Considering a metric space and its Lipschitz dual one defines the notion of M-ideal and HB-subspace of a metric space (with respect to its Lipschitz dual). One obtain some results analogous to these in the theory of M-ideal and HB-subspaces in a normed space. The results in the paper are based on an extension theorem of McShane [2], [3] and on a uniquenese theorem which is similar to one of R.R.Phelps [10], [11].
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Paper coordinates
C. Mustăţa, M-ideals in metric spaces, ”Babeş-Bolyai” University, Faculty of Math. and Physics, Research Seminars, Seminar on Mathematica Analysis, Preprint Nr.7 (1988), 67-74 (MR # 90b: 54019)
[1] Alfsen, D.M., Effross, E., Structure in real Banach spaces, Ann. of Math. 96(1972), 98-173.
[2] Czisper, J.,Geher, L., Extension of Funcitons satisfying a Lipschitz conditions, Acta Math. Acad.Sci. Hungar 6(1955), 213-220.
[3] Fakhoury, E., Selections lineaires associees au Theoreme de Hahn-Banach, J. of Funcitonal analysis 11 (1972), 436-452.
[4] Hennefeld, J., M – ideas, HB – subspaces and Compact Operators, Indiana Univ. Math. J. 28 (6) (1979), 927-934.
[5] Hennefeld, J., A note on M – ideals in B(X), Mat. Soc. 98 (1) (1980), 89-92.
[6] Holmes, R.B., Scrantor, B., Ward, J.D., Approximation from the space of compact operators and other M – ideals Duke Math. J. 42 (1975), 259-269.
[7] Holmes, R.B., Geometric Functional Analysis and its Applications, Springer – Verlag – New York – Heidelberg – Berlin, 1975.
[8] Shane, E.J., Extension of range of funcitons, Bull. Amer. Math. Soc. 40 (1934), 837-842.
[9] Johnson, J.A., Banach Spaces of Lipschitz Functions and vector – valued Lipschitz Functions, Trans. Amer. Math. Soc. 148(1970), 147-169.
[10] Mustata, C., Best Approximation and Uniwuq Extension of Lipschitz Functions, J. Approx. Theory 19 (3) (1977), 222-230.
[11] Phelps, R.R., Uniqueness of Hahn-Banach Extension and Unique Best Approximation, Trans. Amer. Math. Sec. 25 (1960), 238-255.
[12] Oja, E., On the uniquess of the norm preserving extension of linear functional in this Hahn-Banach Theorem, Proc. Acad. Science Esteonian SSR 33 (4) (1984), 422-433 (Russian).
Abstrect. Considering a netric space and its Lipschitz dual one defines the notion of M-ideal and HB-subspace of a metric space (with respect to its Lipschitz dual). One obtain some results analogous to those in the theory of MM-ideals and bar(BB)\overline{B B}-subspaces in a normed space. The rocults in the paper are based on an extension theorem of KCShane [2],[8][2],[8] and on a uniqueness theorem which is similar to one of R.R. Paelps [10], [22].
Introduction. The notion of M-ideal, introduced by E.M. Alfsen and 2. Erross [1], has many applications in functional analysis as, for example, to the problem of best approximation of continuous linear operators by compact operators (see [6]). Following E.M. Alfsen and I. Effross [1], an U-ideal in a normed space XX is a closed sibspace YY of XX whose annihilator Y^(_|_)Y^{\perp} admits a complement GG in X^(**)X^{*} such that ||g+h||=||g||+||h||,g in G,h inI^(_|_)\|g+h\|=\|g\|+\|h\|, g \in G, h \in I^{\perp}. S. Henneleld [4], [5] defined a more general notion t quad At \quad A closed subspace YY of a normed space XX is called an HBH B-subspace if Y^(_|_)Y^{\perp} has a complement GG 1n Z^(**)Z^{*} such that for every f inX^(**),||f|| >= ||h||,||f|| > ||g||f \in X^{*},\|f\| \geqslant\|h\|,\|f\|>\|g\|, whenever f=g+hf= g+h, with g in G,h inY^(_|_),h!=0g \in G, h \in Y^{\perp}, h \neq 0. It is immediate that overy mu-ide-\mu-i d e- al is an HB\mathbb{H B}-subspace and the converse is not true : The space K(i_(1),i)\mathbb{K}\left(i_{1}, i\right) of compact Iinear operators from l_(1)l_{1} to l_(1)l_{1} is an HB-subspace of A(l,n)A(l, n), the space of all continuous linear operators from ℓ_(1)\ell_{1} to ℓ_(1)\ell_{1}, which 1 s not an H-ideal ( see [4], [5]).
The aim of this paper is to transpose these notions to a guneral
metric space XX (without any linear structure) by appealing to the Iipschitz dual of XX, 1.e. a Banach space of Lipschitz functions on XX. The notions of M-ideal and HB-subspace cannot be eztonded automatically to a metric linear space by using its linear dual, for two reasons f first, this dual may be trivial, as is the case for the space I^(p)[0,1],0 < p < 1I^{p}[0,1], 0<p<1, and second, there is no norm on the dual of a motric linear space.
Results. Let ( X,dX, d ) be a metric space, x_(0)x_{0} a fixed point in XX and let XX be a subset of XX such that x_(0)in Yx_{0} \in Y. If XX is a metric linear space we take always x_(0)=0x_{0}=0. A function f:Y rarr Rf: Y \rightarrow R is called a lipschitz function on YY if there exists K >= 0K \geqslant 0 such that
(1) quad|f(x)-f(y)| <= Kd(x,y)\quad|f(x)-f(y)| \leqslant K d(x, y)
for all x,y in Xx, y \in X :
Denote by Lip _(0){ }_{0} the following set
(2) Lip_(0)I={f&IrarrR,f:}\operatorname{Lip}_{0} \mathrm{I}=\left\{\mathrm{f} \& \mathrm{I} \rightarrow \mathbf{R}, \mathrm{f}\right. is a Iipschitz function on {:I,f(x_(0))=0}\left.\mathrm{I}, \mathrm{f}\left(\mathrm{x}_{0}\right)=0\right\}.
Equiped with the usual operations of addition and multiplication by scalars, Lifo YY is a linear space and the application ||||_(I^(')):}\left\|\|_{I^{\prime}}\right. ! Lip _("o ")X rarr_{\text {o }} X \rightarrow R , defined by coscri beanon & II
(3) ||f||_(Z)=s u p{|f(x)-f(y)|//d(x,y)quad,quad x,y in I,x!=y}\|f\|_{Z}=\sup \{|f(x)-f(y)| / d(x, y) \quad, \quad x, y \in I, x \neq y\},
is a nord on Lip _(0)I{ }_{0} I. It is easily seen that ||f||_(Y)\|f\|_{Y} is the shallest of the numbers x >= 0x \geqslant 0 for which the inequality (1) holds. The space ( Lip_(0)^(gamma),||_(1)||_(Y)L i p_{0}{ }^{\gamma},\left\|_{1}\right\|_{\mathbf{Y}} ) is a Banach space (even a dual Banach space , see [9]) and we call it the Lipschitz dual of YY.
For X-=XX \equiv X the space Lip_(0)XL i p_{0} X and the norm ||||_(X):}\left\|\|_{X}\right. are delined similarly.
In the following, one supposes always that the subset YY of XX contains x_(0)x_{0} o the fixed element of xx.
The following Hahn-Banach type extension theorem for Lipachitz functions was proved by Uc Shane [8]. (see also [2] ) :
TREOREM 1. Let (X,d)(X, d) be a metric space, x_(0)x_{0} a fixed point in XX and let YY be a subset of XX such that x_(0)in Ix_{0} \in I. Then every function f in Lp_(0)If \in L p_{0} I has a norm-preserving extension in Ldp_(0)X,1.6L d p_{0} X, 1.6. there exists Z in Itg_(0)XZ \in I \operatorname{tg}_{0} X such that F|_(Y)=f\left.F\right|_{Y}=f and ||||_(X)=||P||_(X)\left\|\left\|_{X}=\right\| P\right\|_{X}.
In fact, He Shane [8], proved this theorem in the case of the space Lip YY and Lip XX of all Lipschitz functions on YY and XX, respectively, but the above formulation is more appropriate for our needs. We shall call sometimes briefly any function FF, as given in Theorem 1, an extension of ff.
In general, the extension of a function f in Kp_(0)Yf \in K p_{0} Y to XX is not unique. The functions
{:[(4)I_(I)(x)=i n f{f(y)+||f||_(I)d(x,y):y in I}],[I_(2)(x)=s u p{f(y)-||f||_(I)d(x,y):y in I}]:}\begin{align*}
& I_{I}(x)=\inf \left\{f(y)+\|f\|_{I} d(x, y): y \in I\right\} \tag{4}\\
& I_{2}(x)=\sup \left\{f(y)-\|f\|_{I} d(x, y): y \in I\right\}
\end{align*}
are two extensions of ff and they are extremal elements of the conver set R(f;Y)\mathbb{R}(f ; Y) of all extensions of ff. Frery extension II of ff verifies the inequalities :
{:(5)F_(2)(x) <= P(x) <= F_(1)(x)quad","quad x in X:}\begin{equation*}
F_{2}(x) \leqslant P(x) \leqslant F_{1}(x) \quad, \quad x \in X \tag{5}
\end{equation*}
Therefore, the function f in Lip_(0)If \in L i p_{0} I has a unique extension in Lip I_(0)I_{0} if and only if F_(1)=F_(2)F_{1}=F_{2}.
DEFIMITION 1. The subset YY of XX is said to have property ( UU ) if every function f in Lip_(0)If \in L i p_{0} I has a unique extension in Lip I_(0)I_{0}.
Hecessary and surficient conditions in order that a subset YY of I have property (U) and relations of this property with the problem of best approximation in Lip _(0){ }_{0} by slements in x^(_|_)^(_|_){x^{\perp}}^{\perp} are given犃 [10].
For X sube XX \subseteq X donote by X^(_|_)X^{\perp} its annihilator in Lig_(0)X\operatorname{Lig}_{0} X, i.e.
(6) quadI^(_|_)={F in Lp_(0)I, quad I|_(I)=0}\quad I^{\perp}=\left\{F \in L p_{0} I,\left.\quad I\right|_{I}=0\right\}.
Obviously, X^(_|_)X^{\perp} is a closed subspace of It p_(0)X\mathrm{p}_{0} X.
DEFINITION 2. A subset VV of a normed space ZZ is called proyminal if for every z in Zz \in Z there exists grad_(0)in V\nabla_{0} \in V such that
(7) quad||z-v_(0)||=d(z, hat(V))\quad\left\|z-v_{0}\right\|=d(z, \hat{V})
Where d(z,V)-=i n f{|z-grad|,grad in grad}d(z, V) \equiv \inf \{|z-\nabla|, \nabla \in \nabla\} denotes the distance from zz to VV. An element V_(0)V_{0} satisfying (7) is called a bost approximation element of zz by elements in VV. If every z in Zz \in Z has a unique best approximation element in VV then the set VV is called Chebysherian .
R.R. Pholps [11] obtained some results concerning the relations between the property (U) and the unicity of best approximation in the dual of a normed space. As was shown in [10] similar results hold also in the Iipschitz case :
THSOREM 2. ([10]) A s joet YY of a netric spacs XX has proporty (V) if and oniy if its annihilator I^(_|_)I^{\perp} is Chebyshorian subonce of Lipo ^(2){ }^{2}.
The proof of this theorem is based on the following lemma, which vill be used in the sequel :
LEVII 1. ([10]). Let XX be a metric space, I a subset of XX and Y^(_|_)Y^{\perp} the annihilator of YY in Lip_(0)X\operatorname{Lip}_{0} X. If T inLip_(0)XT \in \operatorname{Lip}_{0} X then d(F,Y^(_|_))=^('')I^('')Id\left(F, Y^{\perp}\right)= { }^{\prime \prime} I^{\prime \prime} I and an alement delta_(0)inI^(_|_)\delta_{0} \in I^{\perp}. is a best aporoximation element for yy by elements in I^(_|_)I^{\perp} if and only if g_(0)=F-F_(0)g_{0}=F-F_{0}, where I_(0)I_{0} is a norm preserving extension of F|_(I)\left.F\right|_{I} to XX.
The proporty ( UU ) can be charactorized also in terms of some decompositions of the Lipschitz dual Lip _(0)I{ }_{0} \mathrm{I} of X . To give this charactorization we noed first sone definitions and notations.
Lot It Id p_(0)Irarrp_{0} \mathrm{I} \rightarrow Hip p_(0)Ip_{0} \mathrm{I} denote the restriction operator, defined by :
(8) quad x(T)=F|_(I),F in Lipp_(0)X\quad x(T)=\left.F\right|_{I}, F \in \operatorname{Lip} p_{0} X,
and iev e: Liy _(0)I rarr int(:}{ }_{0} I \rightarrow \int\left(\right. IIp {:_(0)X)\left._{0} X\right) denote the extension orerator, definod by :
(9)
e(f)=E(f;I)quad,quad f in LLp^(˙)_(0)X^(˙).e(f)=E(f ; I) \quad, \quad f \in L L \dot{p}_{0} \dot{X} .
Whore I(f;Y)\mathbb{I}(f ; Y) denotes the set of all nom preserving extensions of I to XX. Lot w:I_(ip_(0))I rarr int(I_(ip_(0))I)w: I_{i p_{0}} I \rightarrow \int\left(I_{i p_{0}} I\right) be the composition of the operators rr and e, i.e.
(10)
w-=e@x.w \equiv e \circ x .
Then, for F inLip_(0)IF \in \operatorname{Lip}_{0} I, we have w(F)=>e(r(F))=S(I(F);Y)w(F) \Rightarrow e(r(F))=\mathbb{S}(I(F) ; Y) and I(G)=I(F),quad||G||_(I)=||I(F)||_(I)I(G)=I(F), \quad\|G\|_{I}=\|I(F)\|_{I}, for all G in E(I(F);Y)G \in E(I(F) ; Y) 。
In goneral, the operator ww is nulti-valued and w(F)w(F) is a conver subset of the ball of radius ||I(F)||_(Y)\|I(F)\|_{Y} and center 0 in Lip_(0)XL i p_{0} X.
Te can now state the theorem of characterization of property (U)(U) :
THEOPI 3. If II is a subset of a netric space XX then the following enserions are equivalent : 2^(@)2^{\circ} I has sroperty (V) ; 2^(@)2^{\circ} Ivery function F inL_(j)p_(0)^("can be uniquely written in the ")F \in L_{j} p_{0}{ }^{\text {can be uniquely written in the }}
form
(11) quad F=E+g,E in w(P),g inI^(_|_)\quad F=E+g, E \in w(P), g \in I^{\perp},
and ||I||_(X) > ||I||_(X)\|I\|_{X}>\|I\|_{X}, vinenever g!=0g \neq 0; 3^(@)quadG={H inLip_(0)X,||H||_(Z)=||r(H)||_(Y)}3^{\circ} \quad \mathscr{G}=\left\{H \in \operatorname{Lip}_{0} X,\|H\|_{Z}=\|r(H)\|_{Y}\right\} is the oniy subset of
Lipo XX such that every I in Lip_(0)XI \in L i p_{0} X can be uniquely written in the 응펴 f=H+g,H in G,g inX^(_|_)f=H+g, H \in G, g \in X^{\perp} and ||F||_(X) > ||H||_(X)\|F\|_{X}>\|H\|_{X} if g!=0g \neq 0.
Proof. 1^(0)=>2^(0)1^{0} \Rightarrow 2^{0}. If the set YY has property (U) then the oxtension operator 9 , delined by (9), is single-valued and so is the operator ww defined by (10). For F in Ii_(0)XF \in I i_{0} X the function D(F) in\in Lip _(X)X{ }_{X} X is the ouly norm preserving extension of I(F)I(F) to XX, i.e. w(B)|_(Y)=I(F)\left.w(B)\right|_{Y}=I(F) and ||w(F)||_(X)=||I(F)||_(Y)\|w(F)\|_{X}=\|I(F)\|_{Y}. It follows that g==F-w(F)inY^(_|_)g= =F-w(F) \in Y^{\perp} and F=w(F)+gF=w(F)+g is the unique decomposition of YY With tilde(sigma)inI^(_|_)\tilde{\sigma} \in I^{\perp}. By the definition (3) of Lipschitz norm we have || vec(F)||_(X) >= ||r(F)||_(X)\|\vec{F}\|_{X} \geqslant\|r(F)\|_{X}. The equality ||I||_(X)=||I(F)||_(Y)\|I\|_{X}=\|I(F)\|_{Y} implies that FF is also a norm preserving extension of I(F)I(F) and, by the unicity of the axtension it follows F=w(F)F=w(F), so that B=F-w(F)=0B=F-w(F)=0. Hence ||I||_(X) > ||I(F)||_(Y)\|I\|_{X}>\|I(F)\|_{Y} if g in0g \in 0. 2^(@)=>3^(@)2^{\circ} \Rightarrow 3^{\circ}. Let F inF \in Lip _(0)X{ }_{0} X and let F=H+gF=H+g the decomposition of FF given in 2^(@)2^{\circ}. As H in W(F)=O(I(R))H \in W(F)=O(I(R)) it follows that I(F)-=r(H)I(F) \equiv r(H) and ||H||_(X)=||r(F)||_(Y)\|H\|_{X}=\|r(F)\|_{Y}, i.e. H invarphi_(j)H \in \varphi_{j}. The condition ||F||_(X) > ||E||_(X)\|F\|_{X} >\|E\|_{X}, for g!=0g \neq 0, follows from the similar condition from 2^(0)2^{0}. 3^(@)Longrightarrow2^(@)3^{\circ} \Longrightarrow 2^{\circ}. Let F in Ip_(0)XF \in I p_{0} X and let Z=H+g,H inG,g inI^(_|_)Z=H+g, H \in \mathscr{G}, g \in I^{\perp} be the unique decomposition of RR given in 3^(@)3^{\circ}. Then P-g=BP-g=B, I(H)-=I(F)I(H) \equiv I(F) and, By Lomma 1, ||I-g||_(X)=||E||_(X)=||I(H)||_(Y)=||r(F)||_(Y)==a^(˙)(B,Y^(_|_))\|I-g\|_{X}=\|E\|_{X}=\|I(H)\|_{Y}=\|r(F)\|_{Y}= =\dot{a}\left(B, Y^{\perp}\right), which shows that gg is an element of best approximation for II by elements in I^(_|_)I^{\perp}. If g_(1)g_{1} is an other element of best approximation for TT by elements in Y^(_|_)Y^{\perp} then, appealing again to Lema 1, there exists H_(1)in o(x(F))H_{1} \in o(x(F)) such that B_(1)=F-B_(1)B_{1}=F-B_{1} and ||H_(1)||_(X)=||F-g_(1)||_(X)=d(F,Y^(_|_))=||r(F)||_(Y)=||r(H_(2))||_(Y)\left\|H_{1}\right\|_{X}=\left\|F-g_{1}\right\|_{X}=d\left(F, Y^{\perp}\right)=\|r(F)\|_{Y}=\left\|r\left(H_{2}\right)\right\|_{Y}, which shows that H_(1)inGH_{1} \in \mathscr{G}. Taking into acsount the unicity assumption in 3^(@)3^{\circ}, it follows G-=G_(1)G \equiv G_{1}. Therefore, Y^(_|_)Y^{\perp} is a Chebyshovian subspace of Iido XX and, by Theorem 2, the set Y has property (U).
Theorem 3 is completely proved.
DEFINITION 3. A subset II of a metric space II is called an lifideel If its annihilator I^(1)I^{1} has a complement GG in Lip I^(I)I^{I} such that ||F||_(X)=||G||_(X)+||E||_(X)\|F\|_{X}=\|G\|_{X}+\|E\|_{X}, whonever I=G+EI=G+E, with G inGG \in \mathcal{G} and E inY^(+),{0}E \in Y^{+},\{0\}. The subset YY is said to have property (HB) if Y^(_|_)Y^{\perp} has a comple mentary subspace xi\xi of Lip _(X){ }_{\mathrm{X}} such that ||F||_(X) >= ||E||_(X)\|F\|_{\mathrm{X}} \geqslant\|E\|_{\mathrm{X}}, ||z||_(X) > ||G||_(X)\|z\|_{\mathrm{X}}>\|G\|_{\mathrm{X}}, Whenever F=G+EF=G+E, with G inYG \in \mathscr{Y} and H inSigma^(i),{0}H \in \Sigma^{i},\{0\}, for every function F in Ip^(˙)_(0)IF \in I \dot{p}_{0} I.
THEOREM 4. If a subset YY of a metric space XX has the property (BB) then YY has the property (U) .
Proof. Suppose that YY has the property (HB) and has not the properfy (J). Then there exists a function f inf \in Lip _(0)I{ }_{0} I having two distinct extensions I_(1),I_(2)I_{1}, I_{2} Lip _(0)I{ }_{0} \mathfrak{I} and the subspace I^(_|_)I^{\perp} has a conplementary subspace GG in Lip ^(I){ }^{I} such that the condition in Definition 3 is fulfilled, implying F_(i)=G_(i)+E_(i)F_{i}=G_{i}+E_{i}, with G_(i)inHG_{i} \in \mathcal{H} and E_(i)inY^(_|_)E_{i} \in Y^{\perp} for 1=1,21=1,2. As F_(1)-F_(2)inI^(_|_)F_{1}-F_{2} \in I^{\perp} it follows that G_(1)-G_(2)=F_(1)-I_(2)G_{1}-G_{2}=F_{1}-I_{2} -- (E_(1)-E_(2))inI^(_|_)\left(E_{1}-E_{2}\right) \in I^{\perp}, hence G_(1)=G_(2)=G(zeta nnI^(_|_)={0}:}G_{1}=G_{2}=G\left(\zeta \cap I^{\perp}=\{0\}\right., as xi\xi and I^(_|_)I^{\perp} are complementary subspaces of Ifipo XX ). Therefore F_(1)=G+H_(1)F_{1}=G+H_{1} and F_(2)=G+E_(2)F_{2}=G+E_{2}. Now, if E_(2)!=0E_{2} \neq 0 then ||F_(1)||_(Sigma) > ||G||_(I)\left\|F_{1}\right\|_{\Sigma}>\|G\|_{I} so that ||f||_(I)==||F_(1)||_(Sigma) > ||G||_(Sigma) >= ||r(G)||_(I)=||I||_(Y)\|f\|_{I}= =\left\|F_{1}\right\|_{\Sigma}>\|G\|_{\Sigma} \geqslant\|r(G)\|_{I}=\|I\|_{Y}. If H_(1)-=0H_{1} \equiv 0 then H_(2)!=H_(1)=0H_{2} \neq H_{1}=0, hence G=F_(1)G=F_{1} and, by Desinition 3 , the equality F_(1)=F_(2)+H_(2)F_{1}=F_{2}+H_{2} implies ||P||_(I)=||F_(1)||_(X) > ||F_(2)||_(X)=||S||_(I)\|P\|_{I}=\left\|F_{1}\right\|_{X}>\left\|F_{2}\right\|_{X}=\|S\|_{I}. The obtained contradictions shows that the set II cannot have the propetty (HB). Theorem 4 is completejy proved.
THEOREIX 5. If the subset YY of XX has the property ( HBH B ) and F in Lipp_(0),FF \in \operatorname{Lip} p_{0}, F of 0 , then F inGF \in \mathscr{G} if and only if ||r(F)||_(I)=||F||_(X)\|r(F)\|_{I}=\|F\|_{X}, There zeta\zeta is the complementary subspace of r^(_|_)r^{\perp} given in Definition 3 .
Proof. Let F in xi,F!in0F \in \xi, F \notin 0, and let GG be a norm preserving extension of x(F)x(F) to I_(". ")I_{\text {. }} Let G=G_(1)+H_(1),G_(1)in xi,H_(2)inX^(_|_)G=G_{1}+H_{1}, G_{1} \in \xi, H_{2} \in X^{\perp} be the
decomposition of GG given by Definition 3. Supposing E_(1)!=0E_{1} \neq 0, one obtains the contradiction ||G||_(I) > ||G_(1)||_(I) >= ||I(G_(1))||_(I)=||I(I)||_(I)==||G_(1)||_(2)\|G\|_{I}>\left\|G_{1}\right\|_{I} \geqslant\left\|I\left(G_{1}\right)\right\|_{I}=\|I(I)\|_{I}= =\left\|G_{1}\right\|_{2}. Therefore H_(1)=0H_{1}=0 and G=G_(1)inzeta_(1)G=G_{1} \in \zeta_{1}. As zeta_(1)\zeta_{1} is a subspace of iip _(0){ }_{0} it follows F-G in GF-G \in G. But F-GF-G is in I-I- too, bocauso r(F)=r(G)r(F)=r(G), so that R-G in xi nnX^(_|_)={0}R-G \in \xi \cap X^{\perp}=\{0\}, i.s. F=G inGF=G \in \mathcal{G}.
Conversely, suppose that F in Kp_(0) tilde(x),F!=0F \in K p_{0} \tilde{x}, F \neq 0, is such that ||x(F)||_(x)=||P||_(X)( > 0)\|x(F)\|_{x} =\|P\|_{X}(>0). Lot I=G+HI=G+H with G in xi,H inI^(L)G \in \xi, H \in I^{L}. If E!=0E \neq 0 then ||P||_(I) > ||G||_(X)\|P\|_{I}>\|G\|_{X} and the equality r(I)=r(G)r(I)=r(G) gives the contradic tion ||x(F)||_(Z)=||F||_(Z) > ||G||_(Z)=||x(G)||_(Y)=||r(F)||_(Y)\|x(F)\|_{Z}=\|F\|_{Z}>\|G\|_{Z}=\|x(G)\|_{Y}=\|r(F)\|_{Y}, which shows that H=0H=0 and F=G invarphi_(S)F=G \in \varphi_{S}.
THFOPN 6. If the subset YY of a metric snnce XX has the property (HB) then the subsoace zeta\zeta (given in Definition 3) is dropetrically isomorphic to the space Lip _(0)^(I){ }_{0}{ }^{I}.
Proof. By Theorem 4, tae subset YY has the property (U) , so that the restriction r_(1)r_{1} of the restriction operator rr to xi_(j)\xi_{j} is single-valued and linear. By Theorem 5, ||r_(1)(G)||_(j)=||G||_(X)\left\|r_{1}(G)\right\|_{j}=\|G\|_{X}, for all G inzeta_(xi)G \in \zeta_{\xi}, showing that T_(1)T_{1} is an loometry.
THEORSU 7. If the subset XX of a motric space XX pes the proper ty (HB) , tien the extension operator *∣\cdot \mid ip _(0)Yrarr_{0} \mathrm{Y} \rightarrow Lip _(c)X_{c} \mathrm{X} de linear.
Proof. Let tilde(:)\tilde{:} : Lip Irarrzeta_(j)\mathrm{I} \rightarrow \zeta_{j} be the inverse of the restiliction operator r_(1)=r|_(e)r_{1}=\left.r\right|_{e} & longrightarrow\longrightarrow Lip _(0)I{ }_{0} \mathrm{I} which, by Theorem 6, is an isoattrical isomorpicism between zeta\zeta and Lipo YY. Then is linear and e=j@ tilde(0)e=j \circ \tilde{0} where j:zeta rarrj: \zeta \rightarrow Lip jxj x, denotes the imbedding operator of zeta\zeta into Ling xx.
RAITPRINCES
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Eipschite Functions, J. Appror. Theory 29 (3) (1977) , 222-230 。