A classification of isomorphisms and isometries on functions spaces.

HAL Id: hal-01082632

https://hal.archives-ouvertes.fr/hal-01082632

Preprint submitted on 14 Nov 2014

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

A classification of isomorphisms and isometries on

functions spaces.

Mohammed Bachir

To cite this version:

Mohammed Bachir. A classification of isomorphisms and isometries on functions spaces.. 2014.

�hal-01082632�

fun tions spa es.

Mohammed Ba hir November14, 2014

Laboratoire SAMM 4543, Université Paris 1 Panthéon-Sorbonne, Centre P.M.F.90 rue Tolbia 75634 Paris edex 13

Email : Mohammed.Ba hiruniv-paris1.fr

Abstra t. We establishin thisarti le aformulawhi h will allowto lassifyisometries as well as partial isometries between spa es of fun tions. Our result applies in a non ompa t framework and for abstra t lassof fun tions spa es in luded in the spa e of ontinuous and bounded fun tions on omplete metri spa e. The resultof this arti le isinparti ular ageneralizationoftheBana h-Stonetheorem. Weusethe te hniquesof dierentiabilityasinthe original proofofBana hfor onvexfun tionsgeneralizingthe norm ofuniform onvergen e. Weusealso adualityresultin a non onvex framework.

Keyword, phrase: The Bana h-Stone theorem, isometries and isomorphisms, dif-ferentiability, duality.

1 Itrodu tion.

The lassi al Bana h-Stone theorem say thatif

K

and

L

are ompa t spa es and

T

is an isomorphism from

(C(K), k.k

∞

)

onto

(C(L), k.k

∞

)

(where

C(K)

and

C(L)

denotes the spa es of all real valued ontinuous fun tions on

K

and

L

respe tively) then, the map

T

isan isometry if and only ifthere existsan homeomorphism

π : L → K

and a ontinuous map

ǫ : L → {±1}

su hthat

T ϕ(y) = ǫ(y)ϕ ◦ π(y); ∀y ∈ L, ∀ϕ ∈ C(K).

(1)

TheBana h-Stonetheoremwasinvestedbyseveralauthorsindiversedire tionsmaking waytoseveraladvan esand publi ationsforexample byD.Amir[1℄,J.Araujo[2℄,[3 ℄, J. Araujoand J.Font [4 ℄,E. Behrends [10℄,[11 ℄, M. Cambern [12℄,B. Cengiz, [13 ℄,M. Garrido and J. Jaramillo [18℄, [19℄, K. Jarosz [23 ℄, K. Jarosz and V. Pathak, [24℄, D. Vieiera [26℄, N. Weaver [27 ℄. The list remaining still long, we send ba kto the arti le of M. Garridoand J.Jaramillo [17℄ for ahistory of ontributions anda more omplete list.

of isomorphisms between fun tions spa es resulting totally or partially from homeo-morphisms. We establish our result in the non ompa t framework for abstra t lass of fun tions spa es in luded lassi al spa es. To be more lear, we give the following example. Suppose that

X

and

Y

aretwo Bana h spa es and supposethat thereexists an homeomorphism

π : Y → X

preserving norms i.e

kπ(y)kX

= kykY

for all

y ∈ Y

. Letus denote by

H

k.k

X

,k.k

Y

(Y, X)

the lassof allhomeomorphism from

Y

onto

X

pre-serving norms and by

C

b

(X)

the spa e of all ontinuous bounded fun tionson

X

. Let

T : (Cb(X), k.k∞) → (Cb(Y ), k.k∞)

be anisomorphism.

Question 1. Under what ondition the isomorphism

T

omes anoni ally as in the formula(1)from the lassofthe homeomorphisms

π ∈ H

k.k

X

,k.k

Y

(X, Y )

? The answer to this question is that

T

omes anoni ally form

π ∈ H

k.k

X

,k.k

Y

(X, Y )

ifand only if,itsatisesthe following ondition

sup

y∈Y

{|T ϕ(y)| − kykY

} = sup

x∈X

{|ϕ(x)| − kxkX

} , ∀ϕ ∈ Cb

(X).

(2)

Thus the lass ofall isomorphisms

T

satisfying(2)is in bije tionwith the set

{ǫ : Y → {±1}

ontinuous

} × H

k.k

X

,k.k

Y

(Y, X).

For example, when

X = Y = R

(whi h are in parti ular onne ted spa es), we have that the isomorphisms

T : Cb(R) → Cb(R)

that satisfy

sup

y∈R

{|T ϕ(y)| − |y|} =

sup

_x∈R

{|ϕ(x)| − |x|} , ∀ϕ ∈ Cb

(R)

are exa tly four isometries orresponding to the homeomorphismssatisfying

|π(x)| = |x|

for all

x ∈ R

whi hare

π(x) = x

or

π(x) = −x

for all

x ∈ R

:

(1) T1ϕ(y) = ϕ(y)

forall

ϕ ∈ Cb

(R)

andall

y ∈ R

. Theidentity map.

(2) T2ϕ(y) = −ϕ(y)

for all

ϕ ∈ Cb(R)

and all

y ∈ R

.

(3) T

3

ϕ(y) = ϕ(−y)

for all

ϕ ∈ C

b

(R)

and all

y ∈ R

.

(4) T

4

ϕ(y) = −ϕ(−y)

for all

ϕ ∈ C

b

(R)

and all

y ∈ R

.

Our main result Theorem 1 gives a general lassi ation in the spirit mentioned in the example above. Moregenerally, a lassof homeomorphism

π

given bya fun tional onstraint

f ◦ π = g

for xed lower semi ontinuous and bounded from belowfun tions

f

and

g

, hara terize the lass of isomorphism satisfying

sup

y∈Y

{|T ϕ(y)| − g(y)} =

sup

_x∈X

{|ϕ(x)| − f (x)}

for all

ϕ ∈ C

b

(X)

. In the parti ular ase where

f = 0

on

X

and

g = 0

on

Y

we re over the lassof allisometries for

k.k∞

. Before giving our main result Theorem1 below, we aregoingto introdu e explanatory ommutative diagrams orresponding to our lassi ation.

1.1 The ommutative diagrams for the lassi al Bana h-Stone theo-rem.

Let

T : C(K) → C(L)

be an isomorphism, andlet

H0,0

(L, K)

be the set ofall homeo-morphism from

L

onto

K

ifsu h homeomorphism exists. The left diagram ommutes

C(K)

T

//

k.k

∞

$$

■

■

■

■

■

■

■

■

■

C(L)

k.k

∞



R

⇔

∃π : L

π

//

0

❅

❅

_❅

❅

❅

❅

❅

❅

K

0



R

k.k∞

◦ T = k.k∞

⇔ ∃π ∈ H0,0

(L, K)

su h that

T

omes anoni allyfrom

π

1.2 The ommutative diagrams for the lassi ation result. Let

(X, d)

and

(Y, d

′

₎

be a omplete metri spa es. Let

A ⊂ Cb(X)

and

B ⊂ Cb

(Y )

be aBana hspa essatisfying ertaingeneralaxioms(Thisaxiomsareveriedfor lassi al and known fun tion spa es. See Axioms 1 and the examples in Proposition 2). Let

f : X → R ∪ {+∞}

and

g : Y → R ∪ {+∞}

betwo lower semi ontinuousand bounded frombelowfun tionswithnonemptydomainsi.e

dom(f ) := {x ∈ X : f (x) < +∞} 6= ∅

and

dom(g) 6= ∅

. Wedenoteby

dom(f )

and

dom(g)

the losure of

dom(f )

and

dom(g)

in

X

and

Y

respe tively. Wedenoteby

Hf,g(Y, X)

the lassofallhomeomorphismfrom

dom(g)

onto

dom(f )

ifsu hhomeomorphism exists. Wegivebelowthediagramsofour lassi ation result. The leftdiagram ommutes ifandonly thatof right also.

Let

T : A → B

bean isomorphism,

A

T

//

F

❅

❅

_❅



❅

❅

❅

❅

❅

B

G



R

⇔ ∃π : dom(g)

π

//

g

&&

▲

▲

▲

▲

▲

▲

▲

▲

▲

▲

▲

dom(f )

f



R

_{∪ {+∞}}

G◦T = F ⇔ ∃π ∈ Hf,g

(Y, X)

su hthat

f ◦π = g

on

dom(g)

and T omes anoni ally from

π

on

dom(g).

Where

F(ϕ) := sup

x∈X

{|ϕ(x)| − f (x)} , ∀ϕ ∈ A

G(ψ) := sup

y∈Y

{|ψ(y)| − g(y)} , ∀ψ ∈ B.

We re over the lassi al real valued Bana h-Stone theorem in the ase of omplete metri spa es

X

and

Y

with

A = C

b

(X)

,

B = C

b

(Y )

,

f = 0

on

X

and

g = 0

on

Y

. The situationin Question 1. orrespond to

f = k.kX

and

g = k.kY

. Ourmain resultis then the following theorem.

1.3 The main result.

Theorem 1 Let

X

and

Y

be two omplete metri spa es and

A ⊂ Cb(X)

and

B ⊂

Cb(Y )

be two Bana hspa es satisfyingthe axioms

A1

-

A4

(with the same property

P

β

. See Axioms 1 and the examples in Proposition 2). Let

T : A → B

be an isomorphism and let

f : X → R ∪ {+∞}

and

g : Y → R ∪ {+∞}

be lower semi ontinuous and bounded frombelow withnonempty domains. Then

(1) ⇔ (2)

.

(1)

For all

ϕ ∈ A

, we have

sup

y∈Y

{|T ϕ(y)| − g(y)} = sup

x∈X

{|ϕ(x)| − f (x)}

.

(2)

There exist an homeomorphism

π : dom(g) → dom(f )

and a ontinuous fun tion

ε : dom(g) → {+1, −1}

su h that, for all

y ∈ dom(g)

andall

ϕ ∈ A

we have

T ϕ(y) = ε(y)ϕ ◦ π(y)

and

g(y) = f ◦ π(y).

Remark 1 Note that from

(2)

we obtain

sup

y∈dom(g)

|T ϕ(y)| = sup

x∈dom(f )

|ϕ(x)|

,

∀ϕ ∈ A

. Itfollowsthat, if

dom(f ) = X

and

dom(g) = Y

thenthe ondition

(1)

implies in parti ular that

T

is isometri for the norm

k.k∞

. So using lower semi ontinuous fun tions su h that

dom(f ) = X

and

dom(g) = Y

willpermit to lassify isometries for the norm

k.k∞

. Inthe general ase Theorem1permitto lassifyisomorphismsnotne -essary isometries satisfying

sup

y∈E

|T ϕ(y)| = sup

x∈F

|ϕ(x)|

,

∀ϕ ∈ A

for some

E ⊂ X

and

F ⊂ Y

.

As an immediate onsequen e of the above theorem is the following general Bana h-Stone result.

Corollary 1 Let

X

and

Y

be two omplete metri spa es. Let

A ⊂ Cb(X)

and

B ⊂

C

b

(Y )

be two Bana hspa es satisfyingthe axioms

A

1

-

A

4

(withthe same property

P

β

). Let

E

be a losed subset of

X

and

F

be a losed subset of

Y

. Let

T : A → B

be an isomorphism. Then

sup

y∈F

|T ϕ(y)| = sup

x∈E

|ϕ(x)|

forall

ϕ ∈ A

ifandonlyifthere existsanhomeomorphism

π : F → E

anda ontinuous map

ε : F → {−1, +1}

su h that for all

y ∈ F

and all

ϕ ∈ A

we have

T ϕ(y) =

ε(y)ϕ ◦ π(y)

.

Proof. It su es to apply Theorem 1with the indi ator fun tions

f = i

E

and

g = i

F

where

iE

(respe tively,

iF

) isequal to

0

on

E

(respe tively, on

F

)and

+∞

otherwise. Remark 2 Were overthe lassi alversionoftheBana h-Stonetheoremfromtheabove orollary by taking

E = X

and

F = Y

.

Note thatthere are situation where an isomorphism isnot anisometry but partially isometri . Indeed, Let

K

and

L

betwo metri and ompa t non homeomorphi spa es su h that

C(K)

and

C(L)

areisomorphi andlet

T

1

: C(K) → C(L)

bean isomorphis-m. Thissituation existsfor examplefor

C[0, 1]

and

C([0, 1] × [0, 1])

. A.A.Milutin[25 ℄ proved thatif

K

and

L

are both un ountable ompa t metri spa es, then

C(K)

and

C(L)

arealwayslinearly isomorphi . Let

R

and

S

two homeomorphi omplete metri spa es su h that

R ∩ K = ∅

and

S ∩ L = ∅

and let

π : S → R

an homeomorphism. Letus onsider the map

T : Cb

(K ∪ R) → Cb

(L ∪ S)

dened by

T (ϕ)(y) = T1(ϕ

|K

)(y)

if

y ∈ L

and

T (ϕ)(y) = ϕ ◦ π(y)

if

y ∈ S

for all

ϕ ∈ Cb(K ∪ R)

. Where

ϕ

|K

de-notes the restri tion of

ϕ

to

K

. Themap

T

isan isomorphism not isometri satisfying

sup

_y∈S

|T ϕ(y)| = sup

_x∈R

|ϕ(x)|

for all

ϕ ∈ Cb(K ∪ R)

.

and Theorem 4(See Se tion 2.4) whi h were establishedin [7℄ together with Lemma 1 (SeeSe tion 2.3)whi h isa onsequen e ofthe well knownDeville-Godefroy-Ziler v ari-ational prin iple [14 ℄. We us the general version of this prin iple whi h also worksfor ompletemetri spa esandisdutoR.DevilleandJ.Rivalski[16℄. Alsoletusnotethat a similar result of our main result Theorem 1 an be established in a ve torial frame for abstra t subspa es of the spa e of all ontinuous and bounded

Z

-valued fun tions

C

b

(X, Z)

. Some onditions on the Bana h spa e

Z

are ne essary like the smoothness of the norm of

Z

. The proof being more te hni al and requiring others intermediate results, we omit it in this arti le but we send ba k to the thesis [5℄ (see also [6℄) for a ve torial Bana h-Stone theorem. Let us mention here that the rst resultabout the ve torial Bana h-Stone theorem is due to E. Behrends[10 ℄, [11 ℄. Others results about the ve torvaluedBana h-Stone theoremwasestablishedbyJ.Araujoin[2 ℄,[3 ℄and by K. Jaros in [23 ℄.

Remark 3 In Theorem 1 above, we an have more information about the homeomor-phism

π

. More the spa es

A

and

B

are regular more the homeomorphism

π

is it also. For exemple if

A = C

u

b

(X)

and

B = C

u

b

(Y )

(the spa es of uniformly ontinuous and bounded) we obtain that

π

is uniformly ontinuous as well as

π

−1

. If

A = Lipb(X)

and

B = Lip

b

(Y )

(the spa es of all Lips hitz bounded map) and

X

and

Y

are quasi- onvexe then

π

is bi-Lips hitz. This follow from the fa t that if

ϕ ◦ π

is Lips hitz for all

ϕ ∈ Lipb(X)

then

π

is Lips hitz. See Theorem

44

in the paper of M.Garrido and J. Jaramillo [17℄. It is also known from the paper of J. Gutiérrez and J. Llavona [21℄ (See also [22℄) that if

p ◦ π

is

C

k

smooth fun tion ( in this ase

X

is assumed to be a Bana h spa e) for all

p ∈ X

∗

(the topologi al dual of

X

) then

π

is of lass

C

k−1

. So in the ase where

A = C

k

b

(X)

and

B = C

k

b

(Y )

(the spa e ok

k

-times ontinuously Fré het dierentiable fun tion

f

su h that

f

,

f

′

...,

f

(k)

are uniformly bounded) we an dedu e, using the omposition

α ◦ p

where

p ∈ X

∗

and

α : R → R

be a desirable very smooth fun tion, that

π

is

C

k−1

dieomorphism. Note that if moreover

X

has the S hur propertythenfromthe paperof M.Ba hirandG.Lan ien [8℄we an dedu e that

π

is of lass

C

k

. We draw the attention here on the fa t that the S hur property and the axiom

A

3

(See Axioms1)are notstill ompatible ininnitedimension, forexample the spa e

X = l

1

_(N)

whi h has the S hur property does not admit a Lips hitz and

C

1

smoothbump fun tion (See [15℄).

Thispaper is organised asfollow. In se tion2. we introdu e the axioms whi h we shall use in this arti le and we give examples satisfying them. We also give tools and preliminaries result whi h will permit us to give the proof of our main theorem. In se tion 3. We give the proofof our main resultTheorem 1.

2 Tools and preliminaries results.

Toproveour mainresultweneedtoestablish ertainlemmasandtore allotherresults already established.

Let

(X, d)

bea ompletemetri spa es and

(A, k.k)

a Bana hspa ein luded in

Cb

(X)

. By

δ

wedenotetheDira mapand by

δ

x

the Dira massasso iatedto thepoint

x ∈ X

. By

A

∗

wedenote the topologi al dualof

A

. Wehave

δ : X → A

∗

x → δx

and for all

x ∈ X

,

δx

: A → R

ϕ → ϕ(x).

Denition 1 (The property

P

β

) Let

(X, d)

be a omplete metri spa e and

(A, k.k)

a Bana h spa e in luded in

Cb(X)

. We say that

A

has the property

P

F

(respe tively,

P

G

) if, for ea h sequen e

(xn)n

⊂ X

wehave:

(x

n

)

n

onverges in

(X, d)

if and only if the asso iated sequen e of the Dira masses

(δx

n

)n

onverges in

(A

∗

_{, k.k}

∗

₎

(respe tively, in

(A

∗

_{, w}

∗

₎

). Where

k.k

∗

denotes the dual norm and

w

∗

the weak-startopology.

We denoteby

τ

β

the weak-star topology if

β = G

andthe normtopology if

β = F

. The ru ial property

P

β

is related to the geometry of the Bana h spa e

A

and is onne ted to the dierentiability to the supremum norm

k.k

∞

for more detail see the arti le [7℄. Thefollowing proposition follow easily fromthe above denition.

Proposition 1 Let

(X, d)

be a omplet metri spa e and

(A, k.k)

be a Bana h spa e in luded in

Cb(X)

whi h separate the pointsof

X

.

(1)

Suppose that

A

has the property

P

F

then

δ(X) := {δx

: x ∈ X}

is losed for the norm topology in

A

∗

andthe map

δ : (X, d) → (δ(X), k.k

∗

)

x 7→ δx

is an homeomorphism.

(2)

Supposethat

A

hastheproperty

P

G

then

δ(X) := {δx

: x ∈ X}

issequentially losed for the weak-startopology in

A

∗

andthe map

δ : (X, d) → (δ(X), w

∗

)

x 7→ δx

is an sequential homeomorphism. Themap

δ : X → A

∗

x → δx

isanonlinearanalogoustothe anoni allinearisometry

i : Z → Z

∗∗

where

Z

isBana h spa e and

Z

∗∗

its bidual. The map

δ

permit to linearize the metri spa e

X

in

A

∗

( See thepaperofGodefroy-Kalton[20 ℄ when

A

isthe setofallLips hitzmapon

X

that vani h at some point).

Wegive now the general axioms thatthe spa e

A

hasto satisfyin ourresults.

Axiome 1 In all the arti le

(X, d)

isa omplet metri spa e and

A

isa lass of fun -tionsspa es in luded in

C

b

(X)

su h that :

(A

1

)

the spa e

(A, k.k)

is a Bana h spa e and

k.k ≥ k.k∞

.

(A

2

)

the spa e

A

separate the pointsof

X

and ontain the onstants.

(A

3

)

for ea h

n ∈ N

∗

, there exists a positive onstant

Mn

su h that for ea h

x ∈ X

there exists a fun tion

h

n

: X → [0, 1]

su h that

h

n

∈ A

,

khnk ≤ Mn

,

h

n

(x) = 1

and

diam(supp(hn)) <

_n

1

.

(A4)

the spa e

A

has the property

P

β

(

β = F

or

β = G

).

Remark 4 : These axioms are satised by various and lassi al spa es of fun tions. We give below some examples. Let us mention here, that the axiom

(A

3

)

is related to the variational prin iple of Deville-Godefroy-Zizler [14℄ and Deville-Rivalski [16℄ and the axiom

(A

4

)

was introdu ed and studied in [7℄.

By the spa e

C

u

b

(X)

we denote the Bana h spa e ofall bounded uniformly ontinuous fun tion on

X

, by

Lip

α

b

(X)

the Bana h spa e of all

α

-Holder and bounded fun tions on

X (0 < α ≤ 1)

, by

C

k

b

(X)

the Bana h spa e of all

k

-times ontinuously Fré het dierentiablefun tions

f

su h that

f

,

f

′

, ...,

f

(k)

areuniformly bounded, by

C

1,α

b

(X)

(

0 < α ≤ 1

)the Bana hspa eofallFré het dierentiablefun tions

f

on

X

su hthat

f

and

f

′

areuniformlybounded onX and

f

′

is

α

-Holder. Finallyby

C

1,u

b

(X)

we denote the Bana h spa eof all Fré het dierentiablefun tions

f

on

X

su h that

f

and

f

′

are uniformlybounded onX and

f

′

isuniformly ontinuous. All thesespa es areprovided with their natural norm

k.k

ofBana h spa ethatsatisfy

k.k ≥ k.k∞

(See[7℄).

Proposition 2 We have,

(1)

For every omplete metri spa e

X

, the spa es

Cb(X)

,

C

u

b

(X)

and

Lip

α

b

(X)

(

0 <

α ≤ 1

)satisfy the axioms

(A

1

)

-

(A

4

)

(2)

If

X

is a Bana h spa e having a bump fun tion (that is a fun tion with a non empty and bounded support) in

A = C

k

b

(X)

(

k ∈ N

∗

) (respe tively, in

A = C

1,α

b

(X)

(

0 < α ≤ 1

) or

A = C

1,u

b

(X)

), then

A

satisfy the axioms

(A

1

)

-

(A

4

)

.

Proof. Theproof isan onsequen e ofProposition 3,Proposition4 andProposition 5 below.

Proposition 3 Allofthespa es

A

mentionedintheexamplesabove,satisfytheaxioms

(A

1

)

and

(A

2

)

.

Proof. Theaxiom

(A1)

followfromthe denitions ofthe spa es and therenorms (See [7℄). Thesespa es ontainthe onstants. To sethatthese spa esseparatethe pointsof

X

, let

x 6= y

in

X

. Inthe ase ofthe spa es

C

b

(X)

,

C

u

b

(X)

and

Lip

α

b

(X)

where

(X, d)

is a metri spa e, Letus set

φ(.) := min(d(x, .), 1) ∈ Lip

α

b

(X) ⊂ C

b

u

(X) ⊂ Cb(X)

. We have

φ(x) = 0

and

ϕ(y) = min(d(x, y), 1) > 0

. So

ϕ(x) 6= ϕ(y)

and thus these spa es separate the points of

X

. In the ase where

X

is a Bana h spa e and

A = C

k

b

(X)

(

k ∈ N

∗

) or

C

1,α

b

(X)

(

0 < α ≤ 1

) or

C

1,u

b

(X)

, sin e

x 6= y

, then by the Hanh-Bana h theorem,thereexists

p ∈ X

∗

su hthat

p(x) 6= p(y)

. We onstru tthenamap

γ : R → R

su h that

γ ◦ p ∈ A

and

γ(p(x)) = p(x)

and

γ(p(y)) = p(y)

. Thus

A

separatethe point of

X

.

The following proposition isin [7℄.

Proposition 4 [Proposition2.5and2.6,[7℄℄Let

(X, d)

be a omplete metri spa e. We have :

(1)

If

A = C

b

(X)

or

C

u

b

(X)

then

A

satisfy

P

G

(but not

P

F

).

(2)

If

A = Lip

α

b

(X)

with

0 < α ≤ 1

then

A

satisfy

P

F

.

b)

Suppose that

X

is Bana h spa e and

A = C

k

b

(X)

(

k ∈ N

∗

) or

C

1,α

b

(X)

(

0 < α ≤ 1

) or

C

1,u

b

(X)

then

A

satisfy

P

F

.

We have proved that the above examplesof spa es satisfy the axioms

(A

1

)

,

(A

2

)

and

(A4)

. These examples satisfy also the axiom

(A3)

whenever the spa e

X

has a bump fun tionin

A

(thatisafun tionwithanonemptyandboundedsupport). Theexisten e of a bump fun tion in

C

b

(X)

,

C

u

b

(X)

or in Lip

α

b

(X)

with

0 < α ≤ 1

isalways true by usingthemetri

d

on

X

. Thisisnotealwaysthe asewhen

X

isaBana hspa eforthe spa es of smooth fun tions

A = C

k

b

(X)

(

k ∈ N

∗

) or

C

1,α

b

(X)

(

0 < α ≤ 1

) or

C

1,u

b

(X)

. In the last examples the existen e of a bump fun tion is onne ted to the geometry of the Bana h spa e

X

. For more information on the existen e of a bump fun tion in

C

k

b

(X)

(

k ∈ N

∗

)or

C

1,α

b

(X)

(

0 < α ≤ 1

) or

C

1,u

b

(X)

we referto thebookof R.Deville, G. GodefroyandV.Zizler [15 ℄.

Proposition 5 we have:

(1)

Forevery ompelete metri spa e

X

, thespa es

Cb(X)

,

C

u

b

(X)

and

Lip

α

b

(X)

satisfy the axiom

(A

3

)

(2)

If

X

isa Bana hspa e havinga bum fun tionin

A = C

k

b

(X)

(

k ∈ N

∗

)(respe tively in

C

1,α

b

(X)

(

0 < α ≤ 1

) or

C

1,u

b

(X)

), then

A

satisfythe axiom

(A

3

)

.

Proof. Theproofiseasyand an be foundinRemark2.5(SeealsoProposition1.4)in [16℄.

2.3 A useful Lemmas.

We need the following Lemma 1 that is a onsequen e of the Variational prin iple of Deville-Godfroy-Zizler[14 ℄ generalized byDeville-Rivalski [16℄.

Denition 2 We say that a fun tion

f

has a strong minimum at

x

if

inf

X

f = f (x)

and

d(xn, x) → 0

whenever

f (xn) → f (x)

.

Theorem 2 ( [Deville-Godefroy-Zizler [14℄℄ ; [Deville-Rivalski[16℄℄):

Let

(X, d)

be a omplete metri spa e and

A ⊂ Cb(X)

be a spa e satisfying the axioms

(A

1

)

and

(A

3

)

. Let

f

be a lower semi ontinuous andbounded frombelowfun tion with non empty domain. Then,

{ϕ ∈ A/ f − ϕ

doesnot attain astrong minimum on

X}

is

σ

-porous in parti ular it is of therst Baire ategory.

tinuous and bounded from below withnon empty domain

f : X → R ∪ +∞

(

f 6≡ +∞

) the set

D(f ) := {x ∈ X/ ∃ϕ

x

∈ A : f − ϕx

has a strong minimumat

x}

is densein

dom(f )

.

Proof. Let

x ∈ dom(f )

an

n ∈ N

∗

. Let

hn

∈ A

as in Theorem 2with

hn(x) = 1

. let usset

λ

n

x

:= f (x) − infX

(f ) +

n

3

andapplyingTheorem2to thefun tion

f + λ

n

x

hn

, there exists

xn

∈ X

and

ϕ ∈ A

su hthat

kϕk <

1

n

and

f + λ

n

x

hn

− ϕ

hasastrong minimum at

xn

. Supposethat

d(x, xn) ≥

1

n

. sin e

diam(supp(hn)) <

1

n

and

x ∈ supp(hn)

, then

h(xn) = 0

. Thus, we have

inf

X

(f ) − ϕ(xn) ≤ f (xn) − ϕ(xn)

= f (xn) − λ

n

_x

hn(xn) − ϕ(xn)

< f (x) − λ

n

x

hn(x) − ϕ(x)

= f (x) − λ

n

_x

− ϕ(x)

We dedu t that

λ

n

x

< f (x) − inf

X

(f ) +

_n

2

whi h is a ontradi tion with the hoi e of

λ

n

x

. So

d(x, xn) <

1

n

and

xn

∈ D(f )

. Itfollows that

D(f )

isdense in

dom(f )

. Weneed also the following twoLemmas.

Lemma 2 Let

Z

bea Bana hspa e and

h, k : Z → R

two ontinuous and onvex fun -tions. Suppose that the fun tion

z → l(z) := max(h(z), k(z))

is Fré het (respe tively, Gâteaux) dierentiable at

z

0

. Then

h

is Fré het (respe tively, Gâteaux) dierentiable at

z

0

or

k

is Fré het (respe tively, Gâteaux) dierentiable at

z

0

and

l

′

_(z

0

) = h

′

(z

0

)

or

l

′

(z

0

) = k

′

(z

0

)

.

Proof. Wegive the proof for the fré het dierentiability, the Gâteauxdierentiability is similar. Supposewithout looseof generalitythat

l(z

0

) = h(z

0

)

and letus prove that

h

isFré het dierentiable at

z

0

andthat

l

′

_(z

0

) = h

′

(z

0

)

. For ea h

z 6= 0

we have:

0 ≤

h(z

0

+ z) + h(z

0

− z) − 2h(z0

)

kzk

≤

l(z

0

+ z) + l(z

0

− z) − 2l(z0

)

kzk

Therightmembertendsto

0

when

z

tendsto

0

sin e

l

is onvexandFré hetdierentiable at

z

0

. This implies that

h

is Fré het dierentiable at

z

0

by onvexity. Now, sin e

l(z) ≥ h(z)

for all

∈ Z

and

l(z

0

) = h(z

0

)

, then for all

t > 0

we have

h

′

(z

0

)(z) − l

′

(z

0

)(z) ≤

h

′

(z

0

)(tz) − l

′

(z

0

)(tz) + l(z

0

+ tz) − h(z

0

+ tz) − l(z

0

) + h(z

0

)

t

=

l(z

0

+ tz) − l(z

0

) − l

′

_(z

0

)(tz)

t

+

h(z

0

+ tz) − h(z

0

) − h

′

(z

0

)(tz)

t

So,sending

t

to

0

+

,wehave

h

′

_(z

0

)(z)−l

′

(z

0

)(z) ≤ 0

,forall

z ∈ Z

. Thus

h

′

_(z

Lemma 3 Let

(X, d)

be a omplete metri spa e and

A ⊂ Cb(X)

satisfyingthe axioms

A

1

,

A

2

and

A

4

. Let

(λ)

n

⊂ R

su h that

|λn| = 1

for all

n ∈ N

and let

(x

n

)

n

⊂ X

. Suppose that

λnδx

n

onverges for the topology

τβ

(the norm topology or the weak-star topology) to some point

Q ∈ A

∗

. Then

(λn)n

onverges in

R

to some real number

λ

su h that

|λ| = 1

and

(x

n

)

n

onverges tosome point

x

in

(X, d)

andwe have

Q = λδ

x

. Proof. Sin e

λ

n

δ

x

n

onverges for the topology

τ

β

to some point

Q ∈ A

∗

, then

λnδx

n

(ϕ) → Q(ϕ)

forall

ϕ ∈ A

. Sin e

A

ontainthe onstants,wehave

λn

→ Q(1) := λ

with

|λ| = 1

. Now,sin e

(λn)n

onverges to

λ

and

λnδx

n

onverges forthe topology

τβ

to

Q

. By dividing by

λn

we have that

δx

n

onverges for the topology

τβ

to

Q

λ

∈ A

∗

. Theproperty

P

β

impliesthat

(x

n

)

n

onvergestosomepoint

x ∈ X

andin onsequan e that

δx

n

onverges for the topology

τβ

to

δx

. By the uniqueness of the limit we have that

Q = λδx

.

2.4 Duality results.

We re all from[7 ℄ the following results whi h are the key of the results of this arti le. Let

(X, d)

be a omplete metri spa e and

A

be a lass of fun tions spa e satisfying the axioms

A

1

-

A

4

. Forthe proofofTheorem1,we needthe following resultsaboutthe onjuga y

f

×

of lower semi ontinuousbounded frombelowfun tion

f

dened by

f

×

: A → R

ϕ → sup

x∈X

{ϕ(x) − f (x)}

Wedene the se ond onjuga y of

f

by

f

××

_{: X → R ∪ {+∞}}

;

f

××

(x) := sup

ϕ∈A

ϕ(x) − f

×

_{(ϕ) ; ∀x ∈ X}

Proposition 6 ([ Proposition 2.1, [7℄℄): The map

f

×

is onvex and

1

-Lips hitzon

A

.

The se ond onjugate

f

××

isnot onvexin general, but we have:

Theorem 3 ([ Theorem 2.2, [7℄℄): Le

f

be aboundedfrombelowfun tionon

X

and suppose that

A

satisfy the axiom

A4

. Then,

f

is lower semi ontinuous if and only if

f

××

= f

.

Finally, we need the following duality theorem between dierentiability and the well posedproblems.

Theorem 4 ([Theorem 2.8, [7℄℄): Let

A ⊂ C

b

(X)

be a lass of Bana h spa e satis-fyingthe axioms

A1

,

A2

and the property

P

F

(respe tively

P

G

). Let

ϕ ∈ A

and

f

be a lower semi ontinuous fun tion. Then

(1) ⇔ (2)

.

(1)

the fun tion

f − ϕ

has a strong minimum atsomepoint in

X

.

(2)

the fun tion

f

×

is Fré het dierentiable (respe tively, Gâteaux dierentiable) at

ϕ

on

A

.

Moreover,inthis asetheFré hetderivative (respe tively,theGâteauxderivative)of

f

×

at

ϕ

is

(f

×

₎

′

_{(ϕ) = δ}

Lemma 4 We have

sup

x∈X

{|ϕ(x)| − f (x)} = max(f

×

(ϕ), f

×

(−ϕ))

for all

ϕ ∈ A

.

Proof. Sin e

|t| = max(t, −t)

on

R

,byinverting the supremum andthe maximumwe have

sup

x∈X

{|ϕ(x)| − f (x)} = max(sup

x∈X

{ϕ(x) − f (x)} , sup

x∈X

{−ϕ(x) − f (x)})

= max(f

×

(ϕ), f

×

(−ϕ)).

3 The proof of Theorem 1.

The proof is given after some steps. The part

(2) ⇒ (1)

is easy. We prove the part

(1) ⇒ (2)

.

Let

X

and

Y

be two omplete metri spa es and

A ⊂ C

b

(X)

and

B ⊂ C

b

(Y )

be two Bana h spa es satisfying the axioms

A1

-

A4

with the same property

P

β

. (See Ax-ioms 1 and the examples in Proposition 2 ). Let

T : A → B

be an isomorphism and let

f : X → R ∪ {+∞}

and

g : Y → R ∪ {+∞}

be lower semi ontinuous and bounded from below fun tionswith non emptydomains. Suppose that

sup

y∈Y

{|T ϕ(y)| − g(y)} = sup

x∈X

{|ϕ(x)| − f (x)}

Forall

ϕ ∈ A

and denoteby

T

∗

_{: B}

∗

_{→ A}

∗

the adjoint of

T

.

3.1 The map

T

has the anoni al form.

Lemma 5 : Thereexistsamap

π : dom(g) → dom(f )

andamap

ε : dom(g) → {−1, 1}

su h that for all

y ∈ dom(g)

we have :

T

∗

_δ

y

= ε(y)δ

π(y)

.

Remark 5 Theformulaaboveisequivalent to

T ϕ(y) = ε(y)ϕ ◦ π(y)

for all

y ∈ dom(g)

and all

ϕ ∈ A

.

Proof. By lemma1, the set

D(g)

isdense in

dom(g)

. Let

y ∈ D(g)

and

ψy

˜

∈ B

su h that

g − ˜

ψy

hasastrongminimum at

y

. Let

c ∈ R

su hthat

c >

1

2



g

×

(− ˜

ψy

) − g

×

( ˜

ψy

)



and put

ψ

y

= c + ˜

ψ

y

. The fun tion

g − ψ

y

has also a strong minimum at

y

and satises by the hoi e of

c

the inequality

g

×

_(ψy

_{) > g}

×

_(−ψy

₎

. Sin e

g

×

and so al-so

g

×

_{◦ (−IY}

₎

(where

IY

denotes the identity map on

B

) are ontinuous (Lips hitz) by Proposition 6, there exists an open neighborhood

O(ψ

y

) ⊂ B

of

ψ

y

su h that

g

×

(ψ) > g

×

(−ψ)

for all

ψ ∈ O(ψy)

. Thus, we have

max(g

×

_{(ψ), g}

×

_{(−ψ)) = g}

×

_(ψ)

on the open set

O(ψy)

. Sin e

g − ψy

has a strong minimum at

y

, Theorem 4 guar-antees the

β

-dierentiability of

g

×

at

ψ

y

with

(g

×

₎

′

_(ψ

ψ → max(g

×

(ψ), g

×

(−ψ))

and

ψ → g

×

_(ψ)

oin ide on the open set

O(ψy

)

we on- lude that

ψ → max(g

×

_{(ψ), g}

×

_(−ψ))

is

β

-dierentiable at

ψ

y

with derivative equal to

(g

×

)

′

(ψy

) = δy

. On the other hand, there exists

ϕy

∈ A

su h that

ψy

= T ϕy

by the surje tivity of

T

. The omposition of

β

-dierentiable fun tion with linear an ontinu-ousmapisagain

β

-dierentiable. Thus,wehavethe

β

-dierentiabilityofthe omposite map

ϕ → max(g

×

_{(T ϕ), g}

×

_{(−T ϕ))}

at

ϕy

on

A

and the hain rule formulagive

δy

◦ T

as derivative of the fun tion

ϕ → max(g

×

_{(T ϕ), g}

×

_{(−T ϕ))}

at

ϕy

. But by hypothesis and by Lemma 4,

max(g

×

_{(T ϕ), g}

×

_{(−T ϕ)) = max(f}

×

_{(ϕ), f}

×

_(−ϕ))

for all

ϕ ∈ A

. We dedu e thatthe fun tion

ϕ → max(f

×

_{(ϕ), f}

×

_(−ϕ))

is also

β

-dierentiable at

ϕy

on

A

with the same derivative

δy

◦ T

. Lemma 2implies thateither the fun tion

ϕ → f

×

_(ϕ)

or the fun tion

ϕ → f

×

_(−ϕ)

is

β

-dierentiable at

ϕy

with the derivative given bythe derivative of

ϕ → max(f

×

_{(ϕ), f}

×

_(−ϕ))

at

ϕy

that is here

δy

◦ T

. If it is the fun tion

ϕ → f

×

(ϕ)

,Theorem4assertthatthereexists

π(y) ∈ D(f )

su hthat

(f

×

₎

′

_(ϕy

_{) = δ}

π(y)

. If it is the fun tion

ϕ → f

×

_(−ϕ)

, Theorem 4 assert the existen e of point

π(y) ∈ X

su h that

(f

×

_{◦ (−IX}

₎₎

′

_{(ϕy) = (f}

×

₎

′

_{(−ϕy) ◦ (−IX}

_{) = δ}

π(y)

◦ (−IX

) = −δ

π(y)

where

IX

denotes the identity map on

A

. By identifying the derivatives of the two equal fun -tions

ϕ → max(g

×

_{(T ϕ), g}

×

_{(−T ϕ)) = max(f}

×

_{(ϕ), f}

×

_(−ϕ))

we have:

δ

y

◦ T = δπ(y)

or

δy

◦ T = −δ

π(y)

. Letus put

ε(y) = ±1

. Then we have

∀ y ∈ D(g) ∃ π(y) ∈ D(f )/ T

∗

_δ

y

:= δ

y

◦ T = ε(y)δπ(y)

.

(3)

Now, let

y

beanypointof

dom(g)

,there existsbyLemma 1asequen es

(yn)n

⊂ D(g)

su hthat

yn

→ y

. Theproperty

P

β

(axiom

A

4

)impliesthat

δy

n

τ

β

→ δy

. Sin e

T

∗

is

τβ

to

τβ

ontinue (here

τβ

is the normor the weak-startopology) then

T

∗

_δy

n

τ

β

→ T

∗

δy

. Sin e

(y

n

)

n

⊂ D(g)

, then from (3)there exists

π(y

n

) ∈ D(f )

su h that

T

∗

_δ

y

n

= ε(y

n

)δ

π(y

n

)

. Sowe have

ε(yn)δ

π(y

n

)

τ

β

→ T

∗

_δy

. Lemma 3impliesthe existen eofarealnumber

ε(y) =

±1

andsomepoint

π(y) ∈ X

su hthat

ε(yn) → ε(y)

in

R

and

π(yn) → π(y)

in

X

. Thus

π(y) ∈ D(f ) = dom(f )

. TheLemma3impliesalsothat

T

∗

_δy

_{= ε(y)δ}

π(y)

. Thuswehave proved thatthere existsamap

π : dom(g) → dom(f )

and amap

ε : dom(g) → {−1, 1}

su h thatfor all

y ∈ dom(g)

, we have

T

∗

_δy

_{= ε(y)δ}

π(y)

.

3.2 The map

π

is bije tive. Lemma 5 applied to

T

−1

implies the existen eof a map

π

′

_{: dom(f ) → dom(g)}

and a map

ε

′

_{: dom(f ) → {−1, 1}}

su hthatforall

x ∈ dom(f )

wehave

(T

−1

₎

∗

_δx

_{= ε}

′

_(x)δ

π

′

(x)

. Weobtain then ,

δx

= T

∗

(ε

′

(x)δ

π

′

(x)

)

= ε

′

(x)T

∗

(δ

π

′

(x)

)

= ε

′

(x)ε(π(x))δ

π(π

′

_(x))

By applying the above identity to the ostant fun tion

1

we obtain

ε

′

_{(x)ε(π(x)) = 1}

. Onthe otherhand,sin e thespa e

A

separatethe pointsof

X

(axiom

(A2)

),we obtain

π(π

′

(x)) = x

. Thisreasoningapplies for all

x ∈ dom(f )

. Byinverting the roles of

T

et

T

−1

,wehave also

π

′

_{(π(y)) = y}

3.3 The maps

ε

,

π

and

π

−1

are ontinuous.

Let

yn

∈ dom(g)

su h that

yn

→ y ∈ dom(g)

. Let us prove that

π(yn) → π(y)

in

dom(f )

. Indeed, by the property

P

β

( axiom

A

4

) we have that

δ

y

n

τ

β

→ δy

. Sin e

T

is

τβ

to

τβ

ontinuous then

T

∗

_δy

n

τ

β

→ T

∗

δy

for the

τβ

topology in

A

∗

. In other words

ε(yn)δ

π(y

n

)

τ

β

→ ε(y)δπ(y)

whi h implies by Lemma 3 that

ε(yn) → ε(y)

in

R

and

δ

_π(y

_n

₎

→ δπ(y)

τ

β

in

A

∗

. Again by the property

P

β

, we have

π(yn) → π(y)

in

dom(g)

. Thus,

ε

and

π

are ontinuous. The same argument applied to

T

−1

shows that

π

−1

is also ontinuous.

3.4 The formula

g

= f ◦ π

on

dom(g)

.

This formula follow fromthe previous armationstogether with Theoreme 4. Indeed, byhypothesis we have

sup

y∈Y

{|T ϕ(y)| − g(y)} = sup

x∈X

{|ϕ(x)| − f (x)}

forall

ϕ ∈ A

. Sin e

g

(resp

f

)isequalto

+∞

on

Y \dom(g)

(respe tively,on

X\dom(f )

) we have

sup

y∈dom(g)

{|T ϕ(y)| − g(y)} =

sup

x∈dom(f )

{|ϕ(x)| − f (x)}

for all

ϕ ∈ A

. Sin e

dom(g)

and

dom(f )

arehomeomorphi , then by repla ing

T ϕ(y)

in the aboveformulawith its expression

ε(y)ϕ(π(y))

for

y ∈ dom(g)

, we obtain

sup

y∈dom(g)

{|ϕ(π(y))| − g(y)} =

sup

x∈dom(f )

{|ϕ(x)| − f (x)} ,

for all

ϕ ∈ A

. Sin e

π

is bije tive,we obtainbythe hange ofvariable

x = π

−1

_(y)

sup

x∈dom(f )

|ϕ(x)| − g(π

−1

_{(x)) =}

_sup

x∈dom(f )

{|ϕ(x)| − f (x)} ,

for all

ϕ ∈ A

. The above formulais also truefor the fun tions

ϕ − inf

X

(ϕ) ≥ 0

for all

ϕ ∈ A

sin e

A

ontain the onstants. Repla ing

ϕ

by

ϕ − inf

X

(ϕ) ≥ 0

we obtain

sup

x∈dom(f )

ϕ(x) − g(π

−1

_{(x)) − inf}

X

(ϕ) =

sup

x∈dom(f )

{ϕ(x) − f (x)} − inf

X

(ϕ),

for all

ϕ ∈ A(X)

. So

sup

x∈dom(f )

ϕ(x) − g(π

−1

_{(x)) =}

_sup

x∈dom(f )

{ϕ(x) − f (x)} ,

for all

ϕ ∈ A

. Let us denote by

i

dom(f )

the lower semi ontinuous indi ator fun tion whi h is equal to

0

on

dom(f )

and equal to

+∞

otherwise. Theabove formula an be written asfollow

sup

x∈X

n

ϕ(x) −



g(π

−1

(x)) + i

_{dom(f )}

o

= sup

x∈X

n

ϕ(x) −



f (x) + i

_{dom(f )}

o

,

for all

ϕ ∈ A

. In otherwords, byusing the notation ofthe onjuga y (see se tion1.4)



g ◦ π

−1

+ i

_{dom(f )}

×

(ϕ) = (f + i

_{dom(f )}

)

×

(ϕ)

for all

ϕ ∈ A

. By passingto the se ond onjuga ywe obtain

(g ◦ π

−1

+ i

_{dom(f )}

)

××

(x) =



f + i

_{dom(f )}

××

(x)

for all

x ∈ X

. Sin e the fun tions

f + i

dom(f )

and

g ◦ π

−1

_{+ i}

dom(f )

arebounded from below and lower semi ontinuous on

X

, then by Theoreme 4, ea h of these fun tions oin ide with hisse ond onjuga y. Thus we have

g ◦ π

−1

+ i

_{dom(f )}

= f + i

_{dom(f )}

whi h isequivalent to

g ◦ π

−1

_{= f}

on

dom(f )

aswellas

g = f ◦ π

on

dom(g)

. Remark 6 By imitating the above prof, we obtainthe followingversion.

Theorem 5 Let

X

and

Y

be two omplete metri spa es and

A ⊂ C

b

(X)

and

B ⊂

Cb(Y )

be two Bana hspa es satisfyingthe axioms

A1

-

A4

(with the same property

P

β

. See Axioms 1 and the examples in Proposition 2). Let

T : A → B

be an isomorphism and let

f : X → R ∪ {+∞}

and

g : Y → R ∪ {+∞}

be lower semi ontinuous and bounded frombelow withnonempty domains. Then

(1) ⇔ (2)

.

(1) g

×

_{◦ T = f}

×

(i.e for all

ϕ ∈ A

,

sup

y∈Y

{T ϕ(y) − g(y)} = sup

x∈X

{ϕ(x) − f (x)}

).

(2)

There existan homeomorphism

π : dom(g) → dom(f )

su h that, forall

y ∈ dom(g)

and all

ϕ ∈ A

we have

T ϕ(y) = ϕ ◦ π(y)

and

g(y) = f ◦ π(y).

Referen es

[1℄ D. Amir, On isomorphisms of ontinuous fun tion spa es, Israel J. Math., 3 (1966), 205-210.

[2℄ J. Araujo, Real ompa tness and spa es of ve tor-valued fun tions, Fund. Math., 172 (2002), 27-40.

[3℄ J. Araujo, The non ompa t Bana h-Stone theorem, J. Operator Theory, 55(2) (2006), 285-294.

[4℄ J.Araujo,J.J.Font,Linearisometriesbetweensubspa esof ontinuous fun tions, Trans. Amer.Math. So ., 349 (1997), 413-428.

[5℄ M. Ba hir, Dualité, al ul sous-diérentiel et intégration en analyse non lisse et non onvexe, Ph.D. Dissertation, UniversitéBordeaux I, Fran e, 2000.

A ad.S i. Paris, 330 (2000), 687-690.

[7℄ M.Ba hir,Anon onvexeanaloguetoFen helduality,J.Fun t.Anal.181,(2001) 300-312.

[8℄ M.Ba hirandG.Lan ien,Onthe ompositionofdierentiablefun tions,Canad. Math.Bull. 46 (2003), no.4,481-494.

[9℄ S. Bana h, “Th´eorie des Op´erations Lineaires”, Warszowa 1932. Reprinted, ClelseaPublishing Company, New York1963.

[10℄ E.Behrends,M-stru tureandtheBana h-StoneTheorem,Springer-Verlag,Berlin 1978.

[11℄ E. Behrends, M. Cambern, An isomorphi Bana h-Stone theorem, StudiaMath., 90 (1988), 15-26.

[12℄ M. Cambern, Isomorphisms of

C

0

(Y )

onto

C

0

(X)

, Pa i J. Math., 35 (1970), 307-312.

[13℄ B.Cengiz,AgeneralizationoftheBana h-Stonetheorem,Pro .Amer.Math.So ., 50 (1973), 426-430.

[14℄ R. Deville and G. Godefroy and V. Zizler, A smooth variational prin iple with appli ations to Hamilton-Ja obi equations in innite dimensions,J.Fun t.Anal. 111,(1993) 197-212.

[15℄ R.DevilleandG.GodefroyandV.Zizler,SmoothnessandRenormingsinBana h Spa es, PitmanMonographs No. 64,London: Longman, 1993.

[16℄ R.Deville, J.P. Revalski.Porosity ofill-posed problems,Pro .Amer.Math.So . 128 (2000), 1117–1124.

[17℄ M.I.Garrido,J.A.Jaramillo, Variationsonthe Bana h-StoneTheorem, Extra ta Math.Vol. 17,N.3,(2002) 351-383.

[18℄ M.I. Garrido,J.A. Jaramillo, A Bana h-Stone theorem for uniformly ontinuous fun tions, Monatsh.Math., 131 (2000), 189-192.

[19℄ M.I. Garrido, J. Gómez, J.A. Jaramillo, Homomorphisms on fun tion algebras, Canad. J.Math., 46 (1994), 734-745.

[20℄ G.Godefroy, N.KaltonLips hitz-freeBana hspa es,StudiaMath.,159(1)(2003) 121-141.

[21℄ J. M. Gutiérrez and J. G. Llavona, Composition operators between algebras of dierentiable fun tions,Trans. Amer.Math.So .338 (1993), no.2,769-782. [22℄ J. A. Jaramillo, M. Jimenez-Sevilla, and L. San hez-Gonzalez, Chara terization

of a Bana h-Finsler manifold in termsof the algebras of smooth fun tions, Pro- eedingsof the Ameri anMathemati al So iety. In press.