STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

Acta Mathematica Scientia, 2019, 39B(4): 1163-1172
https:///10.1007/sl0473-019-0418-9
©Wuhan Institute Physics and Mathematics,
Chinese Academy of Sciences, 2019http: 〃
a ct a STABILITY OF A PAIR OF BANACH SPACES FOR
s-ISOMETRIES
** Received March 22, 2018; revised September 18, 2018. Duanxu Dai is supported in part by NSFC (11601264, 11471270 and 11471271), the Fundamental Research Funds for the Central Universities (20720160037), the Outstanding Youth Scientific Research Personnel Training Program of Fujian Province, the High level Talents Innovation and Entrepreneurship Project of Quanzhou City (2017Z032), the Research Foundation of Quanzhou Normal University (2016YYKJ12), and the Natural Science Foundation of Fujian Province of China (2019J05103). Bentuo Zheng is supported in part by NSFC (11628102).
tCorresponding author
Duanxu DAI (戴端旭)十
College of Mathematics and Computer Science, Quanzhou Normal University,
Quanzhou 362000, China
E-mail : dduanxu@l 63. com
Bentuo ZHENG (郑本拓)
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
E-mail : bzheng@memphis. edu
Abstract A pair of Banach spaces (X, Y) is said to be stable if for every £-isometry f : X there exist 7 > 0 and a bounded linear operator T : L(f) —> X with ||T|| < a such that ||T/(x) — x\\ < w for all x E X, where L(/) is the closed linear span of /(X). In this article, we study the stability of a pair of Banach spaces (X, Y) when X is a C(K) space. This gives a new positive answer to Qian's problem. Finally, we also obtain a nonlinear version for Qian's problem.
Key words St a bility; e-isometry; Figiel theorem; Banach space
2010 MR Subject Classification 46B04; 46B20; 54C60
1 Introduction
Mazur and Ulam [18] proved in 1932 that every surjective isometry between two Banach spaces is necessarily affine. When an isometry is not assumed to be onto, we do not get linearity in general. Instead, Figiel [11] showed the following remarkable result(Figiel theorem) in 1968: For every standard isometry / : X —> V, there is a linear operator T : L(/) —> X with ||T|| = 1 so that T o f = Id on X , where L(/) is the closed linear span of /(X) in Y . In this article, we will mainly focus on perturbations of isometries which are called e-isometries.
Definition Let X, Y be two Banach spaces, s > 0, and / : X —> Y be a mapping.
(1) f is said to be an e-isometry if
一旧一训I < E for all X,ye X.(1-1)
g
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1164ACTA MATHEMATICA SCIENTIA Vol.39Ser.B When6?=0,/is an isometry.
(2)We say that an e-isometry f is standard if/(0)=0.
(3)Let q,7>0.A standard s-isometry f is(q,7)-stable if there exists a bounded linear operator T:L(/)—>X with||T||<a such that
||T/(x)—x\\<for all x e X.(1.2)
We say that f is stable if f is(a,7)-stable for some ce,7>0.
(4)A pair of Banach spaces(X,F)is said to be stable if every standard s-isometry f:X—> y is stable.
(5)Let q,7>0.A pair of Banach spaces(X,Y)is called(q,7)-stable if every standard 「isometry f:X Y is(a,7)-stable.
In[13],Hyers and Ulam had an extensive study of s-isometries.They proved that every surjective e-isometry between real Hilbert spaces can be uniformly approximated by an affine surjec t ive isome t ter,people started to consider non-surjective s-isome t ries and st a bility of a pair of Banach spaces(see,for instance,[3,7-10,19,21-23]).The following problem was proposed by Qian[21]in1995.
Problem1・1Is it true that for every pair of Banach spaces(X,/),there exists7>0 such that every standard s-isometry/:X—>Y is(q,7)-s table for some&>0?
However,Qian[21]presented a counterexample by showing that if a separable Banach space Y contains an uncomplemented closed subspace X,then for every s>0,there is a standard e-isometry/:X—>Y which is not stable.So,some conditions must be added to X or Y to get a positive answer of Problem1.1.The following question was proposed in[7].
Problem1・2Is there a characterization for the class of Banach spaces X satisfying given any X E X and Banach space V,the pair(X,Y)is((q,7)-,resp.)st a ble?
Every space X of this class is said to be universally((q,7)-,resp.)left-stable.On one hand, Cheng,Dai,Dong et al[7]proved that every injective Banach space is universally left-stable. On the other hand,the first two authors Cheng and Dai,together with others[3],showed that every universally left-stable space is cardinality injective(that is,complemented in every superspace with the same cardinality)and they also showed that a dual space is injective if and only if it is universally left-stable.When X is a certain type of C(K)spaces,a positive answer for Problem1.1is given in Section3.We also obtain a nonlinear version of stability for £-isometries,that is,Theorem3.14.
We refer interested readers to an excellent article[2]of Aviles-Sanchez-Castillo-Gonzalez-Moreno for further information about(separably)injective Banach spaces.All symbols and notations in this article are standard.We denote X to be a real Banach space and X*to be its dual.Ex,ext(B%.)and Sx are the closed unit ball of X、the set of all extremal points of Bx
*and the unit sphere of X,respectively.For a subset A C X,A and card(A)st a nd for the closure of A and the cardinality of A,respectively.Given a bounded linear operator T:X—>匕71*:Y*t X*is its adjoint operator.
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No.4 D.X.Dai&B.T.Zheng:STAEILITY OF A PAIR OF BANACH SPACES1165
2Preliminaries
Recall that a Banach space X is said to be A-(resp.separably injective)injective if it has the following extension property:Every bounded linear operator T from a closed subspace of
a(resp.separable)Banach space into X can be extended to a bounded operator on the whole space with norm at most A||T||.X is said to be injective(resp.separably injective)if it is
A-(resp.separably injective)injective for some入>1.The following Proposition was proved
by Aviles,Sanchez,Castillo,Gonzalez,and Moreno(see[2,Proposition3.2]).
Proposition2.1(1)If a Banach space X is入-separably injective,then it is3入-complemented in every superspace Y such that Y/X is separable.
(2)If a Banach space X is入-complemented in every superspace Y such that Y/X is separable,then X is入-separably injective.
Recently,Cheng et al[6]showed the following weak st a bility theorem.
Theorem 2.2(Cheng et al)Let X and Y be Banach spaces,and let f:X Y be
a standard s-isometry for some s>0.Then,for every x*€X*,there exists0€/*with
||0||=|k*||such that|〈0J(z)〉—(x*,x)\<2e||ir
*||for all x E X.
Let Q be a w*-dense subset of ext(Bx
*).Then,there is a natural isometrie embedding i
of X into^oo(Q)by=
Lemma2.3Suppose that X,Y are Banach spaces.Let s>0.Assume that f is an
e-isometry from X into Y with/(0)=0.Then,for every w*-dense subset Q C ext(Bx
*),there
is a bounded linear operator T:Y^(Q)such that\\Tf^x)—z(rr)||<2s for all x E X.
*)G Sy*such Proof By Theorem2.2,for every x*e Q,there exists a functional Q(z
that/(x))—{x*,x)\<2e for all x E X.We now define a mapping T:Y by
*)(y)}xa.
T(y)={Q(z
It is clear that T is a bounded linear operator with norm one and
*)f(x)—<2s for all x E X.
\\Tf(x)—迪)||=sup\Q(x
□The following lemma follows from Qian's counterexample in[21].
Lemma2.4Let X be a closed subspace of a Banach space Y.If card(X)=card(Y), then for every£〉0and every bijective mapping g:X By with g(0)=0,there is a standard
e-isometry/:X—>V defined for all x€X by/(x)=x号g(£)such that
(1)L⑴=span/(X)=Y\
(2)X is A-complemented in Y whenever f is(A,7)-stable for some入，丁>0.
3Stability for(C(K),Y)
Recall that a dual Banach space Y*is said to have the point of weak star to norm continuity property(in short,w*-PCP)if every nonempty bounded subset of Y*admits relative weak star neighborhoods of arbitrarily small diameter.For example,if Y is an Asplund space,then Y* has the w*-PCP(see,for instance,[20]).
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1166ACTA MATHEMATICA SCIENTIA Vol.39Ser.B
A set valued mapping F:X is said to be usco at x e X if F is nonempty compact valued and upper semicontinuous at x,that is,for every open set V of V containing F(x),there exists a open neighborhood U of X such that F(U)C V.F is usco if it is usco at each x E X.
A mapping:X—>V is called a selection of F if卩(无)E F(x)for each x E X.We say that0is a continuous(linear)selection of F if92is a continuous(linear)mapping.We denote the graph of F by G(F)三{(x,y)e X x Y:y e F(x)}.We write Fi C F2if G(FJ c G(F2).
A usco mapping F is said to be minimal E=F whenever E is a usco mapping and E C F (see,for instance,[9],[20,page19,102-109]).
Lemma3.1Suppose that X and Y are Banach spaces.Let e>0.Assume that/is a standard s-isometry from X into Y and H is a Baire subspace contained in Sx*with respect to the w*-topology.If we define a set-valued mapping①1:Sx*—>2%"by
①1(Z*)={0W S q(〃：ISJ(z)〉-{x\x}\<2e,for all x e X},
where=span/(X),then
(i)①1is convex w-w*usco at each point of Sx*•
(ii)There exists a minimal convex w-w*usco mapping contained in①「
(iii)If,in addition,Y*has the w*-PCP(especially,if Y is an Asplund space)or Y is separable,then there exists a selection Q of①1such that Q is w-w*continuous on a w*-dense
G§subset of H.
Proof(i)follows easily from[9,Lemma4.2(i)].
(ii)By Zorn's Lemma(see[9,Lemma4.2(ii)]or[20,Prop.7.3,p.103]),there exists a minimal convex w*・w*usco mapping contained in①1.
(iii)By(ii),there is a minimal convex w-w*usco mapping F U①1.Because H is a Baire space with respect to the w*-topology,and V*has the w*-PCP or Y is separable,our conclusion follows from[20,Lemma7.14,p.106-107]and[9,Lemma4.2(iii)].Indeed,if Y*has the w*-PCP,then by the definition of the w*-PCP,for every open subset V,F(V)contains relatively w*open subsets of arbitrarily small diameter because F(V)is bounded.Hence,it follows from[20,Lemma7.14,p.106-107]that there exists a w*-dense G$subset D of H such that F is single-valued at each point of D.Therefore,there exists a selection Q of①1such that Q is w-w*continuous on a w*-dense subset D of H.Secondly,if Y is separable,then there exists a norm-dense countable set{£仇}診]C S l⑴such that the relative w*-topology on every bounded subset A of L(f)
*coincides with a metric defined for all z*,9*€X*by
00
d(0,y*)=刀2-"|仗*-y*,x n)\.
n—1
As for every open subset V,F(V)is bounded,it must contain relatively w*open subsets of arbitrarily small d-diameter.Hence,it follows from[20,Lemma7.14,p.106-107]that there exists a w*-dense subset D of H such that F is single-valued at each point of D.Therefore, there exists a selection Q of①1such that Q is w-w*continuous on a w*-dense G$subset D of H.□
*.
Remark3.2(iii)of Lemma3.1remains valid if we substitute V*by L(/)
In the following lemma,the symbol C(Q)denotes the space of all the bounded and w*-continuous function on Q C ext Bx
*•
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No.4 D.X.Dai&B.T.Zheng：STABILITY OF A PAIR OF BANACH SPACES1167
Lemma3・3Suppose that X and Y are Banach spaces.Let s>0.Assume that/is a standard e-isometry from X into Y.If F*has the w*-PCP or Y is separable,then there exists
*such that there is a bounded linear operator T:Y C(Q) a w*-dense G&subset Q C ext Bx
such that\\Tf(x)—z(x)||<2e for all x E X.
Proof Because ext(Bx
*)is a Baire space in its relative w*-topology(see[12.p.217,line
*)such 17-19]),it follows from Lemma3.1that there is a w*-dense subset Q in ext(B%
that there is a w-w*continuous selection Q of①i on Q satisfying that for every x E X and x*E Q,the following inequality holds:
仗*)J(r)〉-<2e.
Let T:Y^oq(Q)be defined as in the proof of Lemma2.3.Then,T(y)€C(Q)and
\\Tf(x)—z(a;)||<2e,for all x E X.
□In the following parts of this article,the symbol C(Q)denotes the space of all the bounded and continuous function on Q C JC.
Corollary3.4Suppose that X=C(K)for a compact Hausdorff space K and Y*has the w*-PCP or Y is separable.Let>0.Assume that/is a standard e-isometry from X into Y.Then,there exists a dense subset Q of K and a bounded linear operator T:F—> C(Q)C08(Q)so that||T f(x)—z(x)||<2s for all x E X.
*)=:t€K}and{机：t€K}is a compact Proof It suffices to note that ext(B%
Baire space and a norming set of X.By applying Lemma3.1with H={6t：t E K}and Lemma3.3,we get0111conclusion immediately.□Remark3・5The pair(X,Y)of Corollary3.4is(q,2a：)-stable if z(X)is Q-complemented in C(Q),which gives a new positive answer to Problem1.1.
A topological space K is said to have the Stone-Cech property provided that for every dense subset Q U K,there is a dense subset S C Q,such that K is homeomorphic to the Stone-Cech compactification of S.For example,0N,the Stone-Cech compactification of N has the Stone-Cech property because every dense subset of it must contain the discrete space N.
As a consequence of Corollary3.4,we have
Theorem3・6Suppose that X=C(JC),where K has the Stone-Cech property.Assume that for every isometrie embedding J:X—>X,J(X)is a-complemented in X.If either Y* has the w*-PCP or Y is separable,then the pair(X,Y)is(q,2a)・stable.
Proof By Corollary3.4,there exists a dense subset Q of K and a bounded linear operator T:Y C(Q)C so that
||T/(a:)—迪)||<2e,for all x E X.
As K has the Stone-Cech property,there exists a dense subset S U Q such that K is homeomor­phic to the Stone-Cech compactification of S.Therefore,it is easy to see that C(S)=C(0S) and C(0S)=C(K),then we have C(S)=X.In fact,if丁：K—*/3S is an onto homeomor­phism,then T t:C(0S)—>C(JC)is an onto isometry such that T r(/)(A;)=/o r(k)for all
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1168ACTA MATHEMATICA SCIENTIA Vol.39Ser.B
f e C(0S)and k W K.Let R:C(Q)—>C(S)=X be the natural isometrical embedding. Hence,we have
\\RTf(x)—<2e,for all x W X.
As Ri(X)is ct-complemented in X,there is a projection P:X t Ri(X)with||P||<a such that
\\PRTf(x)一Ri(x)\\=\\PRTf(x)—PRi(x)\\<2ae,for all x W X.
Hence,for all z€X,we have||(Ri)-1PRTf(x)—x\\<2qe,where PRT\l^:L(/)—*X with||CR0)-iPkT||<a.□As C(0N)is1-complemented in C(0N)(see Proposition2.2and Remark2.4of[3]),by Theorem3.6we have
Corollary3.7Suppose that X=C(0N),where0N is the Stone・CQch compactification of N.If either Y*has the w*-PCP or Y is separable,then the pair(X,Y)is(1,2)-stable.
We do not know whether K=0N\N has the Stone-Cech property.Instead we will use the following proposition to show the stability of(C(0N\N),y)for any separable space Y,which also gives a new positive answer to Problem1.1.
Proposition3・8(i)If X is a A-separably injective Banach space,then the pair(X,Y) is(3A,6A)stable for every separable Banach space Y.
(ii)If the pair(X,Y)is(A,7)stable for every separable Banach space F,then X is a A-separably injective Banach space.
Proof(i)As Y is separable,it follows from Lemma3.3that for every w*-dense subset Q C ext(£x・)，there is a bounded linear operator T:V—>0oo(Q)such that
\\Tf(x)—i(x)||<2e,for all x E X.
Let Z=span{Tf(X)U i(X)}.As Y is separable,Z/d(X)is separable.As X is A-separably injective,it follows from Proposition2.1that i(X)is3入-complemented in Z.Therefore,there is a bounded linear operator P:Z2(X)with||P||<3A such that for all x E X,
||厂"TV仗)-x||=\\PTf(x)-i(x)||=||PT他)-Fi(x)||<6A e,
where i~x PT\^^:L(f)—>X satisfies that||i_1PT|£/(y)||<3A.
(ii)By Proposition2.1,it suffices to show that X is A-complemented in every superspace Y such that Y]X is separable.As Y/X is separable,card(X)=card(F).It follows from Lemma2.4that there is an e-isometry f:X Y such that Y=L(f)and/(0)=0.Hence, by the assumption,there is a projection P:Y X with||P||<A and we complete the proof.
□As C(0N\N)is1-separably injective(see,for instance,[2,p.202-203],[17]),we have
Corollary 3.9The pair(C(/?N\N),Y)(or equivalently^oo/cq.Y))is(3,6)stable for every separable Banach space Y.
Recall that a compact space K has height n if=0,where K'is the derived set of K and K(n+i)=(K®))‘.As C{K)is(2n—l)-separably injective for every K of height n(see, for instance,[2,p.203]),we have
Corollary3.10For every compact space K of height n,the pair(C(K),Y)is(6n—3, 12n—6)stable for every separable Banach space Y.
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No.4 D.X.Dai&B.T.Zheng：STABILITY OF A PAIR OF BANACH SPACES1169 Johnson-Oikhberg[15]showed that for every family of A-separably injective,spaces(刀Ei)Co
迈A
is2入2-separably injective.So,an immediate consequence of Proposition3.8is the following corollary.
Corollary3.11The pair((工Ei)Co,Y)is(6A2,12A2)stable for every separable Banach space Y,where{Ei}^人is a family of A-separably injective spaces.
Remark3・12There are many other examples of separably injective Banach spaces,such as the Johnson-Lindenstrauss spaces[14],Benyamini-space which is an M-space nonisomorphic to a C(K)-space[4],and the WCG nontrivial twisted sums of co(『)constructed by Argyros, Castillo,Granero,Jimenez and Moreno[1](see,for instance,[2]).
It is natural to ask the following
Problem3・13Do there exis t a const a nt7and a Lipschitz mapping S:L⑴—>X with Lipschiz constant depending only on X and F,such that for all x G X,
呵仗)-创<w?
To avoid the assumption of complementability,we obtain the following nonlinear version of stability for e-isometries.This also gives a positive answer to Problem3.13.
We say that p:X—>Z is a Lipschitz lifting for g:Z—>X if p is Lipschitz and qop=Idx-Let V be a metric space and let Z be a subset of Y.A Lipschitz map r:Y—>Z is called a Lipschitz retraction if it is the identity on Z.When such a Lipschitz retraction exists,we say that Z is a Lipschitz TetTact of A metric space Z is called an absolute Lipschitz retract if it is a Lipschitz retract of every metric space containing it.If the Lipschitz constant of the retraction is r,then we say that Z is an absolute r-Lipschitz retract.
Theorem3.14Let Z be an absolute r-Lipschitz retract.Assume that/:X—>F is an e-isometry.If p:X—>Z is a Lipschitz lifting for q:Z—>X,then there exists a Lipschitz mapping S:Y X with Lipschitz constant at most7=r||g|匕p||p|匕p such that
\\Sf(x)—x\\<2ye for all x E X.
Proof On one hand,by Lemma2.3,for every w*-dense subset Q C ext(B%
*),there is a bounded linear operator T:Y0oo(Q)such that
||T/(x)—迪)||<2s,for all x G X.
On the other hand,it follows from the assumption that there exists a Lipschitz mapping p: i(X)—*Z with the property that qop=where g:Z—>i(X).Because Z is an absolute r-Lipschitz retract(see Proposition 1.2of[5]),p can be extended to a Lipschitz mapping P:08(Q)t Z with\\p\\Li P<r\\p\\Li P-
Hence,for all x6X,we have
II刃V(Z)-两(z)||<2r\\p]\u P e.
Thus,
WqpTf^x)一qpi(x)\\<2r\\q\\Lip\\p\\Lip E.
Therefore,
\\Sf(x)一x\\<2r\\q\\Lip\\p\\L i p e,
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where S=i~1qpT.□It is well-known that co is an absolute2-Lipschitz retract(see[5]),but it remains open whether co(Z)is an absolute Lipschitz retract for every index set I.
Lemma3.15There is a2-Lipschitz retraction from^(/)onto cq⑴.Consequently, co(/)is an absolute2-Lipschitz retract.
Proof For every①=@)G put
d(£)=d(z,c°(/))=inf sup也
UCl,card(U)<oo
and define
(r(”))t=0,\xi\<d(x), (ki|一d(rr))sign(xi),\xi\>d(x).
Now,we show that for each x60g⑴、r(x)€co(Z).It suffices to prove that for every s>0, V={i E I：|(r(x))i|>e}is a finite set.On the contrary,if V is an infinite subset of I,then for each finite subset U C I,
sup(|xj|—d(x))>e.
i《U
Therefore,
…,inf n sup(|x t|-d(i))>£,
UC.I,card(U)<oo
which is a contradiction with£〉0.It is easy to check that r is a2-Lipschitz retraction from /00(/)onto co⑴.By the same proof of Example1.5in[5],then2is the best constant.It suffices to show that for each i E I,x=(xi),y=(?/i)E08⑴，the following inequality
|(讼)b—(r(2/))i|<2||x-训
holds.
(i)If\xi\<d(z)and\yi\<d(y),the inequality is clearly true.
(ii)If\xi\>d(z)and|眺|>d(y),then
l(^W)i一(咖)hl=|(ki|-d(x))sign(xi)—(|s|一d(?/))sign(s)|
<\xi-Vi\+|d@)sign(航)一d(9)sign(s)|.
If sign(血)=sign(?/i),then
|(r@)h一(旷(肌1<\xi一Vi\+|d(x)一d(“)|<\xi-y{\+\\x—y\\<2\\x—y\\.
If sign(叭)=-sign(2/i),then
|(r(x))j-(r(?/))j|<\xi一yi\+|d(x)+d(g)|<\xi一yi\+如+|眺|
=2\xi-yi\<2||①-训.
(iii)If\xi\>d(x)and|如<d(j/),then
|(r(x))i-(r(i/))i|=\(\xi\-d(x))sign(x£)|=\xi一d@)sign@)|
<也-Vi\+|d(2：)sign(xj)一yi\.
If sign(xi)=sign(s),then
|d(z)sign(窃)一饥|=|d(x)一|y』|<\\x一y\\.
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No.4 D.X.Dai&B.T.Zheng:STABILITY OF A PAIR OF BANACH SPACES1171
Hence,
|(r(z))i-(r(y))i|<2||z-y\\.
If sign(窃)=-sign(y<),then
|d@)sign(窃)一yi\=|d(x)+|训<|2：i|+闽=\n-如.
Hence,
|(r(a;))i-(r(y))i|<2问一训.
(iv)If也|<d(x)and\yi\>d(y),then it follows from(iii)that
-(r(2/))i|<2||①-训.
□If X=Z and p=q=Id、then we have the following results.
Example3・16As co(/)is an absolute2-Lipschitz retract for every index set/,by The­orem3.14and Lemma3.15,we show that for every e-isometry f:co(/)—>F,there is a 2-Lipschitz mapping S:Y—>co(I)such that for all x G co(I),
||S/(x)-x||<4s.
If K is a metric space and Cu(K)denotes the space of real-valued,bounded,uniformly continuous functions on K、equipped with the sup norm,then it is known that Cu(K)is an absolute20-Lipschitz retract(see[5]).In particular,C(K)is an absolute2-Lipschitz retract for every compact metric space K(see[16]).
Example3・17If K is a compact metric space,then for every「isometry f:C(K)—*V, there is a2-Lipschitz mapping S:F—>C(K)such that for all x E C(K),
||S/(i)-s||<4e.
Remark3・18If for each metric space F,and for every£-isometry f:X—*F,there is a constant7>0and a Lipschitz mapping S:Y X such that for all x e X,
lisy仗)—创<7£,
then X is an absolute Lipschitz retract.On the contrary,if X is an absolute Lipschitz retract, then for each Banach space Y,for every e-isometry f:X—>V,there are a constant7>0 depending only on X and a Lipschitz mapping S:Y—>X such that for all x E X,
\\Sf(x)-x||<ye.
Let(A/】，di)and(M2,〃2)be metric spaces and suppose that g:Mi t M2is any mapping. We define the modulus of continuity of g by
=sup{d2(g(o:),g(y)):di(rc,y)<t},t>0.
Proposition3.19Let X and Y be two Banach spaces.For every e-isometry/:X—>V, there is a continuous mapping S:Y—>X such that for all x e X,
||Sf(z)-创<w s(2e),
where g:Z qo(Q)t i(X)is a continuous mapping such that g\i(x)=/氏(X).
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1172ACTA MATHEMATICA SCIENTIA Vol.39Ser.B
Proof It follows from Proposition1.19of[5]that i(X)is an absolute retract.Thus,there is a continuous mapping g:^(Q)—>i(X)such that g(t)=t for all t€i(X).Hence,we have for all a:G X,
\\gTf(x)-gi(x)\\<w s(2e).
Therefore,for all x E X,
<3g(2£),
where S=i~x gT.□
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