On the domain of implicit functions in a projective limit setting without additional norm estimates

Jean-Pierre Magnot

doi:10.1515/dema-2020-0008

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On the domain of implicit functions in a projective limit setting without additional norm estimates

Jean-Pierre Magnot

Published/Copyright: July 2, 2020

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From the journal Demonstratio Mathematica Volume 53 Issue 1

Abstract

We examine how implicit functions on ILB-Fréchet spaces can be obtained without metric or norm estimates which are classically assumed. We obtain implicit functions defined on a domain D which is not necessarily open, but which contains the unit open ball of a Banach space. The corresponding inverse function theorem is obtained, and we finish with an open question on the adequate (generalized) notion of differentiation, needed for the corresponding version of the Fröbenius theorem.

Keywords: implicit functions; ILB spaces

MSC 2010: 58C15

1 Introduction

Classical inverse function theorems, implicit function theorems and Fröbenius theorems on Banach spaces are known to be equivalent. There exist numerous extensions to setting on Fréchet or locally convex spaces, and to our knowledge almost all proofs are based on a contraction principle. In order to obtain in the proofs a mapping which is contracting, one needs to assume conditions which are not automatically fulfilled by a mapping on Fréchet spaces, but which are automatically locally fulfilled (on an open set) by a sufficiently regular mapping on Banach spaces. For classical statements, one can see [1,2,3] for an implicit function theorem on Banach spaces, and [1,4,5,6,7] as a non-exhaustive list of generalizations. From all these references, we remark that a contraction theorem is at the heart of all these results, in Banach spaces as well as in generalized frameworks. In order to get contraction properties, extra assumptions (one could say additional estimates on the functions under consideration) are necessary to obtain implicit functions defined in open subsets and then have a safe proof of regularity of implicit functions.

Motivated by [6,7,8,9] where analysis and geometry on Fréchet manifolds are controlled by a sequence of Banach spaces (this is the inverse limit of Banach (ILB) setting described in [7]), we analyze here how a very classical proof of the implicit function theorem can be adapted on an ILB setting, that is, when the Fréchet spaces under consideration are projective limits of Banach spaces, and when the functions f∞ on Fréchet spaces are restrictions of families of bounded functions fi on the sequence of Banach spaces. This is what one may call degree 0 maps, by analogy with the degree of differential operators or along the lines of [7,9], but in our case, additional estimates are not assumed. Then we get Theorem 2.2, an implicit function which is defined on a domain D, which is not a priori open in the Fréchet topology, but which contains the open ball of radius 1, centered at the origin, of a Banach space. This result can be adapted to some functions for which there does not exist any extension to a Banach space. These functions have to be controlled by a family of injective maps, which explains the terminology “tame” (Theorem 2.3). We have to remark that the domain D can be very small. This is the reason why regularity results on implicit functions cannot be stated: differentiability, in a classical sense, requires open domains or at least manifolds. This leads to natural questions for the adequate setting for analysis beyond the Banach setting. Even if not open, following the same motivations as the ones of Kriegl and Michor in [6] when they consider smoothness on non-open domains, the domain D may inherit some kind of generalized setting for differential calculus, such as diffeologies [10] which are used in e.g. [11,12]. This question is left open, because out of the scope of this work: the most adapted (generalized) framework for the extension of the regularity (i.e. differentiability or smoothness) has to be determined. Then, we give an inverse function theorem, which can be stated with the same restrictions as before on the nature of the domain D, and with an obstruction to follow the classical proof of the Fröbenius theorem from [3] where differentiation on D is explicitly needed.

We finish with an example where our implicit function theorem applies. This example appears as an analog of the infinite dimensional structure group of an ILB vector bundle. This viewpoint is fully described in [8] or the reader can check e.g. [7,13] for a short description. This group is actually known as a Frölicher or diffeological Lie group [11,13] and our implicit function theorem highlights an open question on its topological and differentiable structure.

2 Implicit functions: from Banach spaces to projective limits

2.1 Review of the classical proof of implicit function theorem on Banach spaces

Let (E;∥⋅∥E) and (F;∥⋅∥F) be two Banach spaces. The Banach space E×F is endowed with the norm

∥(x,y)∥E×F=max{∥x∥E,∥⋅∥F}.

We denote by D1 and D2 the (Fréchet) differential with respect to the variables in E and F, respectively. Let us first give the statement and a proof of a classical implicit function theorem on Banach spaces, for the sake of extracting key features for generalization. For this, let U be an open neighborhood of 0 in E; let V be an open neighborhood of 0 in F; and let

(2.1)f:U×V→F

be a Cr-function (r≥1) in the Fréchet sense, such that

f(0;0)=0,
D2f(0;0)=IdF.

Theorem 2.1

There exists a constantc>0such that, on the open ballB(0;c)⊂E,there is a unique map

u:B(0;c)→V

such that

(2.2)∀x∈B(0;c),f(x;u(x))=0.

Let us remark that regularity of the function u is left up. We now divide the main arguments of the classical direct proof of this theorem into three lemmas.

Lemma 2.1

There existc0>0andK>0such that

∥(x;y)∥E×F<c0⇒∥D1f(x;y)∥L(E;F)<K.

Proof

Since f is of class C1, we have in particular that

D1f(.;.)∈C0(U×V,L(E,F)).

Hence, if K>∥D1f(0;0)∥L(E,F), there exists a neighborhood W of (0;0)∈E×F such that ∀(x,y)∈W,||D1f(x;y)||L(E,F)<K.□

Lemma 2.2

There existsc1>0such that

(2.3)||(x;y)||E×F<c1⇒||D2f(x;y)−IdF||L(F)<12.

Proof

The map f is of class Cr, with r≥1, so that, the map D2f:(x;y)∈U×V↦D2f(x;y)(.)∈L(F) is of class Cr−1 and in particular of class C0. By the way,

∃c1>0;∥(x;y)∥E×F<c1⇒∥D2f(x;y)−D2f(0;0)∥L(F)<12.□

Lemma 2.3

Letc1be the constant of Lemma 2.2. There existsc2>0such that

(2.4)∥x∥E<c2⇒∥f(x;0)∥F<c14.

Proof

The map f is of class Cr, with r≥1. So that it is in particular of class C0. By the way, there exists a constant c2 such that

(2.5)∥x−0∥E<c2⇒∥f(x;0)−f(0;0)∥F<c14.

□

Lemma 2.4

Letc=min{c0,c1;c2;1}.Let x such that∥x∥<c.Then, the sequence(un)ℕ∈Fℕ, defined by induction by

(2.6)u0=0un+1=un−f(x,un)

is well-defined and converges tou(x)∈V.

Proof

Let us assume that x is fixed. Let g(x;y)=y−f(x,y). By the way, un=gn(u0). Let (y,y′)∈F2 such that both (x,y) and (x;y′) are in B(0,c)⊂B(0;c1).

f(x,y)−f(x,y′)=∫01D2f(x;ty+(1−t)y′)(y′−y)dt.

By the way,

∥g(x,y)−g(x,y′)∥F=∫01D2g(x;ty+(1−t)y′)(y′−y)dtF=∫01(D2f(x;ty+(1−t)y′)−IdF)(y′−y)dtF≤∫01∥D2f(x;ty+(1−t)y′)−IdF∥L(F)⋅∥y′−y∥Fdt≤∥y′−y∥F2

applying estimates of Lemma 2.2.

By the way,

(2.7)gis12−Lipschitz.

Thus, applying Lemma 2.3,

∥f(x,0)∥F=∥u1−u0∥F<c14

and we obtain by induction applying (2.7):

∀n∈ℕ,∥un+1−un∥F<c12n+2.

Hence,

∀n∈ℕ;∥un∥F≤c1/2.

Hence, (un) is a Cauchy sequence, which is in Vℕ and in particular in B(0,c1/2)¯. Thus, (un) is converging to a limit u(x)∈V.□

Proof of Theorem 2.1

By Lemma 2.4, the map x↦u(x) exists for ∥x∥E<c.□

2.2 Generalization to a class of projective limits

Definition 2.1

Let (Ei)n∈ℕ be a sequence of Banach spaces. If the sequence satisfies the following properties:

∀i∈ℕ,Ei+1⊂Ei, which can be summarized by stating that sequence (Ei)i∈ℕ is decreasing for the order given by inclusion (⊂) of sets.
∀i∈ℕ, ∀x∈Ei+1, ∥x∥Ei≤∥x∥Ei+1. In other words, the inclusion map Ei+1↪Ei is continuous (bounded).

We summarize these two properties saying that Ei+1⊂Ei with continuous inclusion.

Example 2.1

Let M be a smooth boundaryless compact manifold. Sequences (Ck(M,ℝ))k∈ℕ (maps with continuous kth differential) and (Wk(M,ℝ))k∈ℕ (Sobolev spaces of order k) are sequences as in Definition 2.1.

We now adapt the results of the previous section to the following setting. Let (Ei)i∈ℕ and (Fi)i∈ℕ be two sequences of Banach spaces which satisfy the properties of Definition 2.1. We then consider U0 and V0 two open neighborhoods of 0 in E0 and F0, respectively, and a function f0 of class Cr with the same properties as in Theorem 2.1. Let us now define, for i∈ℕ, Ui=U0∩Ei and Vi=V0∩Fi. Moreover, let us assume that f0 restricts to Cr-maps fi:Ui×Vi→Fi. Let

E∞=lim←{Ei;i∈ℕ},

let

F∞=lim←{Fi;i∈ℕ},

let

U∞=E∞∩U0andV∞=V0∩F∞.

Finally, let f∞=lim←fi.

Theorem 2.2

There exists a non-empty domainD∞⊂U∞, possibly non-open inU∞, and a function

u∞:D∞→V∞

such that

∀x∈D∞,f∞(x;u∞(x))=0.

Moreover, there exists a sequence(ci)i∈ℕ∈(ℝ+⁎)ℕand a Banach spaceBf∞such that

Bf∞⊂E∞(as a subset)
the canonical inclusion mapBf∞↣E∞is continuous

which is the domain of the following norm (and endowed with it):

||x||f∞=sup||xi||ci|i∈ℕ.

Then,D∞contains the unit ball (of radius 1 centered at 0) ofBf∞.

Proof

Let i∈ℕ. We now consider the maximal domain Di⊂Ui where there exists a unique function ui such that ∀x∈Di,fi(x;ui(x))=0. This domain is not empty since it contains 0∈Ei and, applying Theorem 2.1, there exists a constant ci>0 such that

∥x∥Ei<ci⇒x∈Di.

By the way, any maximal domain Di is an open neighborhood of 0 in Ei, and setting D∞=∩i∈ℕDi we get that D∞ contains 0∈E∞. Of course, D∞ is not a priori open in the projective limit topology. However, let

Bf∞=x∈E∞|supi∈ℕ∥x∥Eici<+∞.

This space is a Banach space for the norm ∥⋅∥f∞=sup∥⋅∥Eici∣i∈ℕ.

Since ∥⋅∥f∞<1⇔∀i∈ℕ;∥x∥Ei<ci, we get that the open ball of radius 1 centered at 0 in Bf∞ is a subset of D∞ which ends the proof.□

Let us now extend it to a class of functions that we call tame with respect to a control function Φ. For this, we define the sequences (Ei)i∈ℕ and (Fi)i∈ℕ as before, as well as E∞ and F∞. We also define a similar sequence (Gi)i∈ℕ of Banach spaces and G∞ the projective limit of this family.

Definition 2.2

Let U0×V0 be an open neighborhood of 0 in E0×F0 and let U∞=U0∩E∞ and V∞=V0∩F∞. Let{Φ(x,y)}(x,y)∈U∞×V∞ be a family of maps from G∞toF∞ such that

∀(x,y)∈U∞×V∞,Φ(x,y)−1(0)={0}.

A map

f:U∞×V∞→G∞

is Φ-tame if and only if the map

f∞:(x,y)∈U∞×V∞↦Φ(x,y)∘f(x,y)

is the restriction of Cr-maps (r≥1)

fi:Ui×Vi→Vi.

Theorem 2.3

Let f be aΦ-tame map, such that,

∀i∈ℕ,D2fi(0;0)=IdFi.

Then, there exists a non-empty domainD∞⊂U∞, possibly non-open inU∞and a functionu∞:D∞→V∞such that,

∀x∈D∞,f(x;u∞(x))=0.

Moreover, there exists a sequence of positive real numbers(ci)i∈ℕsuch thatD∞contains the unit ball of the Banach spaceBf,Φ⊂E∞defined as the domain of the norm

∥x∥f,Φ=sup∥x∥Eici|i∈ℕ.

Proof

We apply Theorem 2.2 to Φ(.,.)∘f(.,.). Then, since Φ(.,.)−1(0)=0,

Φ(x,u∞(x))f(x;u∞(x))=0⇒f(x,u∞(x))=0.□

3 Tentatives for inverse function and Fröbenius theorems

3.1 “Local” inverse theorem

Let (Ei)i∈ℕ be a sequence of Banach spaces as before, i.e., ∀i∈ℕ we have Ei+1⊂Ei with continuous inclusion, and let E∞ be the projective limit of the family (Ei)i∈ℕ. Let U0 be an open neighborhood of 0 in E0 and define for i∈ℕ∪{∞}, Ui=U0∩Ei. Let V0 be an open neighborhood of 0 in E0 and define for i∈ℕ∪{∞}, Vi=V0∩Ei. Let f∞:U∞→V∞ be a Cr-map (r≥1)

such that f(0)=0,
which the restriction of the maps fi:Ui→Vi, for i∈ℕ,
and such that Dfi(0)=IdEi.

Theorem 3.1

There exists a non-empty domainD⊂U∞which contains the open unit ball of a Banach spaceBf∞⊂E∞with norm defined by a sequence(ki)of positive numbers by

∥⋅∥f∞=sup∥⋅∥Eiki|i∈ℕ

such thatf∞|Dis a bijection fromDontof∞(D).

Proof

We apply Theorem 2.2 to g(x,y)=x−f∞(y) for (x,y)∈V∞×E∞. Indeed, we define a Cr-map g:V∞×U∞→E∞, which is the restriction of the maps

gi:(x,y)∈Vi×Ui↦x−fi(y)∈Ei.

We have that

D2gi(0;0)=Dfi(0)=IdEi

so that there exists a domain D∞⊂V∞ and a sequence (ci) of positive real numbers such that D∞ contains the unit open ball of the Banach space Bg∞⊂E∞ with the norm

∥⋅∥g∞=sup∥⋅∥Eici|i∈ℕ

and a function u∞:D∞→U∞ such that

∀x∈D∞,x−f∞(u∞(x))=0.

We set D=u∞(D∞), and hence D∞=f∞(D). Since each fi is a C0-map, there exists a sequence (ki) of positive numbers such that

∥y∥Ei<ki⇒∥fi(y)∥Ei<ci.

Setting y=u∞(x), we also get that ∀y∈D,y−u∞∘f∞(y)=0 and ∀y∈U∞,

sup∥y∥Eiki∣i∈ℕ<1⇒sup∥f∞(y)∥Eici∣i∈ℕ<1⇒f∞(y)∈D∞⇒y=u∞∘f∞(y)∈D.□

3.2 An obstruction for a Fröbenius theorem

Let (Ei)i∈ℕ and (Fi)i∈ℕ be two sequences of Banach spaces such that

∀i∈ℕ,Ei+1⊂EiandFi+1⊂Fi,withcontinuousinclusion,

and let E∞ and F∞ be the projective limits of the two previous families. We then consider U0 and V0 two open neighborhoods of 0 in E0 and F0, respectively. Let us now define, for i∈ℕ, Ui=U0∩Ei, Vi=V0∩Fi and Oi=Ui×Vi. A setting for an adapted Fröbenius theorem would be the following: let

fi:Oi→L(Ei,Fi),i∈ℕ

be a collection of smooth maps satisfying the following condition:

i<j⇒fi|Oj:Oj→L(Ei,Fi)restrictsasalinearmaptofj:Oj→L(Ej,Fj)

and such that,

∀(x,y)∈Oi,∀(a,b)∈(Ei)2,(D1fi(x,y)(a)(b)+(D2fi(x,y))(fi(x,y)(a))(b)=(D1fi(x,y)(b)(a)+(D2fi(x,y))(fi(x,y)(b))(a)

(this condition is the analogous of the Fröbenius condition in a Banach setting). Let us now try to adapt the classical proof [3], with the help of Theorem 2.2. We can assume with no restriction that f(0;0)=0. We consider

Gi=Cb1([0,1],Fi)={γ∈C1([0,1],Fi)|γ(0)=0}

and

Hi=C0([0,1];Fi),

endowed with their usual topologies. Obviously, if i<j, the injections Gj⊂Gi and Hj⊂Hi are continuous. Let B0 be an open ball of U0 centered in 0,B0′ an open ball of V0 centered in 0,B0′ an open ball of G0 centered in 0. We set, for i∈ℕ∪{∞}, Bi=B0∩Ei, Bi′=B0′∩Fi and Bi″=B0″∩Gi. Then, we define, for i∈ℕ∪{∞},

gi:Bi×Bi′×Bi″→Higi(x;y;γ)(t)=dγdt(t)−fi(tx;y+γ(t))(x).

We then apply Theorem 2.2 to

∫0(.)∘g∞:B∞×B∞′×B∞″⊂(E∞×E∞)×G∞→G∞.

There exists a domain D∞ such that we can define the function α∞ as the unique function such that

α(0;0)=0gi(x;y;α∞(x;y))=0∀(x;y)∈D∞

Since we set

J(x;y)=y+α∞(x;y)(1),

uniqueness follows from Theorem 2.2.

Open problem: We are now facing a theoretical impossibility. Classical theory of differentiation is valid for functions on open domains. We need here to consider J which is here defined as D∞ and which is not a priori open. There exist numerous extensions of the classical theory of differentiation, one of them is used in [6] based on [2]. Which one is better for this setting?

4 Application: on a projective limit group of operators

We consider here a sequence of Banach spaces (Ei)i∈ℕ

∀i∈ℕ,Ei+1⊂Ei,withcontinuousinclusion.

We also assume that

∀i∈ℕ,Ei+1(asasubset)isdenseinEi.

Following [7,8], we consider the set of linear maps E∞→E∞ which extend to bounded linear maps Ei→Ei. Let us note it as L∞ and GL∞=∩i∈ℕGL(Ei) is a group known as a topological group [8], and [7] quotes “natural differentiation rules” that are identified in [11,13] as generating a smooth Lie group for generalized differentiation on Frölicher or diffeological spaces. Let i∈ℕ, we define Li as the closure in L(Ei) of the set of a∈L∞ that extend by density to an element of L(Ei).

We equip these spaces with the norms

∥a∥i=max{∥a∥L(Ei−r)|0≤r≤i}.

We apply Theorem 2.2 to the map

f∞:(a,b)∈L∞2↦(Id+a)(Id+b)−Id

for the sequence of Banach spaces (Li) with projective limit L∞. We already know that the maximal domain D∞ of the implicit function obtained will be

D∞⊃{a∈L∞|Id+a∈GL∞}

and the implicit function will be

u∞:a∈D∞↦(Id+a)−1−Id,

where (Id+a)−1 is the right inverse of Id+a. But the main question about GL∞ is the most adequate structure for it: it behaves like a Lie group [7,11], but does not carry a priori charts which allow us only to consider it as a topological group [8]. Applying Theorem 2.2, there exists a Banach subspace B of L∞ defined by the norm

∥a∥=supi∈ℕ∥a∥L(Ei)ci.

But we easily show that each Li is a Banach algebra, so that, ci=1 since its group of the units contains the open ball of radius 1 centered at Id. By the way,

∥a∥=supi∈ℕ∥a∥L(Ei)

and B is a Banach algebra, with group of the units GL(B)⊂GL∞ which is a Banach Lie group. We finish with the special case when (Ei) is an inverse limit of Hilbert (ILH) sequence (i.e., a sequence of Hilbert spaces with continuous inclusion and such that Ei+1 is dense in Ei, see [7]) and when there exists a self-adjoint, positive (unbounded) operator Q such that

(Qia,b)E0=(a,b)Ei.

In this case, there exists (ek)k∈ℕ an orthonormal base in E0 of eigenvectors of Q in E∞, which is also orthogonal in Ei. In this case, the orthogonal projections

a↦(ek,a)E0ek

restrict to operators in B, which shows that B is an infinite dimensional Banach algebra.

Open question: There is a natural right action of GL(B) on GL∞ by composition. What is the (e.g. topological) structure of GL∞/GL(B)?

Acknowledgments

The author would like to thank the three anonymous referees for their efforts and for their precise work, which led to the present version. In particular, their remarks (even if not explicitly given for this issue) led the author to improve (and weaken) the necessary assumptions for Theorem 2.2, which is the key theorem of this work.

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Received: 2019-09-20

Revised: 2020-04-11

Accepted: 2020-04-17

Published Online: 2020-07-02

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https://doi.org/10.1515/dema-2020-0008

Keywords for this article

implicit functions; ILB spaces

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