Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity

http://www.elsevier.com/locate/aim

Advances in Mathematics 187 (2004) 488–520

Linear biseparating maps between spaces ofvector-valued differentiable functions and

automatic continuity

Jesus Araujo1

Departamento de Matematicas, Estadıstica y Computacion, Universidad de Cantabria,

45Facultad de Ciencias, Avda. de los Castros, s. n., E-39071 Santander, Spain

Received 19 April 2002; accepted 24 September 2003

Communicated by Professor Vesentini

Abstract

We give a complete description of linear biseparating maps between spaces of vector-valued

differentiable functions. This description automatically implies continuity of such maps.

r 2003 Elsevier Inc. All rights reserved.

MSC: primary 47B33; secondary 46H40, 47B38, 46E40, 46E25

Keywords: Biseparating map; Disjointness preserving; Automatic continuity; Vector-valued differentiable

functions

1. Introduction

It is well known that an algebraic link between spaces of continuous functions maylead to a topological link between the spaces on which the functions are defined. Forinstance, it turns out that if there exists a ring isomorphism T : CðXÞ-CðYÞ; thenthe realcompactifications of X and Y are homeomorphic [16, pp. 115–118]. Also if h

is the resultant homeomorphism from the realcompactification of Y onto that of X ;then Tf ¼ f 3h for every fACðXÞ; so we have a complete description of it. As aresult, when X and Y are realcompact, we deduce that if both spaces of continuous

ARTICLE IN PRESS

E-mail address: [email protected] partially supported by the Spanish Direccion General de Investigacion Cientıfica y Tecnica

(DGICYT, PB98-1102).

0001-8708/$ - see front matter r 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.aim.2003.09.007

functions CðX Þ and CðYÞ are endowed with the compact-open topology, then everyring isomorphism between them is continuous. In this result, the key point is thatevery ring isomorphism sends maximal ideals into maximal ideals. This implies that agood description of maximal ideals lead to the definition of a map from Y onto X :

Of course, the pattern above has been successfully applied to many other algebrasof functions. However the situation becomes more complicated if we consider spacesof functions which take values in arbitrary Banach spaces. In this context and unlikealgebra or ring homomorphisms, we can still use mappings satisfying the propertyjjTf jj jjTgjj � 0 if and only if jj f jj jjgjj � 0: These maps are called biseparating, andcoincide with disjointness preserving mappings whose inverses preserve disjointnesstoo [1]. In general these maps turn out to be efficacious substitutes forhomomorphisms. Indeed, in [2], we prove that the existence of a biseparatingmapping between a large class of spaces of vector-valued continuous functionsAðX ;EÞ and AðY ;FÞ (E; F are Banach spaces) yields homeomorphisms betweensome compactifications (and even the realcompactifications) of X and Y : Theautomatic continuity of a linear biseparating mapping is also accomplished in somecases (see [3,4]). Related results have also been given recently, for some other familiesof scalar-valued functions, for instance, in [5,15,19,20]. In this paper, we go a stepbeyond and work in a context which does not seem to have made its way into theliterature yet, namely, linear operators between spaces of differentiable functionstaking values in arbitrary Banach spaces.

As for spaces (indeed algebras) of scalar-valued differentiable functions, Myers[25] showed that the structure of a compact differentiable manifold of class Cn isdetermined by the algebra of all real-valued functions on M of class Cn: In this line,Pursell [28] checked that the ring structure of infinitely differentiable functionsdefined on an open convex set of Rn determines such set up to a diffeomorphism.Homomorphisms between algebras of differentiable functions defined on realBanach spaces have been studied by Aron et al. [6]; in their paper a description ofhomomorphisms is given and the automatic continuity is obtained as a corollary inquite a general setting, which includes in particular the case when the Banach spacesare finite-dimensional (see also [17]).

On the other hand, automatic continuity results for algebras of differentiablefunctions have been also given for instance in [7,21,24,26,27]. In particular,automatic continuity of separating maps defined on spaces of differentiablefunctions has been studied by Kantrowitz and Neumann in [22]. For classicalresults and techniques in the study of automatic continuity, see [10,29], and therecent book [11] by Dales.

2. Preliminaries and notation

Let E be a real Banach space. If l ¼ ðl1; l2;y; lpÞ is a p-tuple of nonnegative

integers, we set jlj :¼ l1 þ l2 þ?þ lp: If O is a nonempty open subset of Rp; then

CnðO;EÞ consists of the E-valued functions f in O that are of class Cn; that is, those

ARTICLE IN PRESSJ. Araujo / Advances in Mathematics 187 (2004) 488–520 489

functions whose partial derivatives

@lf :¼ @l1þl2þ?þlp f

@xl1

1 @xl2

2 y@xlpp

exist and are continuous for each l ¼ ðl1; l2;y; lpÞAL; where L :¼flAðN,f0gÞp: jljpng: It is well known that, if for LkðRp;EÞ we denote the spaceof continuous k-R-linear maps of Rp into E; then CnðO;EÞ coincides with the space

of maps f :O-E such that the differential Dkf :O-LkðRp;EÞ exists and iscontinuous for each k ¼ 0;y; n: It is well known that, when it exists at a point aAO;the differential Dkf ðaÞALkðRp;EÞ is a symmetric form of degree k:

Now, given a map fACnðO;EÞ; we define its Taylor polynomial function of degreen at aAO as

TaðxÞ :¼ f ðaÞ þ Df ðaÞðx aÞ þ 1

2D2f ðaÞðx a; x aÞ þ?

þ 1

n!Dnf ðaÞðx a; x a;y; x a|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

n

Þ:

In the case when E ¼ R; Cnc ðO;RÞ will denote the subring of CnðO;RÞ of functions

with compact support.

On the other hand, in the case when O is also bounded CnðO;EÞ denotes thesubspace of CnðO;EÞ of those functions whose partial derivatives up to order n

admit continuous extension to the boundary of O:For a set CCRp; clRp C and intRp C denote its closure and its interior in Rp;

respectively. Given x0ARp and d40; Bðx0; dÞ and Bðx0; dÞ stand for the open andclosed balls of center x0 and radius d; respectively. As for the norm in Rp; if x ¼ðx1; x2;y; xpÞ belongs to Rp; we set jxj :¼ maxj jxjj: For iAf1; 2;y; pg; xi :O-R

will be the projection on the ith coordinate, that is, xiðt1; t2;y; tpÞ ¼ ti for every

ðt1; t2;y; tpÞAO:Notice also that if E is a Banach space over C; it can also be viewed as a real space,

and in this sense we consider the space CnðO;EÞ defined. It is immediate that allresults valid in the real setting hold also in the complex case. But being also complexensures that CnðO;EÞ is both real and complex as a linear space, and consequentlywe can consider both real and complex linear maps from CnðO;EÞ into some othervector spaces. This is the reason why in this paper we will assume that E and F areK-Banach spaces, where K ¼ R or C:

As for the spaces of linear functions, we will denote by L0ðE;FÞ and by I 0ðE;FÞthe sets of (not necessarily continuous) linear maps and bijective linear maps from E

into F ; respectively. LðE;FÞ and IðE;FÞ will denote the spaces of continuous linearmaps and bijective continuous linear maps from E into F :

Also, if eAE; then bee denotes the constant function from O to E taking thevalue e:

ARTICLE IN PRESSJ. Araujo / Advances in Mathematics 187 (2004) 488–520490

The context. From now on we will assume that we are in one of the following twosituations. All definitions, results and comments given in this paper apply to thesetwo situations unless otherwise stated.

* Situation 1. O and O0 are (not necessarily bounded) open subsets of Rp

and Rq; respectively ðp; qANÞ: AnðO;EÞ ¼ CnðO;EÞ and AmðO0;FÞ ¼CmðO0;FÞ ðn;mX1Þ: AnðO;KÞ ¼ CnðO;KÞ and AmðO0;KÞ ¼ CmðO0;KÞ:

* Situation 2. O and O0 are bounded open subsets of Rp and Rq; respectively

(p; qAN), with the property that intRp clRp O ¼ O and intRq clRq O0 ¼ O0:

AnðO;EÞ¼CnðO;EÞ and AmðO0;FÞ ¼ CmðO0;FÞ (n;mX1). AnðO;KÞ ¼ CnðO;KÞ

and AmðO0;KÞ ¼ CmðO0;KÞ:

This means that when we refer to spaces O; O0; AnðO;EÞ; AmðO0;FÞ; AnðO;KÞ;AmðO0;KÞ; we assume that all of them are included at the same time in one of theabove two situations.

The topologies. One of the goals of this paper is to provide some results of automaticcontinuity. This will be done when the spaces of functions are endowed with somenatural topologies.

Definition 2.1. We say that a locally convex topology in AnðO;EÞ is compatible with

the pointwise convergence if the following two conditions are satisfied:

1. when endowed with it, AnðO;EÞ is a Frechet (or Banach) space, and2. if ð fnÞ is a sequence in AnðO;EÞ converging to zero, then ð fnðxÞÞ converges to zero

for every xAO:

Biseparating maps. For a function fAAnðO;EÞ; we denote by cð f Þ the cozero set off ; that is, the set fxAO: f ðxÞa0g:

Definition 2.2. A map T : AnðO;EÞ-AmðO0;FÞ is said to be separating if it is

additive and cðTf Þ-cðTgÞ ¼ | whenever f ; gAAnðO;EÞ satisfy cð f Þ-cðgÞ ¼ |:Besides T is said to be biseparating if it is bijective and both T and T1 areseparating.

Equivalently, we see that an additive map T : AnðO;EÞ-AmðO0;FÞ is separating

if jjðTf ÞðyÞjj jjðTgÞðyÞjj ¼ 0 for all yAO0 whenever f ; gAAnðO;EÞ satisfyjj f ðxÞjj jjgðxÞjj ¼ 0 for all xAO:

Let O1 :¼ O; O01 :¼ O0 when we are in Situation 1, and O1 :¼ O; O0

1 :¼ O0if we are

in Situation 2. A point xAO1 is said to be a support point of yAO01 if, for every

neighborhood U of x in O1; there exists fAAnðO;EÞ satisfying cð f ÞCU such thatðTf ÞðyÞa0:

In [2], biseparating maps are studied in a more general setting. In particular,

applied to our situation, we have that AnðO;EÞ and AmðO0;FÞ are modules over the


strongly regular rings CnðO1;RÞ and CmðO01;RÞ; respectively (for the definition of

strongly regular ring, see [2]; see also [30, Corollary 1.2] or [18, Proposition 2.12.5]).According to [2, Corollary 3.2], we conclude that there exists a homeomorphism h

from O01 onto O1 (which implies that in both Situation 1 and Situation 2 h is a

homeomorphism from O0 onto O). This map h sends each point in O01 into its support

point in O1; and is called support map for T : It turns out that the support map for

T1 is h1 (see the proof of [2, Theorem 3.1]). Rephrasing Lemma 4.4 in [2], we havethe following property.

Lemma 2.1. If yAO01 and fAAnðO;EÞ vanishes on a neighborhood of hðyÞ; then

ðTf ÞðyÞ ¼ 0:

Functions of class s Cn. Suppose that K :O-LðE;FÞ is a continuous map, whereLðE;FÞ is endowed with the topology of the norm. For each eAE; we define Ke :

O-F as KeðyÞ :¼ ðKyÞðeÞ for every yAO: We say that K is of class s C1 if, forevery eAE; the map Ke admits all partial derivatives of order 1 in O; and for eachi ¼ 1;y; p; the map

@s

@xi

K :O-LðE;FÞ;

sending each yAO and each eAE into @@xi

KeðyÞ; is continuous when considering in

LðE;FÞ the strong operator topology, that is, the coarsest topology such that themapping AALðE;FÞ+AeAF is continuous for every eAE:

Definition 2.3. Let nX2: A map J :O-LðE;FÞ is said to be of class s Cn if thefollowing two statements are satisfied:

1. J is of class Cn1 (considering LðE;FÞ as a Banach space).

2. All partial derivatives K :O-LðE;FÞ of order n 1 of J are of class s C1:

Examples. Next we provide two examples of biseparating linear maps between

spaces of vector-valued functions of class C1: The first example will give the mostbasic form of a biseparating linear map, which is essentially the only possible when E

and F are finite-dimensional spaces, that is, the case when the operator through

which it is defined is itself of class C1: In the second example, the biseparating map is

defined through an operator which is of class s C1 but it is not of class C1:

Example 2.4. Let O ¼ O0 ¼ ð1; 1Þ; E ¼ F ¼ R2: For each tAð1; 1Þ; let us consider

Jt :R2-R2 defined as

ðJtÞðx1; x2Þ :¼ ðð2 þ tÞx1; ð2 tÞx2Þ


for ðx1; x2ÞAR2: Clearly each Jt is linear, bijective and continuous, being its inverse

Kt :R2-R2 defined as

ðKtÞðx1; x2Þ ¼x1

2 þ t;

x2

2 t

� �

for every ðx1; x2ÞAR2:

It is easy to see that the derivatives of J : ð1; 1Þ-LðR2;R2Þ and

K : ð1; 1Þ-LðR2;R2Þ exist and, for ðx1; x2ÞAR2; ðJ 0tÞðx1; x2Þ ¼ ðx1;x2Þ and

ðK 0tÞðx1; x2Þ ¼tx1

ð2 þ tÞ2;

tx2

ð2 tÞ2

!:

In this way we see that J and K are of class C1: Consequently the map T :

C1ðð1; 1Þ;R2Þ-C1ðð1; 1Þ;R2Þ; defined as ðTf ÞðtÞ ¼ ðJtÞð f ðtÞÞ is biseparating (seeProposition 6.1).

Example 2.5. Let O ¼ O0 ¼ ð1; 1Þ; E ¼ F ¼ c0 be the space of sequences in K

converging to zero, endowed with the sup norm. For each tAð1; 1Þ; let us define themap Jt : c0-c0 as

ðJtÞðxnÞ :¼ 2 þ 2n 1

2nt

2n2n1

� �xn

� �for each ðxnÞAc0:

It is easy to see that each JtAIðc0; c0Þ; that is, it is linear, continuous and bijective,and that its inverse is the map KtAIðc0; c0Þ defined as

ðKtÞðxnÞ :¼xn

2 þ 2n12n

t2n

2n1

� �for each ðxnÞAc0:

Let us see that the map J : ð1; 1Þ-Lðc0; c0Þ is continuous when Lðc0; c0Þ isendowed with the norm topology. We are going to show that if 0otot0o1; thenjjJt Jt0jjp2jt t0j: It is clear that we just need to prove that if nAN; then

jt02n=2n1 t2n=2n1jp2jt0 tj: Notice that

0p t02n

2n1 t2n

2n1

p t02n

2n1 þ t0t1

2n1 tt01

2n1 t2n

2n1


¼ðt0 tÞðt01

2n1 þ t1

2n1Þ

p 2ðt0 tÞ:

Since a similar reasoning also holds for any t; t0Að1; 1Þ; we conclude that J iscontinuous. In the same way we can prove that K : ð1; 1Þ-Lðc0; c0Þ is continuous.

On the other hand, we have that, for each ðxnÞAc0; the mapsJðxnÞ;KðxnÞ : ð1; 1Þ-c0; defined as JðxnÞðtÞ :¼ ðJtÞðxnÞ and KðxnÞðtÞ :¼ ðKtÞðxnÞ are

derivable, and it can be seen that for each tAð1; 1Þ;

J 0ðxnÞðtÞ ¼ ðt

12n1xnÞ

and

K 0ðxnÞðtÞ ¼ t

12n1xn

2 þ 2n12n

t2n

2n1

� �2

0BBB@1CCCA ¼

J 0ðxnÞðtÞ

2 þ 2n12n

t2n

2n1

� �2

0BBB@1CCCA:

Now, let us define J 0s;K 0

s : ð1; 1Þ-Lðc0; c0Þ as J 0sðtÞðxnÞ :¼ J 0

ðxnÞðtÞ and K 0sðtÞðxnÞ :¼

K 0ðxnÞðtÞ for each tAð1; 1Þ and each ðxnÞAc0: It is straightforward to see that, for

every ðxnÞAc0; the maps J 0ðxnÞ and K 0

ðxnÞ are continuous, which is to say that J 0s and K 0

s

are continuous when Lðc0; c0Þ is endowed with the strong operator topology. We

conclude that J and K are of class s C1:Finally notice that, when Lðc0; c0Þ is endowed with the topology of the norm, then

jjJ 0sðtÞjj ¼ 1 for every tAð1; 1Þ\f0g; which implies that J 0

s is not continuous at t ¼ 0:

If we define now the map T : C1ðð1; 1Þ; c0Þ-C1ðð1; 1Þ; c0Þ as

ðTf ÞðtÞ :¼ ðJtÞð f ðtÞÞ;

it turns out that T is well-defined (see Proposition 6.1), and it is routine matter tocheck that it is biseparating.

3. Some previous results

Lemma 3.1. Suppose that a0; a1;y; akACnðO;EÞ; and that f :O� R-E is a

polynomial in t defined as

f ðx; tÞ ¼Xk

i¼0

aiðxÞti

for every ðx; tÞAO� R: Then fACnðO� R;EÞ:


Proof. It is immediate from the fact that all partial derivatives up to order n existand are continuous. &

Lemma 3.2. Suppose that f ; gACnðO;EÞ and k :O-R satisfy

f ðxÞ ¼ kðxÞgðxÞ

for every xAO: If gðxÞa0 for every xAO; then kACnðO;RÞ:

Proof. Fix x0AO: We are going to prove that k is of class Cn in a neighborhood of

x0: First we take f 0 : E-R linear and continuous, and such that f 0ðgðx0ÞÞa0: Since f 0

and g are continuous, then there exists an open neighborhood U of x0 such that

f 0ðgðxÞÞa0

for every xAU :Now

f 0ð f ðxÞÞ ¼ kðxÞf 0ðgðxÞÞ

for every xAU : This implies that, for every xAU ;

kðxÞ ¼ f 0ð f ðxÞÞf 0ðgðxÞÞ ;

which is the quotient of two real-valued functions of class Cn:This proves that k is of class Cn: &

The proof of the following result is straightforward.

Lemma 3.3. Suppose that fACnðO;EÞ; aAO; and 1pkpn: Then the kth derivative of

the Taylor polynomial function Ta of degree n of f is equal to the Taylor polynomial

function of degree n k of Dkf at a:

The following theorem, known as Whitney’s extension theorem, can be found, forinstance, in [12, Theorem 3.1.14].

Theorem 3.4. Suppose nAN; A is a closed subset of Rp; and to each aAA corresponds a

polynomial function

Pa :Rp-E

with degree Papn: Whenever CCA and d40 let rðC; dÞ be the supremum of the set of

all numbers

jjDiPaðbÞ DiPbðbÞjj � ja bjin � ðn iÞ!

corresponding to i ¼ 0;y; n and a; bAC with 0oja bjpd:


If rðC; dÞ-0 as d-0þ for each compact subset C of A; then there exists a map

g :Rp-E of class Cn such that

DigðaÞ ¼ DiPaðaÞ

for i ¼ 0;y; n and aAA:

Proposition 3.5. Let pX2: For sAR; s40; consider the following compact subsets

of Rp:

A :¼ ðx1; x2;y; xpÞARp:Xp

i¼2

x2i px2

1=9; jx1jps

( );

Aþ :¼ fðx1; x2;y; xpÞAA: x1X0g;

and

A :¼ fðx1; x2;y; xpÞAA: x1p0g:

Suppose that O is an open subset of Rp containing A and that f belongs to

CnðO;EÞ: If

@lf ð0; 0;y; 0Þ ¼ 0

for every lAL; then there exists a function fþACnðO;EÞ with compact support such

that, for lAL;

@lfþðxÞ ¼ @lf ðxÞ

for every xAAþ; and

@lfþðxÞ ¼ 0

for every xAA:

Proof. Suppose that for xAA; Tx stands for the polynomial function of degree n

given in the Taylor formula for f at x: Now for aAA; we consider as Pa the

polynomial identically zero, and for aAAþ; we consider as Pa the polynomial Ta: Asit is seen for instance in [9, Theorem 2.71] or [23, p. 350], if r40 and jb ajor; wehave that

jjTaðbÞ f ðbÞjjpjb ajn

n!sup

jxajpr

jjDnf ðxÞ Dnf ðaÞjj:

Now it is easy to see that, by Lemma 3.3, for iAf1; 2;y; ng;

jjDiTaðbÞ Dif ðbÞjjpjb ajni

ðn iÞ! supjxajpr



This proves that if jb ajor;

jjDiTaðbÞ Dif ðbÞjj � jb ajin � ðn iÞ!p supjxajpr

jjDnf ðxÞ Dnf ðaÞjj

for iAf0; 1;y; ng: Then we define tðrÞ as the supremum of the set of all numbers

supjxajpr


for x; aAA; which is a real number, because A is compact. Clearly, since Dnf iscontinuous, if r tends to zero, tðrÞ tends to zero. Now suppose that a and b belong toA and 0ojb ajpr: Then we have the following possibilities:

* a; bAAþ: Then we have that, by Lemma 3.3, for iAf0; 1;y; ng;

DiPbðbÞ ¼ Dif ðbÞand consequently

jjDiPaðbÞ DiPbðbÞjj ¼ jjDiTaðbÞ Dif ðbÞjj:

* a; beAþ: Then

jjDiPaðbÞ DiPbðbÞjj ¼ 0:

* aeAþ; bAAþ: Note that since we are assuming by hypothesis that

Dif ð0; 0;y; 0Þ ¼ 0; then DiP0ðbÞ ¼ 0 for all iAf0; 1;y; ng: Consequently

jjDiPaðbÞ DiPbðbÞjj ¼ jjDiPbðbÞjj

¼ jjDiP0ðbÞ DiPbðbÞjj

¼ jjDiT0ðbÞ Dif ðbÞjj:* aAAþ; beAþ: Then we have that

jjDiPaðbÞ DiPbðbÞjj ¼ jjDiTaðbÞjj

p jjDiTaðbÞ Dif ðbÞjj þ jjDif ðbÞjj

¼ jjDiTaðbÞ Dif ðbÞjj þ jjDif ðbÞ DiT0ðbÞjj:

Note that in the third and forth cases above, jaj; jbjpjb ajpr: This implies that inthese two cases

jjDiPaðbÞ DiPbðbÞjj � jb ajin � ðn iÞ!p supjxajpr


þ supjxjpr

jjDnf ðxÞ Dnf ð0Þjj:


On the other hand it is easy to see that in the other two cases

jjDiPaðbÞ DiPbðbÞjj � jb ajin � ðn iÞ!p supjxajpr


This facts imply that, if r is defined as in Theorem 3.4, rðA; rÞp2tðrÞ: Also, it is clearthat if C is a compact subset of A; then rðC; rÞprðA; rÞ: Consequently by Theorem3.4, we have that there exists f0ACnðO;EÞ such that, given any lAL;

@lf0ðxÞ ¼ @lf ðxÞ

for every xAAþ; and

@lf0ðxÞ ¼ 0

for every xAA: Also it is clear that if we take g0ACnc ðO;RÞ such that g0 � 1 on an

open neighborhood of A; then fþ :¼ g0 f0ACnðO;EÞ satisfies the requirements of thetheorem. &

4. Biseparating maps: a first approach

In this section we make a first attempt to describe all biseparating maps, but we donot take into account some important details which will be discussed in Section 6. In

this way we characterize all biseparating linear maps from AnðO;EÞ onto AmðO0;FÞas weighted composition bijective maps. Notice that we assume no continuity

properties on T : In fact, we will suppose that our spaces AnðO;EÞ and AmðO0;FÞ arenot endowed with any topologies.

Lemma 4.1. Suppose that O contains the origin, and that T : AnðO;EÞ-AmðO0;FÞ is a

biseparating map. Assume also that fAAnðO;EÞ satisfies that for all lAL;

@lf ð0; 0;y; 0Þ ¼ 0:

If ð0; 0;y; 0ÞAO is the support point of yAO0; then ðTf ÞðyÞ ¼ 0:

Proof. First suppose that p41 and that the closed ball of center 0 and radius s is

contained in O: If we take Aþ; A and fþ as in Proposition 3.5, then fþ and f fþbelong to AnðO;EÞ and satisfy fþðxÞ ¼ 0 and ð f fþÞðxÞ ¼ 0 for every xAA and

for every xAAþ respectively. We have that ðTf ÞðyÞ ¼ ðTfþÞðyÞ þ ðTð f fþÞÞðyÞ:Also, since for any neighborhood U of the origin there exists an open subset V of U

such that fþðxÞ ¼ 0 for every xAV ; then we have that, taking into account that the

support map for T1 is h1; by Lemma 2.1, ðTfþÞðh1ðxÞÞ ¼ 0: Since h :O0-O is ahomeomorphism, we deduce that ðTfþÞðyÞ ¼ 0 and, in the same way, ðTð f fþÞÞðyÞ ¼ 0: We conclude that ðTf ÞðyÞ ¼ 0:


Consider now the case when p ¼ 1: Note that if fAAnðO;EÞ satisfies f ð0Þ ¼ 0 and

0 ¼ f 0ð0Þ ¼ ? ¼ f ðnÞð0Þ; then it is clear that f xðN;0Þ and f xð0;þNÞ belong to

AnðO;EÞ (where xA stands for the characteristic function of A) and, as above,ðTð f xð0;þNÞÞÞðyÞ ¼ 0 ¼ ðTð f xðN;0ÞÞÞðyÞ: We conclude that ðTf ÞðyÞ ¼ 0: &

Proposition 4.2. Suppose that T : AnðO;EÞ-AmðO0;FÞ is a K-linear biseparating

map. Then p ¼ q; n ¼ m; and there exist a diffeomorphism h of class Cn from O0 onto Oand a map J :O0-I 0ðE;FÞ such that for every yAO0 and every fAAnðO;EÞ;

ðTf ÞðyÞ ¼ ðJyÞð f ðhðyÞÞÞ:

Proof. First, the existence of the homeomorphism h (the support map) between O0

and O implies that p ¼ q (see for instance [13, p. 120]).

Note that if fAAnðO;EÞ and yAO0 satisfy

@lf ðhðyÞÞ ¼ 0

for all lAL; then by Lemma 4.1 we have that

ðTf ÞðyÞ ¼ 0:

Now take yAO0 and fix eAE; ea0: If #L stands for the cardinal of L; then we can

define a linear map Sy :R#L-F as follows. Given ðal1;y; al#LÞAR#L; we consider

any fAAnðO;RÞ such that

@lf ðhðyÞÞ ¼ al

for every lAL: Then we define

Syðal1;y; al#LÞ :¼ ðTf eÞðyÞ:

The map Sy is linear and, as we have seen above, does not depend on the function f

we choose. This implies that it is well defined.

Then it is easy to see that there exist functions al from O0 into F ; lAL; such that

for every yAO0 and every fAAnðO;RÞ;

ðTf eÞðyÞ ¼XlAL

alðyÞ@lf ðhðyÞÞ: ð4:1Þ

From now on we consider iAf1; 2;y; pg fixed.Next we define some functions

a0i ; a

1i ;y; anþ1

i


from O0 � R into F : For every yAO0 and tAR;

a0i ðy; tÞ :¼ ðT #eÞðyÞ;

a1i ðy; tÞ :¼ ðTxieÞðyÞ a0

i ðy; tÞt;

2!a2i ðy; tÞ :¼ ðTx2

i eÞðyÞ a0i ðy; tÞt2 2a1

i ðy; tÞt;

3!a3i ðy; tÞ :¼ ðTx3

i eÞðyÞ a0i ðy; tÞt3 3a1

i ðy; tÞt2 6a2i ðy; tÞt;

and in general, for kAf1; 2;y; n; n þ 1g

k!aki ðy; tÞ :¼ðTxk

i eÞðyÞ a0i ðy; tÞtk ka1

i ðy; tÞtk1 kðk 1Þa2i ðy; tÞtk2

? k!ak1i ðy; tÞt:

Claim 4.1. For lAf0; 1;y; n þ 1g; l!ali is a polynomial in t whose coefficients are a

linear combination of T #e;Txie;y;Txlie: Moreover, for yAO0 fixed, the degree of the

polynomial l!aliðy; tÞ is at most l: If we also assume that ðT #eÞðyÞa0; then the degree of

l!aliðy; tÞ is l and its leading coefficient is equal to ð1Þla0

i ðy; tÞ (notice that this term

does not depend on t).

We are going to prove it by applying induction on l: It is clear that this is true forl ¼ 0: Suppose that this relation also holds for lAf0; 1; 2;y; kg for some kpn: Weare going to see that it holds for l ¼ k þ 1: We have that

ðk þ 1Þ!akþ1i ðy; tÞ :¼ðTxkþ1

i eÞðyÞ a0i ðy; tÞtkþ1

ðk þ 1Þa1i ðy; tÞtk ðk þ 1Þka2

i ðy; tÞtk1

ðk þ 1Þkðk 1Þa3i ðy; tÞtk2

? ðk þ 1Þ!aki ðy; tÞt;

which implies that it is a polynomial in t and, for fixed yAO; its coefficient for the

term tkþ1 is a0i ðy; tÞ 1 þ ðk þ 1Þ kþ1

2

� �þ kþ1

3

� �? ðk þ 1Þð1Þk

� �; which is

equal to a0i ðy; tÞ ð1 1Þkþ1 þ kþ1

kþ1

� �ð1Þkþ1

� �; that is, to ð1Þkþ1a0

i ðy; tÞ:Thus the claim is proved.

Now, by Lemma 3.1, we have that for every kAf0; 1;y; n þ 1g; aki belongs to

CmðO0 � R;FÞ:


Next define a0 :¼ T #e; and for kAf1; 2;y; ng; ak :¼ al :O0-F ; where

l ¼ ð0; 0;y; 0; k;|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}i

0;y; 0Þ:

Also, let hi stand for the ith coordinate function of h:

Claim 4.2. For every kAf0; 1;y; ng; and for every yAO0;

akðyÞ ¼ aki ðy; hiðyÞÞ:

First we have from Eq. (4.1) that for kAf0; 1;y; ng and yAO0;

ðTxki eÞðyÞ ¼

XlAL

alðyÞ@lxki ðhðyÞÞ;

which can be written as

ðTxki eÞðyÞ ¼ a0ðyÞhk

i ðyÞ þ ka1ðyÞhk1i ðyÞ þ?þ k!ak1ðyÞhiðyÞ þ k!akðyÞ; ð4:2Þ

because

@lxki ðhðyÞÞ ¼ 0

whenever lAL; lað0; 0;y; j;|fflfflfflfflfflffl{zfflfflfflfflfflffl}i

0;y; 0Þ; jAf0; 1;y; kg:

On the other hand, it is clear that a0ðyÞ ¼ ðT #eÞðyÞ ¼ a0i ðy; hiðyÞÞ for every yAO0:

Also suppose that kon and that ajðyÞ ¼ a ji ðy; hiðyÞÞ for every jAf0; 1;y; kg and

every yAO0: Then, by Eq. (4.2), for yAO0

ðk þ 1Þ!akþ1ðyÞ ¼ ðTxkþ1i eÞðyÞ a0ðyÞhkþ1

i ðyÞ ðk þ 1Þa1ðyÞhki ðyÞ

? ðk þ 1Þ!akðyÞhiðyÞ;

which coincides with ðk þ 1Þ!akþ1i ðy; hiðyÞÞ; and the claim is proved.

On the other hand, notice that in the same way as we obtain Eq. (4.2), we have

ðTxnþ1i eÞðyÞ ¼ a0ðyÞhnþ1

i ðyÞ þ ðn þ 1Þa1ðyÞhni ðyÞ þ?þ ðn þ 1Þ!anðyÞhiðyÞ ð4:3Þ

for every yAO0:

Claim 4.3. Suppose that y0AO0 satisfies a0ðy0Þa0: Then for every open neighborhood

U of y0; there exists a nonempty open subset W 0i of U where hi is of class Cm:


First we define F 0i :O0 � R-F as F 0

i :¼ ðn þ 1Þ!anþ1i ; that is,

F 0i ðy; tÞ ¼ ðTxnþ1

i eÞðyÞ a0i ðy; tÞtnþ1 ðn þ 1Þa1

i ðy; tÞtn

ðn þ 1Þna2i ðy; tÞtn1 ? ðn þ 1Þ!an

i ðy; tÞt;

for every yAO0; tAR:

Then, if jAf1; 2;y; n þ 1g; we define Fj

i :O0 � R-F as

Fj

i ðy; tÞ ¼ @ jF0i

@t jðy; tÞ

for all yAO0; tAR:

Notice that from the definition of F 0i ; Claim 4.2 and Eq. (4.3), we deduce that

F 0i ðy; hiðyÞÞ ¼ 0

for every yAO0: Also, as we stated in Claim 4.1, the coefficients of F 0i as a polynomial

of degree n þ 1 in t are linear combinations of

ðT #eÞðyÞ; ðTxieÞðyÞ;y; ðTxnþ1i eÞðyÞ;

and consequently, by Lemma 3.1, for kAf1; 2;y; ng; Fki belongs to CmðO0 � R;FÞ:

Taking into account that F nþ1i ðy; tÞ ¼ ðn þ 1Þ!ð1Þnþ1a0ðyÞ for every ðy; tÞAO0 �

R; and the fact that a0ðy0Þa0; there exists k0Af0; 1;y; ng such that F k0

i ðy; hiðyÞÞ ¼ 0

for every y in a neighborhood of y0 and Fk0þ1i ðy; hiðyÞÞ takes a value different from 0

for some y in every neighborhood of y0: Suppose then that U is an open

neighborhood of y0 such that Fk0

i ðy; hiðyÞÞ ¼ 0 for every yAU and that y1AU

satisfies F k0þ1i ðy1; hiðy1ÞÞ ¼ fAF ; fa0: Now take f 0 in the dual space F 0 (where F is

viewed as a real Banach space) such that f 0ðfÞa0: According to the Implicit FunctionTheorem [14, p. 148], there exist a neighborhood V of ðy1; hiðy1ÞÞ; an openneighborhood W of y1; and a function f : W-R of class Cm such that fðy1Þ ¼hiðy1Þ and

fðy; tÞAV : f 03Fk0

i ðy; tÞ ¼ 0g ¼ fðy;fðyÞÞ: yAWg:

It is easy to prove that this implies that f � hi on a neighborhood W 0i of y1; that is,

for every open neighborhood U of y0; there exists an open subset W 0i of U where hi is

of class Cm: The claim is proved.

Since both T and T1 are biseparating, we can assume from now on, without lossof generality, that npm:

Claim 4.4. Suppose that U is a nonempty open subset of O0: Then there exists a

nonempty open subset W 0 of U such that the restriction of h to W 0 is a diffeomorphism

of class Cm:


Notice first that the open set fyAO0: a0ðyÞa0g is dense in O0: Otherwise we could

find gAAmðO0;FÞ; ga0; such that cðT #eÞ-cðgÞ ¼ cða0Þ-cðgÞ ¼ |: Since T1 is

separating, this would give us O-cðT1gÞ ¼ cð#eÞ-cðT1gÞ ¼ |; which is impossible.So far we have considered iAf1; 2;y; pg fixed. Of course a similar process can be

done for every iAf1; 2;y; pg: In particular, taking into account the aboveparagraph, by Claim 4.3, there exists an open subset W 0

1 of U such that h1 is of

class Cm in W 01: For the same reason we can find an open subset W 0

2 of W 01 where h2

is of class Cm: Following this process we construct (nonempty) open sets W 01;y;W 0

p

with W 01*W 0

2*?*W 0p such that h is of class Cm in W 0

p: It is clear that a similar

reasoning shows that the map h1 is of class Cn in an open subset V of hðW 0pÞ: Then

our situation is as follows: h1 is of class Cn in V and h is of class Cm in h1ðVÞ: It iswell known that, since mXnX1; this implies that h is a diffeomorphism of class Cm

in W 0 :¼ h1ðVÞ; and we are done.

Claim 4.5. Let W 0 be as in Claim 4.4. For every kAf0; 1;y; ng; the map ak belongs to

CmðW 0;FÞ:

First we have that a0 ¼ T #e belongs to AmðO0;FÞ: It is also clear that ifkAf0; 1;y; n 1g; then as given in Eq. (4.2),

ðk þ 1Þ!akþ1 ¼ Txkþ1i e a0hkþ1

i ðk þ 1Þa1hki ? ðk þ 1Þ!akhi

on O0:Consequently, since hiACmðW 0Þ; if a0; a1;y; ak belong to CmðW 0;FÞ; akþ1 also

belongs to CmðW 0;FÞ and then we are done.

Claim 4.6. For every kAf1; 2;y; ng; ak � 0 in W 0:

Suppose that y0AW 0 and anðy0Þa0: Since by Claim 4.5 an is continuous in W 0;there exists an open neighborhood Uðy0Þ of y0 such that Uðy0ÞCW 0 and anðyÞa0

for every yAUðy0Þ: Then take gACnðR;RÞ such that gðnÞ is not derivable at the pointhiðy0Þ: We define fAAnðO;RÞ as

f ðxÞ :¼ gðxiÞ

for every x ¼ ðx1; x2;y; xpÞAO: In this way we have that

@nþ1f

@xnþ1i

ðhðy0ÞÞ

does not exist. Consequently, using a reasoning similar to that giving Eq. (4.2), wehave that Eq. (4.1) applied to f yields

ðTf eÞðyÞ ¼ a0ðyÞf ðhðyÞÞ þ a1ðyÞ@f

@xi

ðhðyÞÞ þ?þ anðyÞ@nf

@xni

ðhðyÞÞ


for every yAUðy0Þ: Now we analyze the terms in the above equation, taking intoaccount that we are assuming 1pnpm; and that by Claim 4.4, h is a diffeomorphism

of class Cm in W 0: First Tf e is of class C1 in Uðy0Þ: Also, by Claim 4.5, for

kAf0; 1;y; ng; each ak is of class C1 in Uðy0Þ: Finally, f and all of its partial

derivatives up to order n 1 are of class C1 in O:Thus we deduce from the above equation that

an

@nf

@xni

3h

is of class C1 in Uðy0Þ; and by Lemma 3.2 the same applies to the function

@nf

@xni

3h:

But, as we said before, h is a diffeomorphism of class Cm in W 0; and consequently

@nf

@xni

admits a partial derivative with respect to the ith coordinate at the point hðy0Þ; whichis a contradiction. This implies that an � 0 in W 0: In a similar way we can see thatak � 0 in W 0 for kX1; that is, al � 0 in W 0 whenever lAL is of the form

l ¼ ð0; 0;y; k;|fflfflfflfflfflffl{zfflfflfflfflfflffl}i

0;y; 0Þ:

Claim 4.7. For every lAL fð0; 0;y; 0Þg; al � 0 in W 0:

Here our reasoning will be similar to the one given in Claim 4.6. In this way, ifi; jAf1; 2;y; pg; iaj; again by Eq. (4.1), for every yAW 0;

ðTðxixjeÞÞðyÞ ¼ a0ðyÞhiðyÞhjðyÞ þ al10ðyÞ;

where l10 :¼ ðl1

01; l102;y; l1

0pÞ; l10i ¼ 1 ¼ l1

0j and l10k ¼ 0; whenever kai; j: Taking

into account that a0; hi and hj are of class Cm in W 0; we easily deduce that al10is of

class Cm in W 0: Likewise, we can inductively prove that all0

is of class Cm in W 0;

where ll0 :¼ ðll

01; ll02;y; ll

0pÞ; ll0i ¼ l; ll

0j ¼ 1 and ll0k ¼ 0; whenever kai; j;

lAf1; 2;y; n 1g: Suppose that aln10ðy0Þa0 for some y0AW 0: Then, as in the

proof of Claim 4.6, we take an open neighborhood Uðy0Þ of y0 such that Uðy0ÞCW 0

and aln10ðyÞa0 for every yAUðy0Þ:

Also, we take f ðxÞ ¼ gðxiÞ for every x ¼ ðx1; x2;y; xpÞAO; where these functions

meet the same requirements as in the proof of Claim 4.6, and define

dðxÞ :¼ xj f ðxÞ;


for every xAO: Clearly d just depends on the ith and jth coordinates, which impliesthat its only partial derivatives which possibly are not zero at hðyÞAhðUðy0ÞÞ aremaybe those

@ld

for

l ¼ ð0; 0;y; 1;|fflfflfflfflfflffl{zfflfflfflfflfflffl}j

0;y; 0Þ;


0;y; 0Þ;

k ¼ 1; 2;y; n; or

l ¼ ll0;

lAf1; 2;y; n 1g: Taking into account that al � 0 on W 0 for

l ¼ ð0; 0;y; 1;|fflfflfflfflfflffl{zfflfflfflfflfflffl}j

0;y; 0Þ

and


0;y; 0Þ;

Eq. (4.1) gives us, for every yAUðy0Þ;

ðTdeÞðyÞ ¼ a0ðyÞdðhðyÞÞ þ al10ðyÞ @f

@xi

ðhðyÞÞ þ?þ aln10ðyÞ @

n1f

@xn1i

ðhðyÞÞ:

We deduce as in the proof of Claim 4.6 that

@n1f

@xn1i

ðhðyÞÞ

admits a second partial derivative with respect to xi at the point hðy0Þ; which is acontradiction. This implies that aln1

0� 0 in W 0: In the same way we deduce that

all0� 0 in W 0; for lAf1; 2;y; n 2g:

A similar pattern of proof leads us to the fact that al � 0 in W 0 for everylað0; 0;y; 0Þ; lAL:

Claim 4.8. For every lAL fð0; 0;y; 0Þg; al � 0 in O0:


Notice that given an open subset U of O0; in Claim 4.4 we obtain a subset W 0 of U :

Notice also that this process can be done for any open subset of O0 because, as we

saw in the proof of Claim 4.4, cða0Þ is dense in O0: Also in Claim 4.7 we proved that,for lað0; 0;y; 0Þ; al � 0 on all the subsets W 0 obtained in this way. This implies

clearly that all these functions al are equal to 0 on a dense subset of O0:Consequently, to prove Claim 4.8, it is enough to show that all these functions arecontinuous.

We are going to prove it using induction on jlj: First, for jlj ¼ 0; we have that

að0;y;0Þ ¼ T #e belongs to AmðO0;FÞ and, consequently, it is continuous.

Now assume that kpn 1; and whenever jljpk; then al is a continuous function.Then fix l ¼ ðl1;y; lpÞAL with jlj ¼ k þ 1:

Next define fAAnðO;RÞ as

f :¼ xl1

1 xl2

2 yxlpp :

It is clear that given m ¼ ðm1;y; mpÞAL; if mi4li for some iAf1;y; pg; then

@mf ðhðyÞÞ ¼ 0

for every yAO0: This implies that in our situation, Eq. (4.1) can be written as

ðTf eÞðyÞ ¼Xm{l

amðyÞ@mf ðhðyÞÞ;

where m{l means mipli for every iAf1;y; pg:As a consequence, for every yAO0;

l1!ylp!alðyÞ ¼ alðyÞ@lf ðhðyÞÞ ¼ ðTf eÞðyÞ X

m{l;mal

amðyÞ@mf ðhðyÞÞ:

On the other hand, if m{l and mal; then jmjpk; and consequently, taking intoaccount that h is continuous and the hypothesis of induction, we deduce that al iscontinuous, and the claim is proved.

Recall that all the process developed so far concerns functions of the formf eAAnðO;EÞ; where eAE f0g and fAAnðO;RÞ: For this e; we define

ae :¼ a0 ¼ T #e:

Notice that by Claim 4.8, we have

ðTf eÞðyÞ ¼ aeðyÞf ðhðyÞÞ ð4:4Þ

for every yAO0 and every fAAnðO;RÞ:


Next we define a map J :O0-L0ðE;FÞ as ðJyÞð0Þ ¼ 0; and

ðJyÞðeÞ :¼ aeðyÞ

for each yAO0 and eAE f0g:

Claim 4.9. For every fAAnðO;EÞ and yAO0;


Fix yAO0: Suppose that

@lf ðhðyÞÞ ¼ elAE

for each lAL:Let L :¼ flAL: ela0g: Next, for each lAL ; take a function flAAnðO;RÞ such

that

@lflðhðyÞÞ ¼ 1

and

@mflðhðyÞÞ ¼ 0

for every mal; mAL:It is easy to see that, for every mAL;

@mXlAL

flelðhðyÞÞ ¼ em;

if mAL ; and

@mXlAL

flelðhðyÞÞ ¼ 0

if meL :According to Lemma 4.1, this implies that

ðTf ÞðyÞ ¼ TXlAL

flel

!ðyÞ:

Consequently, by Eq. (4.4),

ðTf ÞðyÞ ¼XlAL

aelðyÞflðhðyÞÞ:

But by the way we have constructed the functions fl; we have that flðhðyÞÞ ¼ 0 iflað0; 0;y; 0Þ:


On the other hand, let us denote by 0 the multiindex ð0; 0;y; 0Þ: If 0eL ; that is,if e0 ¼ 0; we conclude from the above equality that ðTf ÞðyÞ ¼ 0 ¼ ae0

ðyÞ ¼ ðJyÞð0Þ:Finally, if 0AL ; taking into account that f0ðhðyÞÞ ¼ 1; we deduce that

ðTf ÞðyÞ ¼ ae0ðyÞf0ðhðyÞÞ ¼ ae0

ðyÞ ¼ ðJyÞðe0Þ;

and we are done.

Claim 4.10. Given yAO0; there exists eAE such that aeðyÞa0:

Notice that T is bijective, so if fAF ; fa0; there exists gAAnðO;EÞ with Tg ¼ #f: Inparticular, by Claim 4.9, we have that ðJyÞðgðhðyÞÞÞ ¼ f: In other words, if we takeyAcðg 3 hÞ and define e :¼ gðhðyÞÞ; we have that aeðyÞ ¼ ðJyÞðeÞ ¼ fa0:

Claim 4.11. h is a function of class Cm:

Fix y0AO0: By Claim 4.10, we can take eAE such that aeðyÞa0: Since ae ¼ T #e; it isa continuous function, and we deduce that for some neighborhood V of y0; aeðyÞa0for every yAV :

Now recall that Eq. (4.4),

ðTf eÞðyÞ ¼ aeðyÞf ðhðyÞÞ;

holds in particular for every yAV and every fAAnðO;RÞ:Consequently, for iAf1; 2;y; pg;

ðTxieÞðyÞ ¼ aeðyÞhiðyÞ

for every yAV : Since aeðyÞa0 for every yAV ; applying Lemma 3.2, we have that hi

is of class Cm in V : Clearly this implies that h is of class Cm; and we are done.

Claim 4.12. n ¼ m and h is a diffeomorphism of class Cm:

Recall that we are assuming that npm: Now, we have that by Claim 4.4, for every

nonempty open set VCO0; there is a nonempty open set V 0CV such that the

restriction of h to V 0 is a diffeomorphism of class Cm: Take y0AO0: By Claim 4.10,there exists eAE and an open neighborhood V of y0 such that aeðyÞa0 for everyyAV : Now, as we mentioned above, there exists an open set V 0; V 0CV ; where therestriction of h is a diffeomorphism of class Cm: Assume now that nom and take

gAAnðO0;RÞ CmðO0;RÞ such that cðgÞCV 0: Next define f :O-E; as

f ðxÞ :¼ gðh1ðxÞÞe;

for each xAO: We are going to prove that g is of class Cm; obtaining a contradiction.


It is immediate that fAAnðO;EÞ; and applying Eq. (4.4), we get

ðTf ÞðyÞ ¼ aeðyÞgðh1ðhðyÞÞÞ;

that is,

ðTf ÞðyÞ ¼ aeðyÞgðyÞ;

for every yAO0:Now we have that aeðyÞa0 for every yAV : Finally, by Lemma 3.2, g is of class Cm

in V ; and so is in O0; which contradicts our assumption. This implies that mpn; andsince we are assuming that npm; the claim is proved.

Claim 4.13. For every yAO0; JyAL0ðE;FÞ is bijective.

Since m ¼ n; all claims above also hold for T1; and this means that there exists

K : O-L0ðF ;EÞ such that for every gAAmðO0;FÞ and xAO; ðT1gÞðxÞ ¼ðKxÞðgðh1ðxÞÞÞ:

Fix yAO0 and fAF f0g: Let x ¼ hðyÞ: Now take gAAmðO0;FÞ with gðyÞ ¼ f:

Then it is clear that f ¼ gðyÞ ¼ ðTðT1gÞÞðyÞ; that is,

f ¼ðJyÞððT1gÞðxÞÞ

¼ ðJyÞððKxÞðgðh1ðxÞÞÞÞ

¼ ðJyÞððKxÞðgðyÞÞÞ

¼ ðJyÞððKxÞðfÞÞ:

This implies that ðJyÞðKxÞ is the identity map on F : In the same way we can provethat ðKxÞðJyÞ is the identity map on E: Consequently, Jy is bijective.

This ends the proof of the proposition. &

5. A result on automatic continuity

In this section we see that, if we endow the spaces with some natural topologies,then we obtain the continuity as a consequence. Notice that, according toProposition 4.2, we can assume in particular that n ¼ m and p ¼ q:

Theorem 5.1. Assume that AnðO;EÞ and AnðO0;FÞ are endowed with any topologies

which are compatible with the pointwise convergence. Suppose that T :

AnðO;EÞ-AnðO0;FÞ is a K-linear biseparating map. Then T is continuous.


Proof. In our proof we will take advantage of the description of T given inProposition 4.2. For this reason we will use the notation given there. We startproving the following claim.

Claim 5.1. Let U be a (nonempty) bounded open subset of O0 with clRp UCO0: Then

the set

A :¼ fyAU : JyAI 0ðE;FÞ is not continuousg

is finite.

Suppose that this is not the case, but there exist infinitely many yAU such that Jy

is not continuous. We are going to construct inductively a sequence of points in A; asequence ðUnÞ of pairwise disjoint open subsets of U ; a sequence of functions ð fnÞ inCn

c ðO;RÞ; and a sequence ðenÞ of norm-one elements of E; satisfying the following

properties:

1. hðynÞAcð fnÞChðUnÞ for every nAN:

2. jj fnjj :¼ maxlAL supxAO j@lfnðxÞj ¼ 1=2n for every nAN:3. jjðJynÞðenÞjjXn=j fnðhðynÞÞj for every nAN:

Take any point y1AA such that there are accumulation points of A in O0 fy1g:Then consider an open subset U1 of U in such a way that y1AU1; and there areinfinitely many points of A outside clRp U1: Next take f1ACn

c ðO;RÞ such that

jj f1jj ¼ 1; and such that hðy1ÞAcð f1ÞChðU1Þ: Since Jy1 is not continuous, thereexists eAE; jjejj ¼ 1; with

jjðJy1ÞðeÞjjX1

j f1ðhðy1ÞÞj:

Next assume that we have fy1; y2;y; yngCA; U1;U2;y;UnCU open and pairwisedisjoint such that there are infinitely many points of A outsideclRp U1,clRp U2,?,clRp Un; f f1; f2;y; fngCCn

c ðO;RÞ with hðyiÞAcð fiÞChðUiÞand jj fijj ¼ 1=2i; for iAf1; 2;y; ng; and e1; e2;y; enAE all of them with norm 1,and such that jjðJyiÞðeiÞjjXi=j fiðhðyiÞÞj for i ¼ 1; 2;y; n:

Now it is easy to see how to take ynþ1; Unþ1; fnþ1; and enþ1 so that Properties 1, 2and 3 above hold.

Since jj fnjj ¼ 1=2n for every nAN; we deduce that the map

g :¼XNn¼1

fnen

belongs to AnðO;EÞ: Consequently, Tg should belong to AnðO0;FÞ: But we know byProposition 4.2 that ðTgÞðynÞ ¼ ðJynÞðgðhðynÞÞÞ for every nAN: This implies, by


Property 3 above,

jjðTgÞðynÞjj ¼ jjðJynÞð fnðhðynÞÞenÞjj

¼ j fnðhðynÞÞj jjðJynÞðenÞjj

X n:

As a consequence Tg is unbounded in U : Since this is not possible, we concludethat the claim is correct.

Next, it is clear that to prove that T is continuous it is enough to show that it isclosed, because we are dealing with Frechet spaces. To prove it, let us consider asequence ð fnÞ in AnðO;EÞ convergent to zero, and assume that ðTfnÞ converges to

gAAnðO0;FÞ: We are going to prove that g ¼ 0:

Take a bounded open subset U of O0 with clRp UCO0: By Claim 5.1 above, wehave that the subset A of points yAU such that Jy is not continuous is finite. So, ifyAU A; Jy belongs to IðE;FÞ: Consequently, since fnðhðyÞÞ goes to zero (becausethe topology in AnðO;EÞ is compatible with the pointwise convergence), then wehave that ðTfnÞðyÞ ¼ ðJyÞð fnðhðyÞÞÞ also goes to zero, that is, gðyÞ must be zero. Buttaking into account that U A is dense in U ; we deduce that g � 0 on U : Theconclusion follows now easily and T is continuous. &

6. Biseparating maps and functions of class s Cn

Our aim in this section is to give a final description of biseparating maps betweenspaces of vector-valued differentiable functions taking into account that we knowthat they must be continuous when the spaces are endowed with some naturaltopologies. Of course these topologies will be compatible with the pointwiseconvergence. Namely, it is well known that by means of the seminorms pK defined as

pKð f Þ :¼ maxlAL

maxxAK

jj@l f ðxÞjj

for fACnðO;EÞ; where K runs through the compact subsets of O; CnðO;EÞ becomes

a locally convex space. In fact it is a Frechet space. In the same way, in CnðO;EÞ wecan consider the norm jj � jj defined as

jj f jj :¼ maxlAL

supxAO

jj@lf ðxÞjj

for fACnðO;EÞ: With this norm, our space CnðO;EÞ is also complete. We assume

that AnðO;EÞ and AnðO0;FÞ are endowed with the above topologies. Remark alsothat, as it follows easily from the Closed Graph Theorem, the topologies compatiblewith the pointwise convergence in our spaces coincide with these topologies.


Next proposition states that when J :O0-LðE;FÞ is of class s Cn; we can define

maps through J from AnðO;EÞ-AnðO0;FÞ in a natural way.

Proposition 6.1. Suppose that n ¼ m; p ¼ q: Let J :O0-LðE;FÞ be a map of class

s Cn; and let h be a diffeomorphism of class Cn from O0 onto O: If, for fACnðO;EÞ;we define ðTf ÞðyÞ :¼ ðJyÞð f ðhðyÞÞÞ for every yAO0; then TfACnðO0;FÞ:

Proof. We consider first the map F : LðE;FÞ � E-F defined as FðA; eÞ :¼ Ae foreach ðA; eÞALðE;FÞ � E: This is clearly bilinear and continuous when we consider inLðE;FÞ the topology of the norm.

Suppose next that LðE;FÞ is endowed again with the topology of the

norm, and that K : O0-LðE;FÞ is a continuous map. Then, given g :O0-E

continuous, the map SgK :O0-LðE;FÞ � E sending each yAO0 into ðKy; gðyÞÞ is

continuous.On the other hand, if we suppose that LðE;FÞ is endowed with the topology of the

norm and that fACnðO;EÞ; then Sf 3h

J is of class Cn1 because both maps J and f 3h

are. Consequently the composition map F3S f 3hJ :O0-F ; mapping each yAO0 into

ðJyÞð f ðhðyÞÞÞAF is of class Cn1:Now we check the form of its first partial derivatives. We just see the partial

derivative with respect to the first coordinate x1: It is easy to check that

@

@x1; ðF3S f 3h

J ÞðyÞ ¼ @

@XFðS f 3h

J ðyÞÞ3 @

@x1Jy þ @

@YFðS f 3h

J ðyÞÞ3 @

@x1ð f 3hÞðyÞ

¼F@

@x1Jy; f ðhðyÞÞ

� �þ F Jy;

@

@x1ð f 3hÞðyÞ

� �¼ F3Sf 3h

@@x1

J

!ðyÞ þ F3S

@@x1

f 3h

J

!ðyÞ:

By an inductive reasoning, we see that the partial derivatives of order n 1 of F3S f 3hJ

are just a sum of terms of the form F3S@mð f 3hÞ@lJ

; where jlj; jmjpn 1: Consequently, to

prove that TfACnðO0;FÞ; we just have to show that each one of the terms

F3S@mð f 3hÞ@lJ

ð jlj; jmjpn 1Þ is of class C1: We suppose that SgK :O0-LðE;FÞ � E is

one of the above S@mð f 3hÞ@lJ

(that is, let us denote g ¼ @mð f 3hÞAC1ðO0;EÞ and

K ¼ @lJ), which is continuous when LðE;FÞ is endowed with the topology of thenorm, as we stated above.

Now we check the form of its first partial derivatives. Take iAf1;y; pg; andassume without loss of generality that i ¼ 1: It is easy to check that

limk-0

ðF3SgKÞðy þ ðk; 0;y; 0ÞÞ ðF3Sg

KÞðyÞk

;


that is,

limk-0

ðKðy þ ðk; 0;y; 0ÞÞÞðgðy þ ðk; 0;y; 0ÞÞÞ ðKyÞðgðyÞÞk

is equal to

limk-0

ðKðy þ ðk; 0;y; 0ÞÞÞðgðy þ ðk; 0;y; 0ÞÞÞ ðKðy þ ðk; 0;y; 0ÞÞÞðgðyÞÞk

þ limk-0

ðKðy þ ðk; 0;y; 0ÞÞÞðgðyÞÞ ðKyÞðgðyÞÞk

;

that is, it is equal to

ðKyÞ @

@x1gðyÞ

� �þ @s

@x1Ky

� �ðgðyÞÞ;

by the definition of @s

@x1Ky and the fact that K is continuous for LðE;FÞ endowed

with the topology of the norm.Applied to our context, we have that

@

@x1ðF3S@mð f 3hÞ

@lJÞðyÞ ¼ ðF3S

@@x1

@mð f 3hÞ@lJ

ÞðyÞ þ ðF3S@mð f 3hÞ@s

@x1

@lJÞðyÞ:

Now, as we noted above, S@@x1

@mð f 3hÞ@lJ

is continuous when considering in LðE;FÞ the

topology of the norm. As a consequence, to obtain the continuity of all partial

derivatives of order n of Tf ; it is enough to see that F3S@mð f 3hÞ@s

@x1

@lJis continuous.

In order to prove this, notice first that, since J is of class s Cn1; then for the

above l the map @s

@x1

@lJ is continuous when LðE;FÞ is endowed with the strong

operator topology. This means that, given eAE; the map @s

@x1

ð@lJÞe :O0-F sending

each yAO0 into @s

@x1

@lJy� �

ðeÞ is continuous. Now take e40 and y0AO0: We are going

to show that there exists d40 such that if jy y0jod; yAO0; then

F3S@mð f 3hÞ@s

@x1

@lJðyÞ F3S@mð f 3hÞ

@s

@x1

@lJðy0Þ

��

��oe:

Let e0 :¼ @mð f 3hÞðy0Þ; and take d140 such that the closed ball Bðy0; d1Þ is contained

in O0: Since @s

@x1

ð@lJÞe0is continuous, there exists an upper bound M for this function

on the compact set Bðy0; d1Þ: Also, there exists d240; d2od1; such that


if jy y0jod2; then

@s

@x1

@l Jy @s

@x1

@lJy0

� �e0ð Þ

�� oe2:

On the other hand, since @mð f 3hÞ is continuous, there exists d40; dod2; such that ifyABðy0; dÞ; then jj@mð f 3hÞðyÞ e0jjoeM=2:

Consequently, if jy y0jod; we have

jjF3S@mð f 3hÞ@s

@x1

@lJðyÞ F3S@mð f 3hÞ

@s

@x1

@lJðy0Þjj

is less than or equal to

@s

@x1

@lJy

� �ð@mð f 3hÞðyÞ e0Þ

�� þ @s

@x1

@lJy @s

@x1

@lJy0

� �ðe0Þ

�� ;which is easily strictly less than e: This proves the continuity of our functions, as itwas to see. &

The following result is a direct consequence of Proposition 4.2 and Theorem 5.1.Roughly speaking, it says that Proposition 6.1 provides the only way to construct

linear biseparating maps from AnðO;EÞ onto AnðO0;FÞ: We state the result in itscomplete form.

Theorem 6.2. Suppose that T : AnðO;EÞ-AmðO0;FÞ is a K-linear biseparating map.

Then p ¼ q; n ¼ m; and there exist a diffeomorphism h of class Cn from O0 onto O and

a map J :O0-LðE;FÞ of class s Cn such that for every yAO0 and every fAAnðO;EÞ;


Moreover, JyAIðE;FÞ for every yAO0:

Proof. We will follow the same notation as in Proposition 4.2, which provided a firstdescription of linear biseparating maps, in particular everything related to thedefinition of J and h: Also, by Proposition 4.2, p ¼ q and n ¼ m:

First, for each lAL; we define a map Jl :O0-L0ðE;FÞ as

ðJlyÞðeÞ :¼ @lðT #eÞðyÞ;

for yAO0 and eAE: Jl is clearly well defined.The rest of the proof will apply just for the case when we are in Situation 1, but it

is easy to see that slight changes in it allow to prove the theorem when we are inSituation 2. We will prove it through several claims.

Claim 6.1. For every yAO0 and every lAL; Jly belongs to LðE;FÞ:


Take yAO0 and a sequence ðenÞ in E converging to zero. We will see thatððJlyÞðenÞÞ goes to zero. First we have that, since T is continuous by Theorem 5.1,the sequence of functions ðT #enÞ converges to zero, which implies in particular that

ðð@lT #enÞðyÞÞ goes to zero. But this last sequence is precisely ððJlyÞðenÞÞ; so the claimis proved.

Claim 6.2. For each lAL; jljpn 1; the map Jl :O0-LðE;FÞ is continuous when

LðE;FÞ is endowed with the topology of the norm.

We will show that if y0AO0; then Jl is continuous at y0: Since T is continuous, we

have that, for r40 such that the closed ball Bðy0; rÞ is contained in O0; there existsM40 such that pBðy0;rÞðT #eÞoM holds for every eAE with jjejjp1: This implies that,

for these e; if jy y0jor; then jjðD@lT #eÞðyÞjjppM: Consequently, as it can be seenfor instance in [8, Theorem 3.3.2], we have that

jjð@lT #eÞðyÞ @lðT #eÞðy0ÞjjppMjy y0j;

for every eAE with jjejjp1: Now, taking into account that ðJlyÞðeÞ ¼ ð@lT #eÞðyÞ for

every yAO0; the result follows, and the claim is proved.

Claim 6.3. For each lAL; jljpn 1; @lJ ¼ Jl:

Of course, the result is clear if n ¼ 1; so we suppose that nX2: We will just provethe claim in the particular case when l ¼ l1 :¼ ð1; 0;y; 0Þ: The proof for all otherlAL is similar and can be achieved inductively.

Take y0AO0 and r40 such that the closed ball Bðy0; rÞCO0: It is clear that ifhAR f0g; jhjor; and if e is in the closed unit ball of E; then

Jðy0 þ ðh; 0;y; 0ÞÞðeÞ ðJy0ÞðeÞh

ðJl1y0ÞðeÞ

�� is equal to

ðT #eÞðy0 þ ðh; 0;y; 0ÞÞ ðT #eÞðy0Þh

ð@l1T #eÞðy0Þ�� ;

which, by Lang [23, Corollary XIII.4.4], is less than

supjyy0jo2h

jjð@l1T #eÞðy0Þ ð@l1T #eÞðyÞjj;

that is, less than

supjyy0jo2h

jjJl1y0 Jl1

yjj:


Clearly this implies that

limh-0

Jðy þ ðh; 0;y; 0ÞÞ Jy

h Jl1

y

�� ¼ 0:

Consequently, the partial derivative of J with respect to the first coordinate exists at

each y0AO0 and is equal to Jl1y0:

Claim 6.4. Take l ¼ ðn1; n2;y; ni;y; npÞAL with jlj ¼ n 1: Then Jl is of class

s C1: Moreover, if for each iAf1;y; ng; mi ¼ ðn1; n2;y; ni þ 1;y; npÞ; then

@s

@xi

Jl ¼ Jmi:

The proof that @s

@xiJl ¼ Jmi

is similar to the proof of Claim 6.3 we have done, taking

into account that Jmiis perhaps no longer continuous when LðE;FÞ is endowed with

its norm but @mi T #e is continuous for every eAE: Consequently, to finish we just haveto show that Jm; jmj ¼ n; is continuous when considering in LðE;FÞ the strong

operator topology. We have to prove that if mAL; and ðynÞ is a sequence in O0

converging to yAO0; then ðJmynÞðeÞ converges to ðJmyÞðeÞ for each eAE: But this is

immediate from the definition of Jm: &

Remark 6.3. Notice that, in the case when we are in Situation 2, in Theorem 6.2 themap J and its partial (s-) derivatives up to order n can also be extended continuously

to the boundary of O0 in a natural way, when considering in LðE;FÞ the strongoperator topology.

In the special case when E ¼ K ¼ F ; we immediately deduce the following result.

Corollary 6.4. Suppose that T : AnðO;KÞ-AmðO0;KÞ is a K-linear biseparating map.

Then p ¼ q; n ¼ m; and there exist a diffeomorphism h of class Cn from O0 onto O and

a map a :O0-K of class Cn which does not vanish at any point of O0; such that for

every yAO0 and every fAAmðO;KÞ;

ðTf ÞðyÞ ¼ aðyÞf ðhðyÞÞ:

We finish with a corollary whose proof is easy from Theorem 6.2.

Corollary 6.5. If AnðO;EÞ and AnðO0;FÞ are endowed with the topology of the

pointwise convergence, then every linear biseparating map T : AnðO;EÞ-AnðO0;FÞ is

continuous.


7. Final remark

Even if in previous sections we consider two possible situations for our spaces ofvector-valued functions, Proposition 4.2 and Theorems 5.1 and 6.2 can be given in abroader context.

Let OCRp and O0CRq be (nonempty) open sets, and consider any of the followingcontexts.

Case (i): O1 :¼ O and O01 :¼ O0:

Case (ii): O and O0 are bounded, intRp clRp O ¼ O; intRq clRq O0 ¼ O0; and O1 :¼clRp O and O1 :¼ clRq O:

Suppose now that ACCnðO;RÞ-CðO1;RÞ and BCCmðO0;RÞ-CðO01;RÞ are

strongly regular rings (see [2] for an appropriate description). Take now

AðO;EÞCCnðO;EÞ-CðO1;EÞ and BðO0;FÞCCmðO0;FÞ-CðO01;FÞ: Namely, to

construct the support map h :O01-O1; we need that subspaces

AðO;EÞCCnðO;EÞ-CðO1;EÞ and BðO0;FÞCCmðO0;FÞ-CðO01;FÞ are a compatible

A-module and a compatible B-module (see [2]). In both cases h will be a

homeomorphism from O0 onto O:On the other hand, apart from these necessary conditions for constructing h; a

careful reading of the proofs shows that the essential requirements that linear

subspaces AðO;EÞ and BðO0;FÞ must meet so as to satisfy Lemma 4.1 and

Proposition 4.2 are: (1) AðO;EÞ and BðO0;FÞ contain the constant functions; (2)

AðO;EÞ (respectively, BðO0;FÞ) contains all functions in CnðO;EÞ (respectively,

CmðO0;FÞ) with compact support; and (3) Cnc ðO;RÞCA and Cm

c ðO0;RÞCB: In the

case when O and O0 are bounded and the coordinate projections xi belong to A andB; then the proof of Proposition 4.2 can be followed step by step with no changes.

Otherwise (even if O and O0 are not bounded), changes are few and natural.Finally, if we want to obtain results similar to Theorems 5.1 and 6.2 (with the same

proofs) for our spaces AðO;EÞ and BðO0;FÞ; besides all the above conditions, theymust be endowed with a suitable norm or family of seminorms providing a topologycompatible with the pointwise convergence. This will be the case, for instance, of thespaces of functions with bounded derivatives.

So we study the case of the spaces Cn ðO;EÞCCnðO;EÞ and Cm

ðO0;FÞCCmðO0;FÞconsisting of all functions such that all partial derivatives up to orders n and m;

respectively, are bounded. The space Cn ðO;EÞ (and similarly Cm

ðO0;FÞ) becomes a

Banach space with the norm defined for each fACn ðO;EÞ as

jj f jj :¼ maxlAL

supxAO

jj@l f ðxÞjj:

This is a suitable norm in the above sense because the proof of Theorem 6.2 can befollowed easily for these spaces equipped with such norm.

First, notice that, since we are assuming that n;mX1; then in particular when O isconvex all functions in Cn

ðO;EÞ admit a continuous extension to the closure of O in

Rp: suppose that ðxnÞ is a sequence in O converging to x0 in the boundary of O; and


that fACn ðO;EÞ; then since the differential Df is bounded on the whole O by an

M40; we have that, by Cartan [8, Theorem 3.3.2],

jj f ðxnÞ f ðxmÞjjpMjxn xmj;

which implies that ð f ðxnÞÞ is a Cauchy sequence. In this way we would define theextension f ðxÞ as the limit of this sequence. It is straightforward to see that the newextended function is continuous in the closure of O:

On the other hand, when O and O0 are bounded and convex, then it is easy to see

that Cn ðO;EÞ is a CnðO;RÞ-module, and a similar statement is also valid for

Cm ðO0;FÞ: As a consequence, by the comments given above, Proposition 4.2 and

Theorems 5.1 and 6.2 can also be stated in this new situation (for O and O0 boundedand convex). Furthermore, in this case, as in Remark 6.3, it is also possible to saythat partial derivatives up to order n 1 of J admit a continuous extension (when

LðE;FÞ is equipped with the strong operator topology) to the boundary of O0: As forthe partial s-derivatives of all partial derivatives of order n 1 of J; an elementaryapplication of the Uniform Boundedness Theorem shows that they are bounded

on O0:What happens if O is for instance not bounded? One might be tempted to follow a

similar pattern as indicated above when trying to describe linear biseparating maps

defined between Cn ðO;EÞ and Cm

ðO0;FÞ: But, in that case, we have that Cn ðO;EÞ is

no longer a CnðO;RÞ-module (as AnðO;EÞ in previous sections was). Anyway, it is aCn

ðO;RÞ-module, and we could try to follow the proof of Proposition 4.2 to get a

similar description of linear biseparating maps, but even if we could manage to adaptthe proof step by step (with some changes), there is a major problem from thebeginning: in general the space Cn

ðO;RÞ is not a strongly regular ring, so our results

cannot be applied, and in particular the existence of the support map h is not clear.Let us see an example where Cn

ðO;RÞ is not a strongly regular ring.

Example 7.1. Suppose that OCR is the open set defined as O :¼S

N

k¼1 ðk 1=k;

k þ 1=kÞ: Let us define K as the closure of N in bO (the Stone–Cech compactifica-tion of O). Now take x0AbO ðK,RÞ: It is clear that if a derivable function

f :O-R satisfies f bO � 0 on K and f bO � 1 on a neighborhood of x0 (where f bO

stands for its extension to bO), then for some sequence ðxnÞ in O going to infinity, thesequence ð f ðxnÞÞ converges to 1. As a consequence from the Mean Value Theorem,

we conclude that the derivative of f cannot be bounded on O; that is, feC1 ðO;RÞ; as

we wanted to show.

We end the paper with some related questions, concerning special cases where ourtechniques cannot be applied.

Problem 1. Assume that there exists a biseparating map T : AnðO;EÞ-AmðO0;FÞwhich is not linear. Can we deduce that the support map h :O0-O is a


diffeomorphism of class n? Remark that the assumption of linearity in Proposition4.2 is necessary for its proof.

Problem 2. Suppose that CNðO;EÞ is the space of E-valued functions which are of

class CN in O; and that CNðO0;FÞ is defined in a similar way. Describe the linear

biseparating maps from CNðO;EÞ onto CNðO0;FÞ: Must such a map be continuous?Notice that by the comments given in the Final Remark above, the construction ofthe support map h is possible, but the proof of Proposition 4.2 is no longer valid.

Problem 3. Let O and O0 be unbounded open subsets of Rp and Rq; respectively.

Describe the linear biseparating maps from Cn ðO;EÞ onto Cm

ðO0;FÞ:

Problem 4. Determine all subspaces AðO;EÞCAnðO;EÞ and BðO0;FÞCAmðO0;FÞsuch that the existence of a (linear) biseparating map from AðO;EÞ onto BðO0;FÞimplies that E and F are isomorphic as Banach spaces.

Acknowledgments

The author wishes to thank three anonymous referees for their valuablesuggestions and, also, Professor R. Aron for drawing his attention to some paperswhich appear in the references.

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Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity