1 /M k → ∞ .For the equivalence of C ω ⊆ C ( M ) and M /kk → ∞ , see Proposition 8.1(4). More-over, M /kk → ∞ implies M k +1 /M k → ∞ , since M /kk is increasing, by log-convexity(see Section 2.1), and thus M k +1 /M k ≥ M /kk . Conversely, if M k +1 /M k → ∞ thenfor each n ∈ N there is k n so that M k /M k − ≥ n for all k ≥ k n . It follows that M k /M k n − ≥ n k − k n +1 and thus M /kk → ∞ . This is needed for the C ( M ) inversefunction theorem (see Sections 2.1 and 9.2).9.2. Tools for C [ M ] -analysis. We collect here results which are needed below (seealso Section 2.1):(1) On open sets in R n , C [ M ] -vector fields have C [ M ] -flows, see [18] and [40].(2) Between Banach spaces, the C [ M ] implicit function theorem holds. This isessentially due to [39], but in [39] only the Roumieu case is treated and the C { M } -conditions are global. So we shall indicate briefly how to obtain theresult we need (cf. [32]): Theorem.
Let M = ( M k ) be log-convex. In the Beurling case C [ M ] = C ( M ) wealso assume M k +1 /M k → ∞ . Let E , F be Banach spaces, U ⊆ E , V ⊆ F open,and f : U → V a C ∞ -diffeomorphism. We have: (3) Let K ⊆ U be compact. If f ∈ C [ M ] K ( U, F ) then f − ∈ C [ M ] f ( K ) ( V, E ) . (4) If f ∈ C [ M ] ( U, F ) then f − ∈ C [ M ] ( V, E ) . Proof.
By Proposition 4.1(5), (3) implies (4). The proof of [39, Thm. 2] withsmall obvious modifications provides a proof of (3) in the Roumieu case (see also[36, 3.4.5]).For the Beurling case let f ∈ C ( M ) K ( U, F ) and L k := 1 k ! sup x ∈ K k f ( k ) ( x ) k L k ( E,F ) . Then L ✁ M and since M k +1 /M k → ∞ there exists a log-convex sequence N = ( N k )satisfying N k +1 /N k → ∞ and such that L ≤ N ✁ M , by [16, Lemma 6]. Thus, f ∈ C { N } K ( U, F ) and, by the Roumieu case, f − ∈ C { N } f ( K ) ( V, E ). Since N ✁ M , wehave f − ∈ C ( M ) f ( K ) ( V, E ), by Proposition 8.1. (cid:3)
The C [ M ] implicit function theorem follows in the standard way.9.3. C [ M ] -manifolds. A C [ M ] -manifold is a smooth manifold such that all chartchangings are C [ M ] -mappings. They will be considered with the topology inducedby the c ∞ -topology on the charts. Likewise for C [ M ] -bundles and C [ M ] Lie groups.A mapping between C [ M ] -manifolds is C [ M ] if and only if it maps C [ M ] -plots (i.e., C [ M ] -mappings from open sets (or unit balls) of Banach spaces into the domainmanifold) to such.Note that any finite dimensional (always assumed paracompact) C ∞ -manifoldadmits a C ∞ -diffeomorphic real analytic structure thus also a C [ M ] -structure. ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 37
Maybe, any finite dimensional C [ M ] -manifold admits a C [ M ] -diffeomorphic realanalytic structure. This would follow from: Conjecture.
Let X be a finite dimensional real analytic manifold. Consider thespace C [ M ] ( X, R ) of all C [ M ] -functions on X , equipped with the (obvious) Whitney C [ M ] -topology. Then C ω ( X, R ) is dense in C [ M ] ( X, R ) . This conjecture is the analogue of [13, Proposition 8]. It was proved in thenon-quasianalytic Beurling case C ( M ) for X open in R n by [28].The proofs of the following results are similar to the proofs given in [26, Section5], using other analytical tools. For the convenience of the reader, we give fullproofs here, sometimes with more details.9.4. Spaces of C [ M ] -sections. Let p : E → B be a C [ M ] vector bundle (possiblyinfinite dimensional). The space C [ M ] ( B ← E ) of all C [ M ] -sections is a convenientvector space with the structure induced by C [ M ] ( B ← E ) → Y α C [ M ] ( u α ( U α ) , V ) s pr ◦ ψ α ◦ s ◦ u − α where B ⊇ U α − u α → u α ( U α ) ⊆ W is a C [ M ] -atlas for B which we assume to bemodeled on a convenient vector space W , and where ψ α : E | U α → U α × V form avector bundle atlas over charts U α of B . Lemma.
Assume Hypothesis 9.1. Let D be the open unit ball in a Banach space.A mapping c : D → C [ M ] ( B ← E ) is a C [ M ] -plot if and only if c ∧ : D × B → E is C [ M ] . Proof.
By the description of the structure on C [ M ] ( B ← E ) we may assume byLemma 5.1 that B is c ∞ -open in a convenient vector space W and that E = B × V .Then we have C [ M ] ( B ← B × V ) ∼ = C [ M ] ( B, V ). Thus the statement follows fromthe exponential law, i.e., Theorem 5.2. (cid:3)
Let U ⊆ E be an open neighborhood of s ( B ) for a section s and let q : F → B be another vector bundle. The set C [ M ] ( B ← U ) of all C [ M ] -sections s ′ : B → E with s ′ ( B ) ⊆ U is c ∞ -open in the convenient vector space C [ M ] ( B ← E ) if B iscompact and thus finite dimensional, since then it is open in the coarser compact-open topology. An immediate consequence of the lemma is the following: If U ⊆ E is an open neighborhood of s ( B ) for a section s and if f : U → F is a fiber respecting C [ M ] -mapping where F → B is another vector bundle, then f ∗ : C [ M ] ( B ← U ) → C [ M ] ( B ← F ) is C [ M ] on the open neighborhood C [ M ] ( B ← U ) of s in C [ M ] ( B ← E ). We have ( d ( f ∗ )( s ) v ) x = d ( f | U ∩ E x )( s ( x ))( v ( x )). Theorem 9.1.
Assume Hypothesis 9.1. Let A and B be finite dimensional C [ M ] -manifolds with A compact and B equipped with a C [ M ] Riemann metric. Thenthe space C [ M ] ( A, B ) of all C [ M ] -mappings A → B is a C [ M ] -manifold modeled onconvenient vector spaces C [ M ] ( A ← f ∗ T B ) of C [ M ] -sections of pullback bundlesalong f : A → B . Moreover, a mapping c : D → C [ M ] ( A, B ) is a C [ M ] -plot if andonly if c ∧ : D × A → B is C [ M ] . If the C [ M ] -structure on B is induced by a real analytic structure, then thereexists a real analytic Riemann metric which in turn is C [ M ] . Proof. C [ M ] -vector fields have C [ M ] -flows by Section 9.2; applying this to thegeodesic spray we get the C [ M ] exponential mapping exp : T B ⊇ U → B of theRiemann metric, defined on a suitable open neighborhood of the zero section. Wemay assume that U is chosen in such a way that ( π B , exp) : U → B × B is a C [ M ] -diffeomorphism onto an open neighborhood V of the diagonal, by the C [ M ] inverse function theorem, see Section 9.2.For f ∈ C [ M ] ( A, B ) we consider the pullback vector bundle A × T B A × B T B ? _ o o f ∗ T B π ∗ B f / / f ∗ π B (cid:15) (cid:15) T B π B (cid:15) (cid:15) A f / / B Then the convenient space of sections C [ M ] ( A ← f ∗ T B ) is canonically isomorphicto the space C [ M ] ( A, T B ) f := { h ∈ C [ M ] ( A, T B ) : π B ◦ h = f } via s ( π ∗ B f ) ◦ s and (Id A , h ) ← h . Now let U f := { g ∈ C [ M ] ( A, B ) : ( f ( x ) , g ( x )) ∈ V for all x ∈ A } ,u f : U f → C [ M ] ( A ← f ∗ T B ) ,u f ( g )( x ) = ( x, exp − f ( x ) ( g ( x ))) = ( x, (( π B , exp) − ◦ ( f, g ))( x )) . Then u f : U f → { s ∈ C [ M ] ( A ← f ∗ T B ) : s ( A ) ⊆ f ∗ U = ( π ∗ B f ) − ( U ) } is a bijectionwith inverse u − f ( s ) = exp ◦ ( π ∗ B f ) ◦ s , where we view U → B as a fiber bundle. Theset u f ( U f ) is c ∞ -open in C [ M ] ( A ← f ∗ T B ) for the topology described above inSection 9.4, since A is compact and the push forward u f is C [ M ] , since it respects C [ M ] -plots, by the lemma in Section 9.4.Now we consider the atlas ( U f , u f ) f ∈ C [ M ] ( A,B ) for C [ M ] ( A, B ). Its chart changemappings are given for s ∈ u g ( U f ∩ U g ) ⊆ C [ M ] ( A ← g ∗ T B ) by( u f ◦ u − g )( s ) = (Id A , ( π B , exp) − ◦ ( f, exp ◦ ( π ∗ B g ) ◦ s ))= ( τ − f ◦ τ g ) ∗ ( s ) , where τ g ( x, Y g ( x ) ) := ( x, exp g ( x ) ( Y g ( x ) )) is a C [ M ] -diffeomorphism τ g : g ∗ T B ⊇ g ∗ U → ( g × Id B ) − ( V ) ⊆ A × B which is fiber respecting over A . The chart change u f ◦ u − g = ( τ − f ◦ τ g ) ∗ is defined on an open subset and it is also C [ M ] , since itrespects C [ M ] -plots, by the lemma in Section 9.4.Finally for the topology on C [ M ] ( A, B ) we take the identification topology fromthis atlas (with the c ∞ -topologies on the modeling spaces C [ M ] ( A ← f ∗ T B )), whichis obviously finer than the compact-open topology and thus Hausdorff.The equation u f ◦ u − g = ( τ − f ◦ τ g ) ∗ shows that the C [ M ] -structure does notdepend on the choice of the C [ M ] Riemannian metric on B .The statement on C [ M ] -plots follows from the lemma in Section 9.4. (cid:3) Corollary 9.2.
Assume Hypothesis 9.1. Let A , A and B be finite dimensional C [ M ] -manifolds with A and A compact. Then composition C [ M ] ( A , B ) × C [ M ] ( A , A ) → C [ M ] ( A , B ) , ( f, g ) f ◦ g is C [ M ] . ENJOY–CARLEMAN MAPPINGS OF BEURLING AND ROUMIEU TYPE 39
Proof.
Composition maps C [ M ] -plots to C [ M ] -plots, so it is C [ M ] . (cid:3) Example 9.3.
The result in Corollary 9.2 is best possible in the following sense:If N = ( N k ) is another weakly log-convex sequence such that C [ N ] ( C [ M ] (orequivalently, inf( N k /M k ) /k = 0 and sup( N k /M k ) /k < ∞ ), then composition C [ M ] ( S , R ) × C [ M ] ( S , S ) → C [ M ] ( S , R ) , ( f, g ) f ◦ g is not C [ N ] with respect to the canonical real analytic manifold structures.Namely, there exists f ∈ C [ M ] ( S , R ) \ C [ N ] ( S , R ). We consider f as a periodicfunction R → R . The universal covering space of C [ M ] ( S , S ) consists of all 2 π Z -equivariant mappings in C [ M ] ( R , R ), namely the space of all g + Id R for 2 π -periodic g ∈ C [ M ] . Thus C [ M ] ( S , S ) is a real analytic manifold and t ( x x + t )induces a real analytic curve c in C [ M ] ( S , S ). But f ∗ ◦ c is not C ( N ) (resp. C { N } )since: ( ∂ kt | t =0 ( f ∗ ◦ c )( t ))( x ) k ! ρ k N k = ∂ kt | t =0 f ( x + t ) k ! ρ k N k = f ( k ) ( x ) k ! ρ k N k which is unbounded in k for x in a suitable compact set and for some (resp. all) ρ >
0, since f / ∈ C ( N ) (resp. f / ∈ C { N } ). Theorem 9.4.
Assume Hypothesis 9.1. Let A be a compact (thus finite dimen-sional) C [ M ] -manifold. Then the group Diff [ M ] ( A ) of all C [ M ] -diffeomorphisms of A is an open subset of the C [ M ] -manifold C [ M ] ( A, A ) . Moreover, it is a C [ M ] -regular C [ M ] Lie group: Inversion and composition are C [ M ] . Its Lie algebra consists of all C [ M ] -vector fields on A , with the negative of the usual bracket as Lie bracket. Theexponential mapping is C [ M ] . It is not surjective onto any neighborhood of Id A . Following [24], see also [23, 38.4], a C [ M ] -Lie group G with Lie algebra g = T e G is called C [ M ] -regular if the following holds: • For each C [ M ] -curve X ∈ C [ M ] ( R , g ) there exists a C [ M ] -curve g ∈ C [ M ] ( R , G ) whose right logarithmic derivative is X , i.e., ( g (0) = e∂ t g ( t ) = T e ( µ g ( t ) ) X ( t ) = X ( t ) .g ( t )The curve g is uniquely determined by its initial value g (0), if it exists. • Put evol rG ( X ) = g (1), where g is the unique solution required above. Thenevol rG : C [ M ] ( R , g ) → G is required to be C [ M ] also. Proof.
The group Diff [ M ] ( A ) is c ∞ -open in C [ M ] ( A, A ), since the C ∞ -diffeomor-phism group Diff( A ) is c ∞ -open in C ∞ ( A, A ), by [23, 43.1], and since Diff [ M ] ( A ) =Diff( A ) ∩ C [ M ] ( A, A ), by Section 9.2. So Diff [ M ] ( A ) is a C [ M ] -manifold and compo-sition is C [ M ] , by Theorem 9.1 and Corollary 9.2. To show that inversion is C [ M ] let c be a C [ M ] -plot in Diff [ M ] ( A ). By Theorem 9.1, the mapping c ∧ : D × A → A is C [ M ] and (inv ◦ c ) ∧ : D × A → A satisfies the Banach manifold implicit equation c ∧ ( t, (inv ◦ c ) ∧ ( t, x )) = x for x ∈ A . By the Banach C [ M ] implicit function theorem,see Section 9.2, the mapping (inv ◦ c ) ∧ is locally C [ M ] and thus C [ M ] . By Theorem9.1 again, inv ◦ c is a C [ M ] -plot in Diff [ M ] ( A ). So inv : Diff [ M ] ( A ) → Diff [ M ] ( A )is C [ M ] . The Lie algebra of Diff [ M ] ( A ) is the convenient vector space of all C [ M ] -vector fields on A , with the negative of the usual Lie bracket (compare with theproof of [23, 43.1]). To show that Diff [ M ] ( A ) is a C [ M ] -regular Lie group, we choose a C [ M ] -plot inthe space of C [ M ] -curves in the Lie algebra of all C [ M ] vector fields on A , thatis c : D → C [ M ] ( R , C [ M ] ( A ← T A )). By the lemma in Section 9.4, the plot c corresponds to a ( D × R )-time-dependent C [ M ] vector field c ∧∧ : D × R × A → T A .Since C [ M ] -vector fields have C [ M ] -flows and since A is compact, evol r ( c ∧ ( s ))( t ) =Fl c ∧ ( s ) t is C [ M ] in all variables, by Section 9.2. Thus Diff [ M ] ( A ) is a C [ M ] -regular C [ M ] Lie group.The exponential mapping is evol r applied to constant curves in the Lie algebra,i.e., it consists of flows of autonomous C [ M ] vector fields. That the exponentialmapping is not surjective onto any C [ M ] -neighborhood of the identity follows from[23, 43.5] for A = S . This example can be embedded into any compact manifold,see [12]. (cid:3) References [1] E. Bierstone and P. D. Milman,
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E-mail address : [email protected] Peter W. Michor: Fakult¨at f¨ur Mathematik, Universit¨at Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
E-mail address : [email protected] Armin Rainer: Fakult¨at f¨ur Mathematik, Universit¨at Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
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