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Integrability of weak distributions on Banach manifolds

This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We give some applications to Banach Lie algebroid and Banach Lie-Poisson manifold. The main results of this paper generalize the works presented in [ChSt], [Nu] and [Gl].

of "local triviality", our result can be seen like generalization as well of results of [Su] and [St], than of results of [Nu] and [Gl]. Note that, our proofs take in account Remarks of Balan in [Ba], about results of [Su] and [St] (see Obervations 3.4) The first section contains the most important definitions and properties about weak distributions. It contains also the first result of equivalence between integrability and invariance (Theorem 1), under local lower triviality assumption. This last property is, in fact, a generalization of the classical notion of "smoothness" for distributions (see Observations 2.7). In the second section, we adapt the arguments used in [ChSt] to our context: under condition of "Lie invariance", we give a generalization of their results about the integrability of distributions (Theorem 2). In the second part, under the assumption of "upper triviality" (which is a general context for anchored bundles), we give some conditions of "local involutivity" which gives rise to an integrability property (Theorem 4). In the last section, we give some applications of these results in the context of Banach Lie algebroids and Banach Poisson manifold ( cf [OdRa1] and [OdRa2]) .

Integrability and invariance 2.1 Preliminaries and context
Let M be a smooth connected Banach manifold modelled on a Banach space E. We denote by C ∞ (M ) the ring of smooth functions on M and by X (M ) the Lie algebra of smooth vector fields on M . A local vector field X on M is a smooth section of the tangent bundle T M defined on an open set of M (denoted by Dom(X)). Let be X L (M ) the set of all local vector fields on M . Such a vector field X ∈ X L (M ) has a flow φ X t which is defined on a maximal open set Ω X of M × R. Using the terminology introduced in [El] or [Pe], a weak submanifold of M is a pair (N, f ) where N is Banach manifold modelled on a Banach space F and f : N → M a smooth map such that: 1. there exists an injective continuous linear map i : F → E between these two Banach spaces; 2. f is injective and the tangent map T x f : T x N → T f (x) M is injective for all x ∈ N .

Remark 2.1
Given a weak submanifold f : N → M , on the subset f (N ) in M we have two topologies: 1. the induced topology from M 2. the topology for which f is a homeomorphism from N to f (N ).
With this last topology, via f , we get a structure of Banach manifold modelled on F . Moreover, the inclusion from f (N ) into M is continuous as map from the Banach manifold f (N ) to M . In particular, if U is open in M , then, f (N )∩U is an open set for the topology of the Banach manifold on f (N ). Note that in [El] and [Pe] the definition of a "weak manifold" imposes that these two topologies are identical. Our definition is somewhat different and is motivated by the notion of "weak immersion" introduced in [OdRa1] and [OdRa2] In this work, a weak distribution on M is a map D : x → D x which, for every x ∈ M , associates a vector subspace D x in T x M (not necessarily closed) endowed with a norm || || x so that (D x , || || x ) is a Banach space (denoted byD x ) and such that the natural inclusion i x :D x → T x M is continuous.

Remark 2.2
When D x is closed, via any chart, we get a norm on T x M which induces a Banach structure on D x . So if D x is closed for all x ∈ M , the previous assumption on the Banach structureD x is always satisfied, and we get the usual definition of a distribution on M (compare with [Gl], [ChSt], [Nu]). In this last situation we always endowD x with this induced Banach structure and we say that D is closed.

Examples 2.3
1. Let l p (resp. l ∞ ) be the Banach space of real sequences (x k ) such that ∞ k=1 |x k | p < ∞ (resp. absolutely bounded) and denote by I p the natural inclusion of l 1 in l p , p > 1 or p = ∞. On the Banach space l p , x → D x = x + I p (l 1 ) is a weak distribution which is not closed.
2. Let E and F be two Banach spaces and T : F → E a continuous operator. Denote bŷ T : F/ ker T → E the canonical quotient bijection associated to T that is We can endow T (F ) with the structure of Banach space such thatT is an isometry. On E, the assignment x → D x = x + T (F ) is a weak distribution. This distribution is closed if and only if T (F ) is closed in E.
3. Let L(F, E) be the set of continuous operators between the Banach spaces F and E. Given a smooth map Ψ : E → L(F, E), we denote by Ψ x the continuous operator associated to x ∈ E. As in (1) denote byΨ x the canonical bijection associated Ψ x and we endow D x = Ψ x (F ) with the Banach structure such thatΨ x is an isometry. Then, x → D x is a weak distribution on E which is closed if and only if D x is closed for any x ∈ E.
A vector field Z ∈ X (M ) is tangent to D, if for all x ∈ Dom(Z), Z(x) belongs to D x . The set of local vector fields tangent to D will be denoted by X D . We say that D is is generated by a subset X ⊂ X L (M ) if, for every x ∈ M , the vector space D x is the linear hull of the set {Y (x) , Y ∈ X , x ∈ Dom(Y )}.
For a weak distribution D on M , we have the following classical properties: • an integral manifold of D through x is a weak submanifold f : N → M such that f (x) = x for somex ∈ N and and Tỹf (TỹN ) = D f (ỹ) for allỹ ∈ N .
• D is called integrable if for any x ∈ M there exists an integral manifold N of D through x.
• if D is generated by X ⊂ X L (M ), then D is called X -invariant if for any X ∈ X , the tangent map T x φ X t sends D x onto D φ X t (x) for all (x, t) ∈ Ω X . If D is X D − invariant we simply say that D is invariant. Now, we introduce two properties of "local triviality" which will play an essential role in the whole paper: • D is (locally) lower trivial (lower trivial for short) if, for each x ∈ M , there exists an open neighbourhood V of x, a smooth map Θ :D x × V → T M (called lower trivialization) such that : there exists a continuous operatorΘ y :D x →D y such that i y •Θ y = Θ y ,Θ y is an isomorphism fromD x onto Θ y (D x ) andΘ x is the identity ofD x • D is called (locally) upper trivial (upper trivial for short) if, for each x ∈ M , there exists an open neighbourhood V of x, a Banach space F and a smooth map Ψ : F × V → T M (called upper trivialization) such that : In this case the map Θ is called the associated lower trivialization.
It is easy to see that D and D ′ are weak distributions on E. Then D is lower trivial but not upper trivial and D ′ is neither lower trivial, nor upper trivial.
For an illustration of the property of lower local triviality and upper local triviality in a more large context, we give the following result, which is a generalization of Example 2.3 n 0 3 Proposition 2.5 Let D : x → D x ⊂ T x M be a field on M of normed subspaces. Suppose that for each x ∈ M , there exists an open neighbourhood V of x, a Banach space F and a smooth map Ψ : F × V → T M such that : for each y ∈ V , Ψ y ≡ Ψ( , y) : F → T y M is a continuous operator such that Ψ y (F ) = D y Then, we have the following properties: 1. D x has a natural structure of Banach space (again denoted byD x ) such that the canonical continuous operatorΨ x : F/ ker Ψ x →D x is an isometry, in particular, D is a weak distribution.
2. There exists a neighbourhood W of x and, for any y ∈ W , a continuous surjective operator Ψ y : F →D y such that i y •Ψ y = Ψ y , where i y :D y → T y M is the natural inclusion.
3. Assume that ker Ψ x is complemented (i.e. F = ker Ψ x ⊕ S). Then there exists an open neighbourhood W of x such that the restriction θ y of Ψ y to S is injective for any y ∈ W , and then Θ(u, y) = (θ y • [θ x ] −1 (u), y) is a lower trivialization of D. So, D is upper trivial, and Ψ is an upper trivialization of D.
The context of Proposition 2.5 can be found in the framework of Banach Poisson manifold (M, { , }) where Ψ : T * M → T M ⊂ T * * M is the canonical morphism associated to the Poisson structure (see for instance [OdRa1] and [OdRa2]).

Corollary 2.6
Let π : F → M be a Banach fiber bundle over M with typical fiber F and Ψ : F → T M a morphism of bundle. If the kernel of Ψ is complemented in each fiber, then D is an upper trivial weak distribution.

Observations 2.7
1. Recall the definition of "differentiability" (resp. "smoothness") of a closed and complemented distribution D, introduced in [ChSt] (resp. [Nu]): there exists some neighbourhood V of x ∈ M on which T M is trivializable and there exists on V a smooth field y →Θ y of For a closed distribution, we haveD ≡ D and so Θ(u, y) =Θ y (u) (u, y) ∈ D x × V is a lower trivialization and so, the property of lower triviality is nothing but a generalization of "differentiability". Note that, the property of "smoothness" for a closed distributions, introduced in [Gl] and [Nu], is also a generalization of the property of "differentiability" given in [ChSt] in the same way.
2. Consider an anchored bundle on M , that is a Banach bundle π : F → M over M so that we have a bundle morphism Ψ : F → T M . In this situation, we are going to see that the distribution D = Ψ(F ) is "continuous local lower trivial" in some sense, which is quite different of our previous definition. At first, recall that, from [Mi], if p : F → F/K is the canonical projection of a Banach space F onto the Banach quotient F/K , there always exists a continuous selection σ : F/K → F such that σ(λu) = λσ(u) for any λ ∈ R and u ∈ F/K. However, σ is not linear. Of course, there exists such a linear selection σ, if and only if K is complemented in F . Come back to the situation of an anchored bundle Ψ : F → T M . We fix some x ∈ M , and choose a continuous selection θ Then θ is continuous and in general not linear in the first variable, and is smooth in the second variable. So, we can consider θ as a kind of local lower trivialization which, in general is only continuous and not linear on the fiber D x .
On the other hand, this construction can be considered as a kind of "smoothness" of D: via such a (local) map θ, each u ∈ D x can be extended to a smooth vector field which is tangent to D (compare with [OdRa1] p 35).
Of course if θ x is linear, we get a lower trivialization associated to Ψ (see the proof of Proposition 2.5).
However, in this context, our criteria of integrability works only when we have a lower trivialization associated to an upper trivialization. For this reason, we have introduced the property of upper triviality.
3. In finite dimension, the definition of "differentiability" given in [St] is the same definition as that given in [ChSt]. On the other hand, in [Su], a distribution is called smooth, if there exists a subset X which generates D. In this context, for any x ∈ M there exists a neighbourhood V of x and vector fields X 1 , · · · , X p which are defined and linearly independent on V such that {X 1 (x), · · · , X p (x)} is a basis of D x so D is lower trivial. On the other hand, for any x ∈ M , denote by X Dx the module of germs vector fields at x which are tangent to D. If X Dx is finitely generated, then D is upper trivial.
We end this section with the proof of Proposition 2.5 and its Corollary. For this proof, we need the following Lemma which will be also used later: Lemma 2.8 1. Consider two Banach spaces E 1 and E 2 and i : E 1 → E 2 an injective continuous operator.
Let y → Θ y be a smooth field of continuous operators of L(E 1 , E 2 ) on an open neighbourhood V of x ∈ E 1 such that Θ x = i. Then there exists a neighbourhood W of x in V such that for each y ∈ W , Θ y is an injective operator.
2. Let f : U → V be a C 1 map from two open sets U and V in Banach spaces E 1 and E 2 respectively, such that T u f is injective at u ∈ U . Then there exists an open neighbourhood W of u in U such that the restriction of f to W is injective.
Proof of Lemma 2.8 There exists an open ball B(x, r) included in V such that ||Θ y −Θ x || ≤ K||y−x]] for any y ∈ B(x, r). We can suppose that r < 1. Assume that the conclusion of Lemma 2.8 (1) is not true. So, for each n ∈ N * , there exists x n ∈ B(x, r/n) and h n ∈ E 1 with ||h n || = 1 such that Θ xn (h n ) = 0. We have of course: for all α ∈ E * 2 . It follows that we have: On the other hand, Θ x = i is a continuous bijective operator from the Banach space E 1 onto the normed subspace F = i(E 1 ) in E 2 . So, the transpose operator i * ∈ L(F * , E * 1 ) is a monomorphism 1 with a dense range (see [Ha], [HaMb]). According to Hahn-Banach Theorem, there exists β n ∈ E * 1 such that < β n , h n >= 1 with ||β n || = 1. From the density of i * (F * ), there exists α n ∈ F * such . From these inequalities we get: • as ||i * (α n )|| ≤ 5 4 and as i * is a monomorphism, we have ||i * (α n )|| ≥ k||α n || for some k > 0 and finally we get ||α n || ≤ 5 4k .
On the other hand we can write: for any n. From Hahn-Banach Theorem, we obtain the same relation (4) with α n ∈ E * 2 . But from (3) we get: 4k for any n. So we get a contradiction with (4) for n large enough. So we have completed the proof of the part (1).
Let f : U → V be a map of class C 1 . As the problem is local, without loss of generality, we can suppose that U is an open ball of center 0 ∈ E 1 . As f is of class C 1 , there exists an open ball B(0, r) such that Moreover, we can choose r so that r < 1.
From our assumption it follows that, for each n ∈ N * , there exists u n and v n in B(x, r/n) so that u n = v n but with f (u n ) = f (v n ). So from the previous argument, for any α ∈ E * 2 , we have for some u α,n ∈]u n , v n [ and with h n = v n − u n ||v n − u n || From (5) and (7), we get for any α ∈ E * 2 . Now, we can use the same argument as in part (1) and we get again a contradiction. △

Proof of Proposition 2.5
At first, for any x ∈ M , we have a natural Banach structure on D x (again denoted byD x ) such that the natural morphismΨ x : F/ ker Ψ x →D x is an isometry. On the other hand, take a local trivialization of T M on a neighbourhood W of x; so we have T M ≡ E × W . In this context, on W , Ψ can be identified with a smooth field of continuous operators Ψ y : F → E such that D y = Ψ y (F ) × {y} ⊂ E × {y} ≡ T y M . Let us consider the following commutative diagram: where q is the natural projection andΨ y is the natural bijection induced by Ψ y . So, if we consider the Banach structureD y , we get a continuous operatorΨ y =Ψ y • q : F →D y so that Ψ y = i y •Ψ.
Assume that F = ker Ψ x ⊕ S, for some Banach space S ⊂ F . Let θ y be the restriction to S of Ψ y for any y ∈ W . Clearly, θ(u, y) = (θ y (u), y) defines a smooth map from On the other hand, letθ y be the restriction ofΨ y to S, then,θ y is a continuous operator from S toD y so that θ y = i y •θ y for any y ∈ W . Of course,θ x : S →D x is an isometry and, in particular, it is an isomorphism. As, θ x is injective, according to Lemma 2.8, without loss of generality, we can suppose that θ y is injective for any y ∈ W . It follows thatθ y is a continuous injective operator from S intoD y . As θ y is injective, we have ker Ψ y ∩ S = {0}. It follows that q 1 = q |S is an isomorphism onto q(S) ⊂ F/ ker Ψ y . Of course the restriction q 2 of the isomorphism Ψ y : F/ ker Ψ y →D y to q(S) is an isomorphism ontoθ y (S) such thatθ y = q 2 • q 1 . Soθ y is an isomorphism of S ontoθ y (S).
In this context, in restriction to V , the morphism Ψ can be identified, as a map Ψ : F × V → T M which satisfies assumption (i) and (ii) of Proposition 2.5 △

Results
Let D be a lower trivial distribution on M . For any x ∈ M and any lower trivialization Θ :D x × V → T M and any u ∈D x ,we consider Of course we also have X(z) = i z •Θ(u, z) whereΘ(u, z) =Θ z (u) and X is a local vector field on M tangent to D whose domain is V . Moreover, the set of all such local vector fields spans D.
A lower (local) section of a lower trivial weak distribution D is a map of type (9) for any lower trivialization Θ any u ∈D x and any x ∈ M . Note that the domain of all lower section defined by such a lower trivialization Θ :D x × V → T M is the open set V . The set of such lower sections will be denoted by X − D .
The following Proposition gives a relation between integral manifolds and X − D −invariant weak distributions: In this context, we obtain the following version of Stefan-Sussmann Theorem: Theorem 1 Let D be a lower trivial weak distribution on a Banach manifold M .

D is integrable if and only if it is
Then R is an equivalence relation and the equivalence class L(x) of x has a natural structure of connected Banach manifold modelled onD x . Moreover (L(x), i L(x) ), is a maximal integral manifold of D in the following sense: for any Taking into account Remark 2.2, the property of lower triviality of a weak distribution corresponds to the usual assumptions on the distribution that we find in [ChSt], [Nu], [Gl]. When D x is closed (resp. complemented) in T x M the following Corollary of Theorem 1 gives exactly the main result of integrability of distributions we can find in [Gl] (resp. [ChSt], [Nu]):

Corollary 2.10
For a lower trivial closed distribution the following propositions are equivalent: We end this section with the proof of Proposition 2.9 : Consider a lower section X(y) = Θ(u, y), associated to a lower trivialization Θ :D x ×V → T M . So Dom(X) = V . Fix such such a lower section X and the associated lower trivialization Θ. Denote by △ (resp.△) the subspace D x of T x M ≡ E (resp. the Banach spaceD x ). Let be △ y = Θ y (△) and△ y the natural Banach structure induced byD y .
. According to Remark 2.1, without loss of generality, we may assume that N is an open set inG and M is an open set in E. In these identifications, f is the natural inclusion i N of N in M , that is the restriction to N of the natural inclusion i :G → E. In this context, on i(N ) ⊂ M , y → Θ ′ y is a smooth field of continuous linear operator from△ z ⊂G into D y ≡ G × {y}. Moreover,Θ ′ y is an isomorphism between△ z ⊂G and △ y ⊂G × {y} for any y ∈ i(N ).

Lemma 2.11
With the previous notations, the map y →Θ ′ y from N to L(△ z ,G) is smooth (for the topology induced by△ z on N ).
From Lemma 2.11,Ỹ =Θ ′ y (v) is a smooth vector field on the Banach manifold N , and, moreover, and we have X(i(y)) = T y i[Ỹ )(y)] = (i * Y )(y) on i(N ). So the flow φ X t satisfies the relation φ X t • i = i • φỸ t on a small neighbourhood W of z and for all t such that φỸ t is defined on N . Of course for any y ∈ W and t such that φỸ t (y) is defined, we have Using the previous argument, by induction, we obtain: We deduce that integrability implies X − D −invariance.
Now, if moreover D is closed, given an integral manifold f : N → M and any local section X of D whose domain intersects f (N ), then X induces, by restriction on f (N ), a smooth vector fields on N . So the same arguments used last part of in the previous proof works too. ( see [Gl]). △

Proof of Lemma 2.11
From convenient analysis (see [KrMi]), recall that for a map f from an open set U in a Banach space E 1 to a Banach space E 2 we have the equivalent following properties (iv) for any smooth curve c : R 2 → U , all partial derivatives of f •c exist and are locally bounded.
Fix some v ∈△ z . Note that, for any α ∈G * we have Take any smooth curve c : R → N ⊂△ z . As the inclusion of△ z into △ z is linear continuous, c is also a smooth map from R to N ⊂ △ z , the map h • c is a smooth map from R to [△ z ] * . We conclude that h is a smooth map from N ⊂△ z to [△ z ] * So from (10), we see that the map y →< i * α,Θ ′ y (v) > is a smooth from N ⊂△ z to R, for any α ∈ G * . As i * (G * ) is dense inG * , given any β ∈G * there exists a sequence α n ∈ G * so that i * (α n ) converges to β inG * . For simplicity, we set g(y) =Θ ′ y (v). Consider any smooth curve c : R → N ⊂△ z . Now on any compact K ⊂ R, and for any p ∈ N we have: It follows that < β, g • c > is a smooth map for any β ∈G * . On one hand, we have proved that the map y →Θ ′ y (v)is smooth for any v ∈△ z . On the other hand, we know thatΘ ′ y is a continuous operator from△ z toG. I It follows from (iv) that the map y →Θ ′ y is a smooth map from N ⊂△ z into L(△ z ,G). △

Proof of Theorem 1
Proof of Part (1) At first, according to the Proposition 2.9, integrability implies X − D −invariance. So we have to prove the converse. In fact, this proof is an adaption of arguments of Chillingworth and Stefan used in [ChSt]. Given x ∈ M , we may assume that M is an open set of E and T M ≡ E ×M . We denote by △ (resp.△ ) the normed space (resp. the Banach space) D x (resp.D x ). From the property of lower triviality, after restricting this open if necessary, we have a smooth fields y → Θ y of continuous operators from△ to E. Consider the family {X u (y) = Θ y (u), u ∈△} of smooth vector fields on M . By standard argument (see [ChSt] proof of Corollary 4.2), we can choose an open ball ∈ Ω Xu and all u ∈ B, then we have: where x(t) = φ(t, y, u).

Proof
At first, we fix y ∈ W and u ∈ B, and we set : Of course, A (resp. B) is a smooth field of operators in L(E, E) (resp. L(△, E)). In fact, we have B(t) = Θ y(t) . So, in the Banach space L(△, E)), the linear differential equatioṅ as an unique solution with initial condition U (0) = 0 given by where Γ s is the solution of the differential equatioṅ 3) and (10.7.4) of [Di], we obtain that φ is smooth in the third variable and we have We now look for the integral curve x(t) through x. In this case, Γ s is in fact the t → φ Xu t (x) (see [Di] (10.8.5)), taking in account our assumption of invariance by φ Xu t (x), we have: On the other hand, from the assumption of lower triviality, we have Θ and moreover by integration we also have Finally, using (14) and (12), we obtain the announced result.

△
We are now in situation to give a sufficient condition of the existence of an integral manifold through x ∈ M :

Proposition 2.13
Consider the map: There exists δ > 0 such that Φ : ∈ Ω Xu and all u ∈ B, then, for δ > 0 small enough, (B(0, δ), Φ) is an integral manifold of D through x It is clear that Proposition 2.13, ends the proof of part (1) of Theorem 1.
We now end this subsection with the proof of the previous Proposition.
for all t such that (x, t) ∈ Ω Xu and all u ∈ B. From Lemma 2.12, for any u ∈ B ⊂△, we have: for all u ∈ B. Now, according to the assumption of invariance, we have We In particular, Λ u is a continuous operator from the Banach space△ to the normed space △. The part (1) will be a consequence of the following Lemma: Lemma 2.14 Let be E 1 (resp. E 2 ) a Banach space (resp. a normed space). Suppose that the set L s (E 1 , E 2 ) of surjective operators in L(E 1 , E 2 ) is non empty. Then, L s (E 1 , E 2 ) is an open set.

Proof
The first part of this proof is an adaptation of an argument which can be found in [QuZu].
In the Banach space E 1 , the series of general term ||α n−1 x n || converges. So, there exists z ∈ E 1 such that z = is an open set (see [Ha], [HaMb]), so the Lemma is proved. △ Coming back to the proof of part (1), the map T 0 Φ is the inclusion map ofF in F and [T x φ X0 1 ] = Id E so Λ 0 is surjective. From Lemma 2.14, for δ > 0 small enough, Λ u is surjective for all u ∈ B(0, δ); in particular, we get an equality in (16) which ends the proof of Proposition 2.13. (2) In this subsection, we will use the notations introduced in the previous one. In particular, for any x ∈ M , we associate an integral manifold (B(0, δ), Φ) build in Proposition 2.13. Such an integral manifold will be called a slice through x. At first, we must prove that the relation R is transitive. This fact is a direct consequence of the following Lemma:

Proof
We fix any x ∈ f (N ). Note that N is a connected Banach manifold modelled on△ ≡D x . As the problem is local, according to Remark 2.1, we can assume that N is an open subset of△, M is an open subset of E ≡ T x M and f is the natural inclusion i of△ into E (restricted to N ). Consider a lower trivialization Θ :△ × V → M around x. Given any u ∈△, according to the arguments used in the proof of Proposition 2.9, (with Θ instead of Θ ′ ), we get that the restriction of X u = Θ(u, ) to i(N ) induces a vector fieldỸ u on i(N ) relative to its natural Banach manifold structure. It follows that the integral curve t → Φ Xu t (x) of X u through x lies in i(N ). So, for δ small enough, Φ[B(0, δ)] is contained in i(N ) ⊂ i(△) ⊂ E. But as sets, we have i(△) =△ = △. So using the same arguments used in the proof of part (1) of Theorem 1, but in the Banach space△, we can prove that Φ is a local diffeomorphism of B(0, δ) into N for δ small enough. In particular is an open subset for the topology of the Banach structure on i(N ), which ends the proof of part (1).
Let be (N, f ) and (N ′ , f ′ ) integral manifolds through x ∈ M . Note that N and N ′ are connected Banach manifold modelled on△ ≡D x . Applying part (1) (1), For any z ∈ L there exists a slice (B(0, δ), Φ) such that Φ(0) = z so we get a covering of L by slices. On the other hand, if we have two slices (B(0, δ), Φ) and (B(0, δ ′ ), Φ ′ ) so that Φ(B(0, δ)) ∩ Φ ′ (B(0, δ ′ )) is not empty, then in keeping with part (1) . So we get a structure of connected Banach manifold on L, modelled on△. Moreover, by construction, the natural inclusion i L : L → M is injective. As each slice is an integral manifold of D modelled on△, it follows that, that (L, i L ) is an integral manifold of D. △ It remains to show that any equivalent class L(x) of x ∈ M carries a natural structure of connected Banach manifold modelled onD x . Note that L(x) is the union of all the subset f (N ) where (N, f ) any integral manifold through x. So L(x) is connected. Moreover, as in the proof of part(2) Lemma 2.15, we can cover L(x) by slices and this gives rise to a natural structure of connected Banach manifold on L(x). Again, (L(x), i L(x) ) is an integral manifold of D through x, which is maximal by construction.

Case of lower trivial weak distribution
In this section we shall adopt the material and arguments used in [St] and [ChSt].
Let be X and Y be smooth vector fields on an open set U in a Banach space E. Given any integral curve γ : I = [α, β] → U of X it is well known that the Lie bracket at some γ(t) is given by where DX is the differential of X. Note that, for any diffeomorphism φ : U → V , according to the "chain rule" in differentiation, the same type of formula forX = φ * X,Ȳ = φ * (Y ) and the integral On the other hand, (17), depends only of the map of Y • γ. It follows that the following definition does not depend of the choice of the chart: Definition 3.1 Let be X an vector field on M and γ : I = [α, β] → U ⊂ M an integral curve of X whose range is contained in a chart domain U of a chart (U, φ). Given any vector field Y along γ (i. e. Y : Note that the previous definition do not depends of the choice of the chart. So given any X ∈ X (M ) and any integral curve γ : I →Dom(X), the Lie bracket [X, Y ] is well defined along γ, for any vector field Y along γ.
On the other hand, given any smooth curve γ : I = [α, β] → M , we denote by T γ M the restriction of T M to γ(I). For any Banach space G, we denote by L γ (G, T M ) the bundle, over γ, of morphisms between the trivial bundle G × I and T γ M .
Let D be a lower trivial weak distribution on M . Consider a local vector field X and an integral curve γ : [α, β] → M of X.

Given an upper trivialization ψ as in part 1 , for each
. The Lie derivative of ψ by X along γ is defined by: Remark that, with the previous notations, the map v → (L X ψ) t (v) is linear so we get a smooth map L X ψ : I → L γ (G, T M ) so that for t ∈ I, (L X ψ) t ∈ L(G, T γ(t) M ).

Definition 3.3
Let D be a lower trivial weak distribution.
1. Let Θ :D x × V → T M , be a lower trivialization around x, and X u = Θ(u, ) a lower section on V . We say that the weak distribution D is Lie invariant by X u , if , for any y ∈ V , we can find ε > 0, so that, for all 0 < τ < ε, there exists: • an upper trivialization ψ : 2. The weak distribution D is called Lie invariant if for any x ∈ M there exists a lower trivialization Θ :D x × V → T M such that, for any u ∈ D x , the weak distribution D is Lie invariant by X u = Θ(u, ).
As in [ChSt], we have the following Theorem but without the assumption of closeness and existence of a complement for all subspaces D x Theorem 2 Let D be a lower trivial weak distribution. The following properties are equivalent: Observations 3.4 1. On finite dimensional manifold M consider a "smooth" or "differential" distribution D (see Observations 2.7 part 3). In a local context the follwing version of Theorem 4.7 of [Ba] (which is its central result) can be reformulated in the following way (Ballan's proof leads exactly to this result).

Theorem 3.5 [Ba]
we have equivalence between integrability of D and the following assumptions: for any X tangent to D and any x ∈Dom(X), there exists an open neighbourhood V ⊂Dom(X) and ε > 0, a finite set {X 1 , · · · , X p } defined on V and tangent to D and smooth functions is generated by X 1 (γ(t)), · · · , X p (γ(t)).
We will show that, under the previous assumptions (a), (b), (c), the condition 2 of Theorem 2 is fulfilled and so, we get that D is integrable. Note that using Proposition 2.9, the property of lower triviality and the stability by Lie bracket on an integral manifold we obtain easily the inverse implication.
Indeed, assume that previous assumptions (a), (b), (c), are fulfilled. At first consider a lower trivialization Θ : D x × V → T M associated to a family of linear independent vector fields {Z 1 , · · · , Z n } on a neighbourhood V of x in M (see Observations 2.7 part 3). Fix some lower section X u = n j=1 u j Z j for some u = n j=1 u j Z j (x) ∈ D x and consider the integral curve γ(t) = φ Xu t (x) on an interval ] − ε, ε[ as in the previous assumptions. Denote by G the vector space R p and {e 1 , · · · , e p } its canonical basis. Choose some 0 < τ < ε. On [−τ, τ ] , we consider the smooth field of linear operators : t → ψ t , from G to T γ(t) M , defined by: From assumption (c) and Observations 2.7 part 3, we get an upper trivialization of D along γ . From assumption (b), we can write Consider the field t → Λ t of endomorphisms of G defined by But we have So, we get the Lie invariance of D and we can apply Theorem 2. △ 2. In the same context of finite dimensional manifolds, recall the remark of Balan about the proof of Theorem 4.1 of [Su]. Consider the two following conditions : "(e)" for any x ∈ M there exists vector fields X 1 , · · · X p defined on some neighbourhood of V of x such that (1) D x is generated by X 1 (x), · · · X p (x) (2) for every vector field X defined on V , there exists ε > 0 and smooth functions λ ij : Balan points out that the implication "(e) implies (f )" is wrong and, of course, the implication "(e) implies the integrability of D" is wrong too.
The condition (e) can be be replaced by the previous assumptions (a), (b) and (c) proposed by Balan in [Ba] (see part 1 of this obervation) which gives rise to a correct version of this Theorem. In fact in [Ba], Balan proposes to replace condition (e) by the following condition (e') for any x ∈Dom(X), there exists an open neighbourhood V ⊂Dom(X) and ε > 0, a finite set {X 1 , · · · , X k } defined on V and tangent to D such that (a) X 1 (x), · · · , X k (x) span D x (b) for any vector field X defined on V and tangent to D, there exists smooth functions The essential difference between (e) and (e') is that the parameter "ε" in (e') is "maximal" while in (e), it is not the case. The corrected proof proposed by Balan, for the implication "(e') implies the integrability of D", contains implicitly the fact that (e') implies more or less the previous assumptions (a), (b) and (c).
3. Given a set X of vector fields on a finite dimensional manifold M , in [St], Stefan defines the locally subintegrability of X at some x ∈ M if there exists an open neighbourhood V of x in M and a subset X ♭ of X such that (LS1) D x = (D ♭ ) x and D ♭ is integrable on V where D (resp. D ♭ ) is the distribution generated by X (resp. X ♭ ) (LS2) for any X ∈ X there exists ε > 0 such that for any t ∈] − ε, ε[.
On the other hand, denote by X ♯ the set of finite linear combinations of type λ i X i where λ i are smooth functions and X i belongs to X . Then the Theorem 4 of [St] says: D is integrable if and only if X ♯ is sub integrable According to Balan's remark, the condition (LS2) is not sufficient , again because "ε > 0" given in (LS2) depends of X. Moreover, Balan gives a contre example to this Theorem.
Proof of Theorem 2.
Assume that D is X − D -invariant. Let x ∈ M be a fixed point and choose a lower trivialization Θ :D x × V → T M . Consider a lower section X u = Θ(u, ) and y ∈ V . Note that there exists ε > 0 such that the integral curve t → φ Xu t (y) of X u through y is defined for all t ∈] − ε, ε[. Choose any 0 < τ < ε and set γ(t) = φ Xu t (y) for t ∈ [−τ, τ ]. From our assumption, we have T y φ Xu t (D y ) = D γ(t) . If i y :D y → D y is the natural inclusion, denote by ψ t = T y φ Xu t • i y . Set Γ t = T y φ Xu t . It is clear that ψ is an upper trivialization of D over γ with G =D y . On the other hand, we have: Lemma 2.12). So we obtain L Xu ψ = 0 on [−τ, τ ]. Taking Λ = 0 in (19) we get Lie invariance for X u For the converse, as in [St] [ChSt] and [Nu], we need the following result whose proof is somewhat different (each space D x can be not closed here) Lemma 3.6 Let X be a local vector field and ψ an upper trivialization of D defined over an integral curve γ :] − ε, ε[→ V =dom(X). Moreover we assume that, for any 0 < τ < ε there exists a smooth field Λ : Then, we have T y φ X t [D y ] = D φ X t (y) for all 0 < |t| < ε Now, assume that D is Lie invariant by X u ; let us fix some y ∈ V = Dom(X u ), and take a maximal integral curve γ :]α y , β y [→ V of X u . Consider the set I = {t ∈]α y , β y [: This set is clearly open according to Lemma 3.6. Take a sequence (t n ) in I which converges to some t ∈]α y , β y [. From the assumption (2) of the Theorem, and Lemma 3.6 applied at the point φ Xu t (y), we have φ Xu s (D φ Xu t (y) ) = D φ X t+s (y) for s in some neighbourhood ]t − η, t + η[ of t. As we have some t n which belongs to ]t − η, t + η[, we get that I is closed. So I =]α y , β y [ and finally we deduce that D is invariant by X u . △

Proof of Lemma 3.6
Let ψ : [−τ, τ ] → L γ (G, T M ) be an upper trivialization of D over an integral curve γ of X such that γ(0) = y ∈ V . Consider any smooth field of operators σ : [−τ, τ ] → L(G, G) and setψ = ψ •σ. On a chart domain, we have Assume that L X ψ = ψ • Λ for some smooth field of operators Λ : [−τ, τ ] → L(G, G). Then we have: Consider the solution (again denoted by σ) of the linear equationσ = (−Λ)•σ with initial condition σ 0 = Id G . So σ is a smooth field of isomorphisms of G and in particular, for this choice of σ, we haveψ(t)[G] = D γ(t) for any t ∈ [−τ, τ ]. Moreover, using (21), we have L Xψ = 0. In fact the relation (21) do not depends of the choice of the chart, so we can get the same result even if γ([−τ, τ ]) is not contained in a chart domain. So we can assume that there exists an upper trivialization ψ : On a chart domain we havė According to (17) and (18), on [−τ, τ ] we have Now, (22) do not depend of the choice of the chart. As Γ andΣ are intrinsically defined, so even if γ([−τ, τ ]) is not contained in a chart domain, we can obtain the same relation. Now from our assumption L X ψ = 0, as Γ t is an isomorphism, we must have Σ t = Σ 0 = ψ 0 . We conclude that, for any t ∈ [−τ, τ ], we have [Γ t ] −1 • ψ t (G) = ψ 0 (G) = D y and finally Now, according to the assumption in this Lemma, there exists ε > 0 such that, we are in the previous situation for any interval [−τ, τ ] with 0 < τ < ε. So (23) is true for any 0 < |t| < ε △

Case of upper trivial weak distribution
, v ∈ F } generates D y for all y ∈ V . Moreover, if F has a Schauder basis {e α , α ∈ A}, then the convex hull of {Z eα (y), α ∈ A} is dense inD y 3. If Θ :D x × T M is the lower trivialization associated to the upper trivialization Ψ then each lower section X u = Θ(u, .) can be written X u = Θ(Ψ(u, x), ) and so the set X − D (V ) of such lower sections is contained in X + D (V ).
Let D be a upper trivial weak distribution on M . Let V be the domain of a chart around x ∈ M . Consider a upper trivialization Ψ : F × V → T M and Θ :D x × V → T M the associated lower section. Given any smooth function σ : V → F , let Z σ = Ψ(σ, ) be the associated vector field on V . Consider γ : [−τ, τ ] → V an integral curve of Z σ , then, Ψ γ , defined by Ψ γ (t)[v] = Ψ(v, γ(t)), is an upper trivialization of D along γ, according to Definition 3.2. So the Lie derivative of Ψ by Z σ along γ is L Zσ Ψ γ which we simply denoted by L Zσ Ψ. We can also define directly L Zσ Ψ by: Definition 3.8 An upper trivial weak distribution D is called Lie bracket invariant if, for any x ∈ M , there exists an upper trivialization Ψ : F × V → T M such that for any u ∈ F , there exists ε > 0, such that, for all 0 < τ < ε, there exists a smooth field of operators Λ : [−τ, τ ] → L(F, F ) with the following property along the integral curve t → φ Xu t (x) on [−τ, τ ] of any lower section X u = Θ(Ψ(u, x), ).
With these definitions we have:

Theorem 3
Let D be an upper trivial weak distribution. The following propositions are equivalent: Remark 3.10 1.
The assumption "the kernel of Ψ is complemented in each fiber" is automatically satisfied if the kernel of Ψ is finite dimensional or finite codimensional in each fiber, or in the context of Hilbert manifold.
2. Recall that when M is a finite dimensional manifold, if, for any x ∈ M , each module X Dx of germs of vector fields is finitely generated then D is upper trivial(see Observations 2.7 part 3.) So, the formulation Theorem 4.7 of [Ba] can be given in its original way: if each X Dx is finitely generated for any x ∈ M , then D is integrable if and only if D satisfies the properties (a), (b) and (c) in Theorem 3.5.
Of course, this result is a direct consequence of this last Theorem, but, we can easily see that this Theorem 4.7 can be directly deduced from Theorem 3. This proof is left to the reader.
Coming back to the context of Corollary 2.6, let Π : F → M be a Banach fiber bundle over M with typical fiber F , Ψ : F → T M a morphism of bundle whose kernel is complemented in each fiber. We denote by S(F ) the set of local sections of Π : F → M , that is smooth maps The maximal such open set is called the domain of σ and denoted Dom(σ).
A subset S of S(F ) is called a generating upper set of D if for any x ∈ M , the set X S = {Ψ • σ, σ ∈ S} contains X + D . Of course S(F ) is a maximal generating upper set. We introduce some condition on X S which will be used in the next theorem: X S satisfies the condition (LB) if: for any local section σ ∈ S, there exists an open set V ⊂Dom(σ) on which F is trivializable and for any x ∈ V we have the following property: given any integral curve γ :] − ε, ε[→ V of X = Ψ • σ with γ(0) = x, there exists a smooth field Λ :] − ε, ε[→ L(F x , F x ) such that [Ψ • σ, Ψ(u, )](γ(t)) = Ψ(Λ t (u), γ(t)) for any t ∈] − ε, ε[ for any u ∈ F x Then, using Theorem 3 we get the following Theorem

Theorem 4
Let Π : F → M be a Banach fiber bundle over M with typical fiber F and Ψ : F → T M a morphism of bundles such that the kernel of Ψ is complemented in each fiber and denote D = Im Ψ.

Proofs of Theorem 3 and Theorem 4
Proof of Theorem 3 According to Theorem 1, we have only to prove (1) ⇐⇒ (2).
For the converse, we will use the following Lemma: Lemma 3.11 Assume that D is integrable. Let Ψ : F × V → T M be a upper trivialization, and σ : V → F a smooth map and let X = Ψ(σ, ) be the associated vector field on V . Consider an integral curve γ :] − ε, ε[→ V of X such that γ(0) = x. Then there exists a smooth field Λ :] − ε, ε[→ L(F, F ) such that : So, for σ(y) = (u, y), with u ∈ S, the vector field Z σ is the lower section X u for u ∈ S et clearly Lemma 3.11 proves (1) =⇒ (2). △

Proof of Lemma 3.11
Recall that we have assumed that D is integrable. Fix some x ∈ M . Take an upper trivialization Ψ : F × V → T M around x and let Θ :D x × V → T M be the associated lower trivialization. We can choose V such that T M |V ≡ E × V . Recall that we have a decomposition F = ker Ψ x ⊕ S, and Θ = (θ y • [θ x ] −1 , ) where θ y is the restriction to S of Ψ y (see the proof of Proposition 2.5). At first note that any lower section X u can be written X u = Θ(Ψ(u, x), ) for some u ∈ F and also X u = θ(u, ) but with u ∈ S. On the other hand, according to Lemma 2.15,(B(0,δ), Φ) is an integral manifold of D through x (associated to Θ). In particular,θ y is an isomorphism from S toD y . Given u ∈ F , there exists an unique v ∈ S such that Ψ y (u) = θ y (v) so u ∈ ker Ψ y ⊕ S. It follows that F = ker Ψ y ⊕ S.
Set N = Φ(B(0, δ)) ⊂ M endowed with its Banach manifold structure. Without loss of generality, we can identify S withθ x (S) =D x and so we can consider N as an open set ofD x . Denote by i :D x → T x M ≡ E the canonical inclusion. We have T y N ≡ S × {y}. On N , eachΨ y can be considered as an element of L(F, S). By arguments similar to those used in the proof of Lemma 2.11, we can show that y →Ψ y is a smooth field of operators in L(F, S). So, y →θ y is a smooth field of isomorphisms of S. We get a smooth field π y = [θ y ] −1 •Ψ y of operators in L(F, S) with the following properties: ker π y = kerΨ y = ker Ψ y (31) Take any smooth map σ : V → F . Then Z σ (y) = Ψ y • σ(y) for y ∈ V (resp.Z σ (y) =Ψ y • σ(y) for y ∈ N ) is a smooth vector field on V (resp. on N ) and we have the relations: Ψ(σ(y), y) = Z σ (i(y)) = i[Z σ (y)] = i •θ i(y) • π y (σ(y)) = θ i(y) • π y (σ(y)) = θ(π y (σ(y)), y) (33) Consider the integral curves γ(t) = φ Zσ t (x) andγ(t) = φZ σ t (x) for t ∈] − ε, ε[. Of course we have γ(t) = i •γ(t). For simplicity, we set: σ(γ(t)) = σ(t) and σ(γ(t)) =σ(t) P (t) = πγ (t) . Note that , using (33) we have Now, in keeping with (25), for any v ∈ S, we have: As we have Z σ = i * Zσ and X v = i * Xv , on the Banach manifold N , we get It follows that we have: Without loss of generality, we can choose δ > 0 small enough such that on N = Φ(B(0, δ)) we have : On the other hand, we have: From (37), we have : In the second member of (38), the same majoration is true for Clearly, t → Λ(t) is a smooth field of endomorphisms of S and taking in account (36) we get Now from (34), with the same argument (20) used in the proof of Lemma 3.6, we get: But, according to the definition of P and (32), we have P •Ṗ =Ṗ . So using (41), we get: which ends the proof of Lemma 3.11 by setting Λ =Λ +Ṗ . △

Proof of Theorem 4
(1) According to the context of the proof of Corollary 2.6, for any given x ∈ M we consider a local trivialization of F on an open set V around x, so that the morphism Ψ can be identified, as a map Ψ : F x × V → T M and F x = ker Ψ x ⊕ S x and let Θ : S x × V → T M be the associated lower trivialization. In this context, taking any σ(y) = (u, x), for any u ∈ S x in (LB) for any x ∈ M , we get property (2) of Theorem 3 so (LB) is a sufficient condition for integrability of D. Assume now that D is integrable and consider an upper generating set S of D and any section σ ∈ S defined on an open set U . Fix any x ∈ U . From Lemma 3.11 we get (28) with Λ :]−ε, ε[→ L(F x , S x ) So in particular, if (LB) is true for some Λ :]−ε, ε[→ L(F x , F x ), then D is integrable and by Lemma 3.11 we can find Λ ′ :] − ε, ε[→ L(F x , S x ) which satisfies (28) .
(2) If now D is a closed distribution, then in each fiber F x = ker Ψ x ⊕ S x , θ x is a continuous bijective morphism between both Banach space S x and D x so θ x is an isomorphism. In particular, D x and D x are equivalent as Banach spaces. Coming back to the previous local context of the upper trivialization Ψ : F x × V → T M , the map y → θ y is a smooth field of isomorphisms from S x to Ψ γ(t) (S x ). If D is integrable, from (28) and the properties of Λ we obtain (29). For the converse, it is sufficient to set to get (28). △

Banach Lie Algebroid
The concept of Lie algebroid was first introduced by J. Pradines in relation with Lie groupoids (cf [Pr]). The theory of algebroids was developped by A. Weinstein ( [We]) and, independently, by M. Karasev ([Ka]), in view of the symplectization of Poisson manifolds. This theory has also an important role as models in mechanic and mathematical physic (for a survey, see [Li] and [Kos2] for instance). On the other hand, this concept of Lie algebroid can be extend in the infinite dimensional case: in [KisLe] the authors build variational Lie algebroids of the infinite jet bundles over a vector bundle over a finite dimensional manifolds. This construction can be situated in the Frechet manifold framework . In fact, this context is very special : infinite jet bundles over a vector bundle over finite dimensional manifolds are projective limit of finite dimensionnal Banach spaces so we get a set of coordinate on such a space. The existence of these coordinates is crucial in this construction of the variational Lie algebroid. In this paper, we look for the Banach manifold context and in this framework we do not have (local) coordinates in general. According to the classical definition of a Lie algebroid in finite dimension we introduce: The quadruplet (A, Ψ, M, { , }) is called a Banach Lie algebroid and { , } (resp. Ψ) is called the Lie bracket on A, (resp. the anchor morphism).
As in finite dimension, the Jacobi identity and the Leibniz property implies that Ψ gives rise to a Lie algebra morphism from S(A) into X (M ) i. e.
Given some open set U in M , we denote by A U the restriction of the Banach bundle Π : A → M to the Banach manifold U : A U = Π −1 (U ); the set of sections of A U will be denote by S(A U ).
In finite dimension, it is classical that a bracket { , } on a Lie algebroid (A, Ψ, M, { , }) is compatible the sheaf of sections of Π : A → M or, for short, is localizable (see for instance [Ma]). By this property, we mean the following : Using the same arguments as in finite dimension, when M is smooth regular, we can prove that, for any Lie algebroid (A, Ψ, M, { , }), its bracket is localizable (see [CaPe]). But, if M is not smooth regular, we can no more used this argument. Unfortunately, we have no example of Lie algebroid for which the Lie bracket is not localizable. Note that, according to [KrMi] sections 32. 1, 32.4, 33.2 and 35.1, this problem is similar to the problem of localization (in an obvious sense) of global derivations of the module of smooth functions on M or the module of differential forms on M . In [KrMi] and, to our known, more generally in the literature, there exists no example of such derivations which are not localizable. On the other hand, even if M is not regular, the classical Lie bracket of vector fields on M is localizable. So, there always exists an anchored bundle A = T M and a Lie bracket algebroid (T M, Id, M, [; , .]) for which its Lie bracket is localizable. Moreover, in Examples 4.3 we do not assume that M is regular but, nevertheless, these Lie brackets are also localizable. Thus, in the Definition 4.1, we moreover impose that, if M is not regular, then the Lie bracket of the Lie algebroid is localizable even if this assumption could be, in fact, unnecessary when M is not regular.

Remark 4.2 :
In finite dimension we have many equivalent definitions of a Lie algebroid: a Lie algebroid structure on a vector bundle A → M may be characterized by: • a Lie bracket on an anchored bundle (A, ρ); • a linear Poisson structure on A * ; • a linear Schouten structure on the exterior algebra Λ • A * ; • a differential operator d on the module of sections S(Λ • A * ) with d • d = 0. This last approach can be interpreted in the context of supermanifolds (see [Kos1]). It is precisely this last aspect which is used in [KisLe] for the construction of variational Lie algebroids. However, in the context of Banach manifolds, we have many obstructions in the generalization of the previous equivalent definitions. For instance, when the typical fiber of A is an infinite dimensional Banach space, in general, such a differential operator d could be not localizable (see [KrMi] section 35.1). Moreover, according to the fact that the bidual E * * of a Banach space E may contains strictly E, we must impose complementary conditions on d, to get a Lie algebroid structure on A by this way. On the other hand, the set of sections of the graded algebra Λ • A is choice of f and g).
For each x, on the quotient T * x M/ ker Ψ x we get a skew-symmetric bilinear formω x . On the other hand, letΨ x : T * x M/ ker Ψ x →D x be the canonical isomorphism associated to Ψ x between Banach spaces. Finally we get a skew-symmetric bilinear formω x onD x such that : According to [OdRa2], a symplectic leaf of D is a weak submanifold (L, i) where L ⊂ M and i : L → M is the natural inclusion with the following properties: (i) (L, i) is a maximal integral manifold of D (in the sense of Theorem 1 part (2)); (ii) on L we have a weak symplectic form ω L such that (ω L ) x =ω x for all x ∈ L Remark 4.4 As in the context of Lie Banach algebroids, we will say that a Lie bracket { , } on C ∞ (M ) is localizable if { , } is compatible with the sheaf of germs of functions on M . From our definition of Banach Poisson manifold, the Lie bracket associated to a Poisson anchor Ψ is always localizable. On the other hand, given any Lie bracket { , } on C ∞ (M ), when M is regular, we can prove that { , } is localizable, and then we have a morphism Ψ : T * M → T * * M naturally associated (see [CaPe]). If moreover, Ψ(T * M ) ⊂ T M , then we get the previous definition of Banach Poisson manifold (see for instance [OdRa1] or [OdRa2]).

Theorem 6
Let be Ψ : T * M → T M a Poisson anchor. If the kernel of Ψ is complemented in each fiber, then the associated characteristic distribution D is integrable. Moreover, each maximal integral manifold has a natural structure of weak symplectic leaf.
For an illustration of this result, the reader will find many examples of Banach Poisson manifolds in [OdRa1] and [OdRa2].

Proof of Theorem 6
At first, we can observe that the set S = {Ψ(df ) : f ∈ C ∞ (U ), U any open set in M } is an upper generating set for D: given any x ∈ M , modulo any local chart around x, we can suppose that M is an open subset of E and T * M ≡ E * × M ; for any α ∈ E * the function f α (x) =< α, x > is a smooth map on M such that df α (y) = α for any y ∈ M ; so Z α = Ψ(α, y) = Ψ(df α (y)) is an upper section. For any smooth local function f : U → R, we set Z f = Ψ(df, ). From the Jacobi identity in C ∞ (M ) we have [Z f , Z g ] = Ψ(d{f, g}, ) for any f, g ∈ C ∞ (M ) According to Theorem 4, to prove the integrability of D, we have only to prove (LB) for the generating upper set S. As (LB) is a local property, again we assume that M is an open set in E. So fix some smooth function f : M → R and consider an integral curve γ(t) = φ Z f t (x) through x ∈ M defined on ] − ε, ε[. For any α ∈ E * , using (43), we have: [Z f , Z α ](γ(t)) = [Ψ(df ), Ψ(df α )](γ(t)) = Ψ(d{f, f α }(γ(t))) But, using the same arguments as the ones used in the proof of Lemma 3.11 we can show that the map y → [α → d{f, f α }(y)] is a smooth field of continuous operators from E * to E * . It follows that D satisfies (LB), and then, D is integrable.
Assume now that D is integrable and choose any maximal leaf L. As T x L =D x , on T x L we have the skew-symmetric bilinear formω x previously defined. We will show thatω x defines a closed 2-form ω L on L, which is a weak symplectic form.
As we know thatθ x is an isomorphism from S x to T x L it follows thatω x is a weak symplectic 2-form on the Banach space T x L. On one hand, locally, in keeping to Lemma 2.11, it follows that ω L defined by (ω L ) x =ω x is a smooth differential 2-form on L. On the other hand, for any smooth function f defined on an open set U ⊂ M , we setf = f • i. So for any smooth functions f , g and h defined on U , the Jacobi identity is satisfied forf ,g andh on the open set i −1 (U ) ⊂ L. So, by classical arguments of Poisson bracket (see for instance [LiMa], [OdRa1], [OdRa2]), we get: dω L (i * Z f , i * Z g , i * Z h ) = 0 for any choice of functions f , g and h. So ω L is closed and the proof of Theorem 6 is complete. △