M-ideals in metric spaces

Costica Mustata
(original), Approximation Theory, paper

Abstract


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].

Authors

Costica Mustata
“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)

PDF

https://ictp.acad.ro/wp-content/uploads/2025/07/1988-Mustata-Seminar-UBB-Idelas-in-metric-spaces.pdf

About this paper

Journal
Publisher Name
DOI
Print ISSN
Online ISSN

MR # 90b: 54019

google scholar link

[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).

Paper (preprint) in HTML form

1988-Mustata-Seminar-UBB-Idelas-in-metric-spaces
 "BABES-BOLYAI" UNIVERSITY  Faculty of Mathematics and Physics  Resoarch Seminars  Seminar on Mathematical Analyeis  Preprint Nr.7, 2988, pp. 65-74. 

M - IDEALS IN MRTRIC SPACES

Costică Mustăta

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 M M MMM-ideals and B B ― B B ¯ bar(BB)\overline{B B}BB―-subspaces in a normed space. The rocults in the paper are based on an extension theorem of KCShane [ 2 ] , [ 8 ] [ 2 ] , [ 8 ] [2],[8][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 X X XXX is a closed sibspace Y Y YYY of X X XXX whose annihilator Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ admits a complement G G GGG in X ∗ X ∗ X^(**)X^{*}X∗ such that ‖ g + h ‖ = ‖ g ‖ + ‖ h ‖ , g ∈ G , h ∈ I ⊥ ‖ g + h ‖ = ‖ g ‖ + ‖ h ‖ , g ∈ G , h ∈ I ⊥ ||g+h||=||g||+||h||,g in G,h inI^(_|_)\|g+h\|=\|g\|+\|h\|, g \in G, h \in I^{\perp}‖g+h‖=‖g‖+‖h‖,g∈G,h∈I⊥. S. Henneleld [4], [5] defined a more general notion t A t A t quad At \quad AtA closed subspace Y Y YYY of a normed space X X XXX is called an H B H B HBH BHB-subspace if Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ has a complement G G GGG 1n Z ∗ Z ∗ Z^(**)Z^{*}Z∗ such that for every f ∈ X ∗ , ‖ f ‖ ⩾ ‖ h ‖ , ‖ f ‖ > ‖ g ‖ f ∈ X ∗ , ‖ f ‖ ⩾ ‖ h ‖ , ‖ f ‖ > ‖ g ‖ f inX^(**),||f|| >= ||h||,||f|| > ||g||f \in X^{*},\|f\| \geqslant\|h\|,\|f\|>\|g\|f∈X∗,‖f‖⩾‖h‖,‖f‖>‖g‖, whenever f = g + h f = g + h f=g+hf= g+hf=g+h, with g ∈ G , h ∈ Y ⊥ , h ≠ 0 g ∈ G , h ∈ Y ⊥ , h ≠ 0 g in G,h inY^(_|_),h!=0g \in G, h \in Y^{\perp}, h \neq 0g∈G,h∈Y⊥,h≠0. It is immediate that overy μ − i d e − μ − i d e − mu-ide-\mu-i d e-μ−ide− al is an H B H B HB\mathbb{H B}HB-subspace and the converse is not true : The space K ( i 1 , i ) K i 1 , i K(i_(1),i)\mathbb{K}\left(i_{1}, i\right)K(i1,i) of compact Iinear operators from l 1 l 1 l_(1)l_{1}l1 to l 1 l 1 l_(1)l_{1}l1 is an HB-subspace of A ( l , n ) A ( l , n ) A(l,n)A(l, n)A(l,n), the space of all continuous linear operators from ℓ 1 ℓ 1 ℓ_(1)\ell_{1}ℓ1 to ℓ 1 ℓ 1 ℓ_(1)\ell_{1}ℓ1, 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 X X XXX (without any linear structure) by appealing to the Iipschitz dual of X X XXX, 1.e. a Banach space of Lipschitz functions on X X XXX. 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 < 1 I p [ 0 , 1 ] , 0 < p < 1 I^(p)[0,1],0 < p < 1I^{p}[0,1], 0<p<1Ip[0,1],0<p<1, and second, there is no norm on the dual of a motric linear space.
Results. Let ( X , d X , d X,dX, dX,d ) be a metric space, x 0 x 0 x_(0)x_{0}x0 a fixed point in X X XXX and let X X XXX be a subset of X X XXX such that x 0 ∈ Y x 0 ∈ Y x_(0)in Yx_{0} \in Yx0∈Y. If X X XXX is a metric linear space we take always x 0 = 0 x 0 = 0 x_(0)=0x_{0}=0x0=0. A function f : Y → R f : Y → R f:Y rarr Rf: Y \rightarrow Rf:Y→R is called a lipschitz function on Y Y YYY if there exists K ⩾ 0 K ⩾ 0 K >= 0K \geqslant 0K⩾0 such that
(1) | f ( x ) − f ( y ) | ⩽ K d ( x , y ) | f ( x ) − f ( y ) | ⩽ K d ( x , y ) quad|f(x)-f(y)| <= Kd(x,y)\quad|f(x)-f(y)| \leqslant K d(x, y)|f(x)−f(y)|⩽Kd(x,y)
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X :
Denote by Lip 0 0 _(0){ }_{0}0 the following set
(2) Lip 0 I = { f & I → R , f Lip 0 I = f & I → R , f Lip_(0)I={f&IrarrR,f:}\operatorname{Lip}_{0} \mathrm{I}=\left\{\mathrm{f} \& \mathrm{I} \rightarrow \mathbf{R}, \mathrm{f}\right.Lip0I={f&I→R,f is a Iipschitz function on I , f ( x 0 ) = 0 } I , f x 0 = 0 {:I,f(x_(0))=0}\left.\mathrm{I}, \mathrm{f}\left(\mathrm{x}_{0}\right)=0\right\}I,f(x0)=0}.
Equiped with the usual operations of addition and multiplication by scalars, Lifo Y Y YYY is a linear space and the application ‖ ‖ I ′ ‖ I ′ ||||_(I^(')):}\left\|\|_{I^{\prime}}\right.‖‖I′ ! Lip o X → o  X → _("o ")X rarr_{\text {o }} X \rightarrowo X→ R , defined by coscri beanon & II
(3) ‖ f ‖ Z = sup { | f ( x ) − f ( y ) | / d ( x , y ) , x , y ∈ I , x ≠ y } ‖ f ‖ Z = sup { | f ( x ) − f ( y ) | / d ( x , y ) , x , y ∈ I , x ≠ y } ||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\}‖f‖Z=sup{|f(x)−f(y)|/d(x,y),x,y∈I,x≠y},
is a nord on Lip 0 I 0 I _(0)I{ }_{0} I0I. It is easily seen that ‖ f ‖ Y ‖ f ‖ Y ||f||_(Y)\|f\|_{Y}‖f‖Y is the shallest of the numbers x ⩾ 0 x ⩾ 0 x >= 0x \geqslant 0x⩾0 for which the inequality (1) holds. The space ( L i p 0 γ , ‖ 1 ‖ Y L i p 0 γ , 1 Y Lip_(0)^(gamma),||_(1)||_(Y)L i p_{0}{ }^{\gamma},\left\|_{1}\right\|_{\mathbf{Y}}Lip0γ,‖1‖Y ) is a Banach space (even a dual Banach space , see [9]) and we call it the Lipschitz dual of Y Y YYY.
For X ≡ X X ≡ X X-=XX \equiv XX≡X the space L i p 0 X L i p 0 X Lip_(0)XL i p_{0} XLip0X and the norm ‖ ‖ X ‖ X ||||_(X):}\left\|\|_{X}\right.‖‖X are delined similarly.
In the following, one supposes always that the subset Y Y YYY of X X XXX contains x 0 x 0 x_(0)x_{0}x0 o the fixed element of x x xxx.
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 ) (X,d)(X, d)(X,d) be a metric space, x 0 x 0 x_(0)x_{0}x0 a fixed point in X X XXX and let Y Y YYY be a subset of X X XXX such that x 0 ∈ I x 0 ∈ I x_(0)in Ix_{0} \in Ix0∈I. Then every function f ∈ L p 0 I f ∈ L p 0 I f in Lp_(0)If \in L p_{0} If∈Lp0I has a norm-preserving extension in L d p 0 X , 1.6 L d p 0 X , 1.6 Ldp_(0)X,1.6L d p_{0} X, 1.6Ldp0X,1.6. there exists Z ∈ I tg 0 X Z ∈ I tg 0 X Z in Itg_(0)XZ \in I \operatorname{tg}_{0} XZ∈Itg0X such that F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f and ‖ ‖ X = ‖ P ‖ X X = P X ||||_(X)=||P||_(X)\left\|\left\|_{X}=\right\| P\right\|_{X}‖‖X=‖P‖X.
In fact, He Shane [8], proved this theorem in the case of the space Lip Y Y YYY and Lip X X XXX of all Lipschitz functions on Y Y YYY and X X XXX, respectively, but the above formulation is more appropriate for our needs. We shall call sometimes briefly any function F F FFF, as given in Theorem 1, an extension of f f fff.
In general, the extension of a function f ∈ K p 0 Y f ∈ K p 0 Y f in Kp_(0)Yf \in K p_{0} Yf∈Kp0Y to X X XXX is not unique. The functions
(4) I I ( x ) = inf { f ( y ) + ‖ f ‖ I d ( x , y ) : y ∈ I } I 2 ( x ) = sup { f ( y ) − ‖ f ‖ I d ( x , y ) : y ∈ I } (4) I I ( x ) = inf f ( y ) + ‖ f ‖ I d ( x , y ) : y ∈ I I 2 ( x ) = sup f ( y ) − ‖ f ‖ I d ( x , y ) : y ∈ I {:[(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*}(4)II(x)=inf{f(y)+‖f‖Id(x,y):y∈I}I2(x)=sup{f(y)−‖f‖Id(x,y):y∈I}
are two extensions of f f fff and they are extremal elements of the conver set R ( f ; Y ) R ( f ; Y ) R(f;Y)\mathbb{R}(f ; Y)R(f;Y) of all extensions of f f fff. Frery extension I I III of f f fff verifies the inequalities :
(5) F 2 ( x ) ⩽ P ( x ) ⩽ F 1 ( x ) , x ∈ X (5) F 2 ( x ) ⩽ P ( x ) ⩽ F 1 ( x ) , x ∈ X {:(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*}(5)F2(x)⩽P(x)⩽F1(x),x∈X
Therefore, the function f ∈ L i p 0 I f ∈ L i p 0 I f in Lip_(0)If \in L i p_{0} If∈Lip0I has a unique extension in Lip I 0 I 0 I_(0)I_{0}I0 if and only if F 1 = F 2 F 1 = F 2 F_(1)=F_(2)F_{1}=F_{2}F1=F2.
DEFIMITION 1. The subset Y Y YYY of X X XXX is said to have property ( U U UUU ) if every function f ∈ L i p 0 I f ∈ L i p 0 I f in Lip_(0)If \in L i p_{0} If∈Lip0I has a unique extension in Lip I 0 I 0 I_(0)I_{0}I0.
Hecessary and surficient conditions in order that a subset Y Y YYY of I have property (U) and relations of this property with the problem of best approximation in Lip 0 0 _(0){ }_{0}0 by slements in x ⊥ ⊥ x ⊥ ⊥ x^(_|_)^(_|_){x^{\perp}}^{\perp}x⊥⊥ are given犃 [10].
For X ⊆ X X ⊆ X X sube XX \subseteq XX⊆X donote by X ⊥ X ⊥ X^(_|_)X^{\perp}X⊥ its annihilator in Lig 0 X Lig 0 X Lig_(0)X\operatorname{Lig}_{0} XLig0X, i.e.
(6) I ⊥ = { F ∈ L p 0 I , I | I = 0 } I ⊥ = F ∈ L p 0 I , I I = 0 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\}I⊥={F∈Lp0I,I|I=0}.
Obviously, X ⊥ X ⊥ X^(_|_)X^{\perp}X⊥ is a closed subspace of It p 0 X p 0 X p_(0)X\mathrm{p}_{0} Xp0X.
DEFINITION 2. A subset V V VVV of a normed space Z Z ZZZ is called proyminal if for every z ∈ Z z ∈ Z z in Zz \in Zz∈Z there exists ∇ 0 ∈ V ∇ 0 ∈ V grad_(0)in V\nabla_{0} \in V∇0∈V such that
(7) ‖ z − v 0 ‖ = d ( z , V ^ ) z − v 0 = d ( z , V ^ ) quad||z-v_(0)||=d(z, hat(V))\quad\left\|z-v_{0}\right\|=d(z, \hat{V})‖z−v0‖=d(z,V^)
Where d ( z , V ) ≡ inf { | z − ∇ | , ∇ ∈ ∇ } d ( z , V ) ≡ inf { | z − ∇ | , ∇ ∈ ∇ } d(z,V)-=i n f{|z-grad|,grad in grad}d(z, V) \equiv \inf \{|z-\nabla|, \nabla \in \nabla\}d(z,V)≡inf{|z−∇|,∇∈∇} denotes the distance from z z zzz to V V VVV. An element V 0 V 0 V_(0)V_{0}V0 satisfying (7) is called a bost approximation element of z z zzz by elements in V V VVV. If every z ∈ Z z ∈ Z z in Zz \in Zz∈Z has a unique best approximation element in V V VVV then the set V V VVV 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 Y Y YYY of a netric spacs X X XXX has proporty (V) if and oniy if its annihilator I ⊥ I ⊥ I^(_|_)I^{\perp}I⊥ is Chebyshorian subonce of Lipo 2 2 ^(2){ }^{2}2.
The proof of this theorem is based on the following lemma, which vill be used in the sequel :
LEVII 1. ([10]). Let X X XXX be a metric space, I a subset of X X XXX and Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ the annihilator of Y Y YYY in Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X. If T ∈ Lip 0 X T ∈ Lip 0 X T inLip_(0)XT \in \operatorname{Lip}_{0} XT∈Lip0X then d ( F , Y ⊥ ) = ′ ′ I ′ ′ I d F , Y ⊥ = ′ ′ I ′ ′ I d(F,Y^(_|_))=^('')I^('')Id\left(F, Y^{\perp}\right)= { }^{\prime \prime} I^{\prime \prime} Id(F,Y⊥)=′′I′′I and an alement δ 0 ∈ I ⊥ δ 0 ∈ I ⊥ delta_(0)inI^(_|_)\delta_{0} \in I^{\perp}δ0∈I⊥. is a best aporoximation element for y y yyy by elements in I ⊥ I ⊥ I^(_|_)I^{\perp}I⊥ if and only if g 0 = F − F 0 g 0 = F − F 0 g_(0)=F-F_(0)g_{0}=F-F_{0}g0=F−F0, where I 0 I 0 I_(0)I_{0}I0 is a norm preserving extension of F | I F I F|_(I)\left.F\right|_{I}F|I to X X XXX.
The proporty ( U U UUU ) can be charactorized also in terms of some decompositions of the Lipschitz dual Lip 0 I 0 I _(0)I{ }_{0} \mathrm{I}0I of X . To give this charactorization we noed first sone definitions and notations.
Lot It Id p 0 I → p 0 I → p_(0)Irarrp_{0} \mathrm{I} \rightarrowp0I→ Hip p 0 I p 0 I p_(0)Ip_{0} \mathrm{I}p0I denote the restriction operator, defined by :
(8) x ( T ) = F | I , F ∈ Lip p 0 X x ( T ) = F I , F ∈ Lip p 0 X quad x(T)=F|_(I),F in Lipp_(0)X\quad x(T)=\left.F\right|_{I}, F \in \operatorname{Lip} p_{0} Xx(T)=F|I,F∈Lipp0X,
and iev e: Liy 0 I → ∫ ( 0 I → ∫ _(0)I rarr int(:}{ }_{0} I \rightarrow \int\left(\right.0I→∫( IIp 0 X ) 0 X {:_(0)X)\left._{0} X\right)0X) denote the extension orerator, definod by :
(9)
e ( f ) = E ( f ; I ) , f ∈ L L p ˙ 0 X ˙ . e ( f ) = E ( f ; I ) , f ∈ L L p ˙ 0 X ˙ . 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} .e(f)=E(f;I),f∈LLp˙0X˙.
Whore I ( f ; Y ) I ( f ; Y ) I(f;Y)\mathbb{I}(f ; Y)I(f;Y) denotes the set of all nom preserving extensions of I to X X XXX. Lot w : I i p 0 I → ∫ ( I i p 0 I ) w : I i p 0 I → ∫ I i p 0 I 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)w:Iip0I→∫(Iip0I) be the composition of the operators r r rrr and e, i.e.
(10)
w ≡ e ∘ x . w ≡ e ∘ x . w-=e@x.w \equiv e \circ x .w≡e∘x.
Then, for F ∈ Lip 0 I F ∈ Lip 0 I F inLip_(0)IF \in \operatorname{Lip}_{0} IF∈Lip0I, we have w ( F ) ⇒ e ( r ( F ) ) = S ( I ( F ) ; Y ) w ( F ) ⇒ e ( r ( F ) ) = S ( I ( F ) ; Y ) w(F)=>e(r(F))=S(I(F);Y)w(F) \Rightarrow e(r(F))=\mathbb{S}(I(F) ; Y)w(F)⇒e(r(F))=S(I(F);Y) and I ( G ) = I ( F ) , ‖ G ‖ I = ‖ I ( F ) ‖ I I ( G ) = I ( F ) , ‖ G ‖ I = ‖ I ( F ) ‖ I I(G)=I(F),quad||G||_(I)=||I(F)||_(I)I(G)=I(F), \quad\|G\|_{I}=\|I(F)\|_{I}I(G)=I(F),‖G‖I=‖I(F)‖I, for all G ∈ E ( I ( F ) ; Y ) G ∈ E ( I ( F ) ; Y ) G in E(I(F);Y)G \in E(I(F) ; Y)G∈E(I(F);Y) 。
In goneral, the operator w w www is nulti-valued and w ( F ) w ( F ) w(F)w(F)w(F) is a conver subset of the ball of radius ‖ I ( F ) ‖ Y ‖ I ( F ) ‖ Y ||I(F)||_(Y)\|I(F)\|_{Y}‖I(F)‖Y and center 0 in L i p 0 X L i p 0 X Lip_(0)XL i p_{0} XLip0X.
Te can now state the theorem of characterization of property ( U ) ( U ) (U)(U)(U) :
THEOPI 3. If I I III is a subset of a netric space X X XXX then the following enserions are equivalent :
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ I has sroperty (V) ;
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ Ivery function F ∈ L j p 0 can be uniquely written in the F ∈ L j p 0 can be uniquely written in the  F inL_(j)p_(0)^("can be uniquely written in the ")F \in L_{j} p_{0}{ }^{\text {can be uniquely written in the }}F∈Ljp0can be uniquely written in the 
form
(11) F = E + g , E ∈ w ( P ) , g ∈ I ⊥ F = E + g , E ∈ w ( P ) , g ∈ I ⊥ quad F=E+g,E in w(P),g inI^(_|_)\quad F=E+g, E \in w(P), g \in I^{\perp}F=E+g,E∈w(P),g∈I⊥,
and ‖ I ‖ X > ‖ I ‖ X ‖ I ‖ X > ‖ I ‖ X ||I||_(X) > ||I||_(X)\|I\|_{X}>\|I\|_{X}‖I‖X>‖I‖X, vinenever g ≠ 0 g ≠ 0 g!=0g \neq 0g≠0;
3 ∘ G = { H ∈ Lip 0 X , ‖ H ‖ Z = ‖ r ( H ) ‖ Y } 3 ∘ G = H ∈ Lip 0 X , ‖ H ‖ Z = ‖ r ( H ) ‖ Y 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\}3∘G={H∈Lip0X,‖H‖Z=‖r(H)‖Y} is the oniy subset of
Lipo X X XXX such that every I ∈ L i p 0 X I ∈ L i p 0 X I in Lip_(0)XI \in L i p_{0} XI∈Lip0X can be uniquely written in the 응펴 f = H + g , H ∈ G , g ∈ X ⊥ f = H + g , H ∈ G , g ∈ X ⊥ f=H+g,H in G,g inX^(_|_)f=H+g, H \in G, g \in X^{\perp}f=H+g,H∈G,g∈X⊥ and ‖ F ‖ X > ‖ H ‖ X ‖ F ‖ X > ‖ H ‖ X ||F||_(X) > ||H||_(X)\|F\|_{X}>\|H\|_{X}‖F‖X>‖H‖X if g ≠ 0 g ≠ 0 g!=0g \neq 0g≠0.
Proof. 1 0 ⇒ 2 0 1 0 ⇒ 2 0 1^(0)=>2^(0)1^{0} \Rightarrow 2^{0}10⇒20. If the set Y Y YYY has property (U) then the oxtension operator 9 , delined by (9), is single-valued and so is the operator w w www defined by (10). For F ∈ I i 0 X F ∈ I i 0 X F in Ii_(0)XF \in I i_{0} XF∈Ii0X the function D(F) ∈ ∈ in\in∈ Lip X X X X _(X)X{ }_{X} XXX is the ouly norm preserving extension of I ( F ) I ( F ) I(F)I(F)I(F) to X X XXX, i.e. w ( B ) | Y = I ( F ) w ( B ) Y = I ( F ) w(B)|_(Y)=I(F)\left.w(B)\right|_{Y}=I(F)w(B)|Y=I(F) and ‖ w ( F ) ‖ X = ‖ I ( F ) ‖ Y ‖ w ( F ) ‖ X = ‖ I ( F ) ‖ Y ||w(F)||_(X)=||I(F)||_(Y)\|w(F)\|_{X}=\|I(F)\|_{Y}‖w(F)‖X=‖I(F)‖Y. It follows that g == F − w ( F ) ∈ Y ⊥ g == F − w ( F ) ∈ Y ⊥ g==F-w(F)inY^(_|_)g= =F-w(F) \in Y^{\perp}g==F−w(F)∈Y⊥ and F = w ( F ) + g F = w ( F ) + g F=w(F)+gF=w(F)+gF=w(F)+g is the unique decomposition of Y Y YYY With σ ~ ∈ I ⊥ σ ~ ∈ I ⊥ tilde(sigma)inI^(_|_)\tilde{\sigma} \in I^{\perp}σ~∈I⊥. By the definition (3) of Lipschitz norm we have ‖ F → ‖ X ⩾ ‖ r ( F ) ‖ X ‖ F → ‖ X ⩾ ‖ r ( F ) ‖ X || vec(F)||_(X) >= ||r(F)||_(X)\|\vec{F}\|_{X} \geqslant\|r(F)\|_{X}‖F→‖X⩾‖r(F)‖X. The equality ‖ I ‖ X = ‖ I ( F ) ‖ Y ‖ I ‖ X = ‖ I ( F ) ‖ Y ||I||_(X)=||I(F)||_(Y)\|I\|_{X}=\|I(F)\|_{Y}‖I‖X=‖I(F)‖Y implies that F F FFF is also a norm preserving extension of I ( F ) I ( F ) I(F)I(F)I(F) and, by the unicity of the axtension it follows F = w ( F ) F = w ( F ) F=w(F)F=w(F)F=w(F), so that B = F − w ( F ) = 0 B = F − w ( F ) = 0 B=F-w(F)=0B=F-w(F)=0B=F−w(F)=0. Hence ‖ I ‖ X > ‖ I ( F ) ‖ Y ‖ I ‖ X > ‖ I ( F ) ‖ Y ||I||_(X) > ||I(F)||_(Y)\|I\|_{X}>\|I(F)\|_{Y}‖I‖X>‖I(F)‖Y if g ∈ 0 g ∈ 0 g in0g \in 0g∈0.
2 ∘ ⇒ 3 ∘ 2 ∘ ⇒ 3 ∘ 2^(@)=>3^(@)2^{\circ} \Rightarrow 3^{\circ}2∘⇒3∘. Let F ∈ F ∈ F inF \inF∈ Lip 0 X 0 X _(0)X{ }_{0} X0X and let F = H + g F = H + g F=H+gF=H+gF=H+g the decomposition of F F FFF given in 2 ∘ 2 ∘ 2^(@)2^{\circ}2∘. As H ∈ W ( F ) = O ( I ( R ) ) H ∈ W ( F ) = O ( I ( R ) ) H in W(F)=O(I(R))H \in W(F)=O(I(R))H∈W(F)=O(I(R)) it follows that I ( F ) ≡ r ( H ) I ( F ) ≡ r ( H ) I(F)-=r(H)I(F) \equiv r(H)I(F)≡r(H) and ‖ H ‖ X = ‖ r ( F ) ‖ Y ‖ H ‖ X = ‖ r ( F ) ‖ Y ||H||_(X)=||r(F)||_(Y)\|H\|_{X}=\|r(F)\|_{Y}‖H‖X=‖r(F)‖Y, i.e. H ∈ φ j H ∈ φ j H invarphi_(j)H \in \varphi_{j}H∈φj. The condition ‖ F ‖ X > ‖ E ‖ X ‖ F ‖ X > ‖ E ‖ X ||F||_(X) > ||E||_(X)\|F\|_{X} >\|E\|_{X}‖F‖X>‖E‖X, for g ≠ 0 g ≠ 0 g!=0g \neq 0g≠0, follows from the similar condition from 2 0 2 0 2^(0)2^{0}20.
3 ∘ ⟹ 2 ∘ 3 ∘ ⟹ 2 ∘ 3^(@)Longrightarrow2^(@)3^{\circ} \Longrightarrow 2^{\circ}3∘⟹2∘. Let F ∈ I p 0 X F ∈ I p 0 X F in Ip_(0)XF \in I p_{0} XF∈Ip0X and let Z = H + g , H ∈ G , g ∈ I ⊥ Z = H + g , H ∈ G , g ∈ I ⊥ Z=H+g,H inG,g inI^(_|_)Z=H+g, H \in \mathscr{G}, g \in I^{\perp}Z=H+g,H∈G,g∈I⊥ be the unique decomposition of R R RRR given in 3 ∘ 3 ∘ 3^(@)3^{\circ}3∘. Then P − g = B P − g = B P-g=BP-g=BP−g=B, I ( H ) ≡ I ( F ) I ( H ) ≡ I ( F ) I(H)-=I(F)I(H) \equiv I(F)I(H)≡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 == a ˙ B , Y ⊥ ||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)‖I−g‖X=‖E‖X=‖I(H)‖Y=‖r(F)‖Y==a˙(B,Y⊥), which shows that g g ggg is an element of best approximation for I I III by elements in I ⊥ I ⊥ I^(_|_)I^{\perp}I⊥. If g 1 g 1 g_(1)g_{1}g1 is an other element of best approximation for T T TTT by elements in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ then, appealing again to Lema 1, there exists H 1 ∈ o ( x ( F ) ) H 1 ∈ o ( x ( F ) ) H_(1)in o(x(F))H_{1} \in o(x(F))H1∈o(x(F)) such that B 1 = F − B 1 B 1 = F − B 1 B_(1)=F-B_(1)B_{1}=F-B_{1}B1=F−B1 and ‖ H 1 ‖ X = ‖ F − g 1 ‖ X = d ( F , Y ⊥ ) = ‖ r ( F ) ‖ Y = ‖ r ( H 2 ) ‖ Y H 1 X = F − g 1 X = d F , Y ⊥ = ‖ r ( F ) ‖ Y = r H 2 Y ||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}‖H1‖X=‖F−g1‖X=d(F,Y⊥)=‖r(F)‖Y=‖r(H2)‖Y, which shows that H 1 ∈ G H 1 ∈ G H_(1)inGH_{1} \in \mathscr{G}H1∈G. Taking into acsount the unicity assumption in 3 ∘ 3 ∘ 3^(@)3^{\circ}3∘, it follows G ≡ G 1 G ≡ G 1 G-=G_(1)G \equiv G_{1}G≡G1. Therefore, Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is a Chebyshovian subspace of Iido X X XXX and, by Theorem 2, the set Y has property (U).
Theorem 3 is completely proved.
DEFINITION 3. A subset I I III of a metric space I I III is called an lifideel If its annihilator I 1 I 1 I^(1)I^{1}I1 has a complement G G GGG in Lip I I I I I^(I)I^{I}II such that ‖ F ‖ X = ‖ G ‖ X + ‖ E ‖ X ‖ F ‖ X = ‖ G ‖ X + ‖ E ‖ X ||F||_(X)=||G||_(X)+||E||_(X)\|F\|_{X}=\|G\|_{X}+\|E\|_{X}‖F‖X=‖G‖X+‖E‖X, whonever I = G + E I = G + E I=G+EI=G+EI=G+E, with G ∈ G G ∈ G G inGG \in \mathcal{G}G∈G and E ∈ Y + , { 0 } E ∈ Y + , { 0 } E inY^(+),{0}E \in Y^{+},\{0\}E∈Y+,{0}. The subset Y Y YYY is said to have property (HB) if Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ has a comple mentary subspace ξ ξ xi\xiξ of Lip X X _(X){ }_{\mathrm{X}}X such that ‖ F ‖ X ⩾ ‖ E ‖ X ‖ F ‖ X ⩾ ‖ E ‖ X ||F||_(X) >= ||E||_(X)\|F\|_{\mathrm{X}} \geqslant\|E\|_{\mathrm{X}}‖F‖X⩾‖E‖X, ‖ z ‖ X > ‖ G ‖ X ‖ z ‖ X > ‖ G ‖ X ||z||_(X) > ||G||_(X)\|z\|_{\mathrm{X}}>\|G\|_{\mathrm{X}}‖z‖X>‖G‖X, Whenever F = G + E F = G + E F=G+EF=G+EF=G+E, with G ∈ Y G ∈ Y G inYG \in \mathscr{Y}G∈Y and H ∈ Σ i , { 0 } H ∈ Σ i , { 0 } H inSigma^(i),{0}H \in \Sigma^{i},\{0\}H∈Σi,{0}, for every function F ∈ I p ˙ 0 I F ∈ I p ˙ 0 I F in Ip^(˙)_(0)IF \in I \dot{p}_{0} IF∈Ip˙0I.
THEOREM 4. If a subset Y Y YYY of a metric space X X XXX has the property (BB) then Y Y YYY has the property (U) .
Proof. Suppose that Y Y YYY has the property (HB) and has not the properfy (J). Then there exists a function f ∈ f ∈ f inf \inf∈ Lip 0 I 0 I _(0)I{ }_{0} I0I having two distinct extensions I 1 , I 2 I 1 , I 2 I_(1),I_(2)I_{1}, I_{2}I1,I2 Lip 0 I 0 I _(0)I{ }_{0} \mathfrak{I}0I and the subspace I ⊥ I ⊥ I^(_|_)I^{\perp}I⊥ has a conplementary subspace G G GGG in Lip I I ^(I){ }^{I}I such that the condition in Definition 3 is fulfilled, implying F i = G i + E i F i = G i + E i F_(i)=G_(i)+E_(i)F_{i}=G_{i}+E_{i}Fi=Gi+Ei, with G i ∈ H G i ∈ H G_(i)inHG_{i} \in \mathcal{H}Gi∈H and E i ∈ Y ⊥ E i ∈ Y ⊥ E_(i)inY^(_|_)E_{i} \in Y^{\perp}Ei∈Y⊥ for 1 = 1 , 2 1 = 1 , 2 1=1,21=1,21=1,2. As F 1 − F 2 ∈ I ⊥ F 1 − F 2 ∈ I ⊥ F_(1)-F_(2)inI^(_|_)F_{1}-F_{2} \in I^{\perp}F1−F2∈I⊥ it follows that G 1 − G 2 = F 1 − I 2 G 1 − G 2 = F 1 − I 2 G_(1)-G_(2)=F_(1)-I_(2)G_{1}-G_{2}=F_{1}-I_{2}G1−G2=F1−I2 -- ( E 1 − E 2 ) ∈ I ⊥ E 1 − E 2 ∈ I ⊥ (E_(1)-E_(2))inI^(_|_)\left(E_{1}-E_{2}\right) \in I^{\perp}(E1−E2)∈I⊥, hence G 1 = G 2 = G ( ζ ∩ I ⊥ = { 0 } G 1 = G 2 = G ζ ∩ I ⊥ = { 0 } G_(1)=G_(2)=G(zeta nnI^(_|_)={0}:}G_{1}=G_{2}=G\left(\zeta \cap I^{\perp}=\{0\}\right.G1=G2=G(ζ∩I⊥={0}, as ξ ξ xi\xiξ and I ⊥ I ⊥ I^(_|_)I^{\perp}I⊥ are complementary subspaces of Ifipo X X XXX ). Therefore F 1 = G + H 1 F 1 = G + H 1 F_(1)=G+H_(1)F_{1}=G+H_{1}F1=G+H1 and F 2 = G + E 2 F 2 = G + E 2 F_(2)=G+E_(2)F_{2}=G+E_{2}F2=G+E2. Now, if E 2 ≠ 0 E 2 ≠ 0 E_(2)!=0E_{2} \neq 0E2≠0 then ‖ F 1 ‖ Σ > ‖ G ‖ I F 1 Σ > ‖ G ‖ I ||F_(1)||_(Sigma) > ||G||_(I)\left\|F_{1}\right\|_{\Sigma}>\|G\|_{I}‖F1‖Σ>‖G‖I so that ‖ f ‖ I == ‖ F 1 ‖ Σ > ‖ G ‖ Σ ⩾ ‖ r ( G ) ‖ I = ‖ I ‖ Y ‖ f ‖ I == F 1 Σ > ‖ G ‖ Σ ⩾ ‖ r ( G ) ‖ I = ‖ I ‖ Y ||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}‖f‖I==‖F1‖Σ>‖G‖Σ⩾‖r(G)‖I=‖I‖Y. If H 1 ≡ 0 H 1 ≡ 0 H_(1)-=0H_{1} \equiv 0H1≡0 then H 2 ≠ H 1 = 0 H 2 ≠ H 1 = 0 H_(2)!=H_(1)=0H_{2} \neq H_{1}=0H2≠H1=0, hence G = F 1 G = F 1 G=F_(1)G=F_{1}G=F1 and, by Desinition 3 , the equality F 1 = F 2 + H 2 F 1 = F 2 + H 2 F_(1)=F_(2)+H_(2)F_{1}=F_{2}+H_{2}F1=F2+H2 implies ‖ P ‖ I = ‖ F 1 ‖ X > ‖ F 2 ‖ X = ‖ S ‖ I ‖ P ‖ I = F 1 X > F 2 X = ‖ S ‖ I ||P||_(I)=||F_(1)||_(X) > ||F_(2)||_(X)=||S||_(I)\|P\|_{I}=\left\|F_{1}\right\|_{X}>\left\|F_{2}\right\|_{X}=\|S\|_{I}‖P‖I=‖F1‖X>‖F2‖X=‖S‖I. The obtained contradictions shows that the set I I III cannot have the propetty (HB). Theorem 4 is completejy proved.
THEOREIX 5. If the subset Y Y YYY of X X XXX has the property ( H B H B HBH BHB ) and F ∈ Lip p 0 , F F ∈ Lip p 0 , F F in Lipp_(0),FF \in \operatorname{Lip} p_{0}, FF∈Lipp0,F of 0 , then F ∈ G F ∈ G F inGF \in \mathscr{G}F∈G if and only if ‖ r ( F ) ‖ I = ‖ F ‖ X ‖ r ( F ) ‖ I = ‖ F ‖ X ||r(F)||_(I)=||F||_(X)\|r(F)\|_{I}=\|F\|_{X}‖r(F)‖I=‖F‖X, There ζ ζ zeta\zetaζ is the complementary subspace of r ⊥ r ⊥ r^(_|_)r^{\perp}r⊥ given in Definition 3 .
Proof. Let F ∈ ξ , F ∉ 0 F ∈ ξ , F ∉ 0 F in xi,F!in0F \in \xi, F \notin 0F∈ξ,F∉0, and let G G GGG be a norm preserving extension of x ( F ) x ( F ) x(F)x(F)x(F) to I . I .  I_(". ")I_{\text {. }}I.  Let G = G 1 + H 1 , G 1 ∈ ξ , H 2 ∈ X ⊥ G = G 1 + H 1 , G 1 ∈ ξ , H 2 ∈ X ⊥ 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}G=G1+H1,G1∈ξ,H2∈X⊥ be the
decomposition of G G GGG given by Definition 3. Supposing E 1 ≠ 0 E 1 ≠ 0 E_(1)!=0E_{1} \neq 0E1≠0, one obtains the contradiction ‖ G ‖ I > ‖ G 1 ‖ I ⩾ ‖ I ( G 1 ) ‖ I = ‖ I ( I ) ‖ I == ‖ G 1 ‖ 2 ‖ G ‖ I > G 1 I ⩾ I G 1 I = ‖ I ( I ) ‖ I == G 1 2 ||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}‖G‖I>‖G1‖I⩾‖I(G1)‖I=‖I(I)‖I==‖G1‖2. Therefore H 1 = 0 H 1 = 0 H_(1)=0H_{1}=0H1=0 and G = G 1 ∈ ζ 1 G = G 1 ∈ ζ 1 G=G_(1)inzeta_(1)G=G_{1} \in \zeta_{1}G=G1∈ζ1. As ζ 1 ζ 1 zeta_(1)\zeta_{1}ζ1 is a subspace of iip 0 0 _(0){ }_{0}0 it follows F − G ∈ G F − G ∈ G F-G in GF-G \in GF−G∈G. But F − G F − G F-GF-GF−G is in I − I − I-I-I− too, bocauso r ( F ) = r ( G ) r ( F ) = r ( G ) r(F)=r(G)r(F)=r(G)r(F)=r(G), so that R − G ∈ ξ ∩ X ⊥ = { 0 } R − G ∈ ξ ∩ X ⊥ = { 0 } R-G in xi nnX^(_|_)={0}R-G \in \xi \cap X^{\perp}=\{0\}R−G∈ξ∩X⊥={0}, i.s. F = G ∈ G F = G ∈ G F=G inGF=G \in \mathcal{G}F=G∈G.
Conversely, suppose that F ∈ K p 0 x ~ , F ≠ 0 F ∈ K p 0 x ~ , F ≠ 0 F in Kp_(0) tilde(x),F!=0F \in K p_{0} \tilde{x}, F \neq 0F∈Kp0x~,F≠0, is such that ‖ x ( F ) ‖ x = ‖ P ‖ X ( > 0 ) ‖ x ( F ) ‖ x = ‖ P ‖ X ( > 0 ) ||x(F)||_(x)=||P||_(X)( > 0)\|x(F)\|_{x} =\|P\|_{X}(>0)‖x(F)‖x=‖P‖X(>0). Lot I = G + H I = G + H I=G+HI=G+HI=G+H with G ∈ ξ , H ∈ I L G ∈ ξ , H ∈ I L G in xi,H inI^(L)G \in \xi, H \in I^{L}G∈ξ,H∈IL. If E ≠ 0 E ≠ 0 E!=0E \neq 0E≠0 then ‖ P ‖ I > ‖ G ‖ X ‖ P ‖ I > ‖ G ‖ X ||P||_(I) > ||G||_(X)\|P\|_{I}>\|G\|_{X}‖P‖I>‖G‖X and the equality r ( I ) = r ( G ) r ( I ) = r ( G ) r(I)=r(G)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 ||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}‖x(F)‖Z=‖F‖Z>‖G‖Z=‖x(G)‖Y=‖r(F)‖Y, which shows that H = 0 H = 0 H=0H=0H=0 and F = G ∈ φ S F = G ∈ φ S F=G invarphi_(S)F=G \in \varphi_{S}F=G∈φS.
THFOPN 6. If the subset Y Y YYY of a metric snnce X X XXX has the property (HB) then the subsoace ζ ζ zeta\zetaζ (given in Definition 3) is dropetrically isomorphic to the space Lip 0 I 0 I _(0)^(I){ }_{0}{ }^{I}0I.
Proof. By Theorem 4, tae subset Y Y YYY has the property (U) , so that the restriction r 1 r 1 r_(1)r_{1}r1 of the restriction operator r r rrr to ξ j ξ j xi_(j)\xi_{j}ξj is single-valued and linear. By Theorem 5, ‖ r 1 ( G ) ‖ j = ‖ G ‖ X r 1 ( G ) j = ‖ G ‖ X ||r_(1)(G)||_(j)=||G||_(X)\left\|r_{1}(G)\right\|_{j}=\|G\|_{X}‖r1(G)‖j=‖G‖X, for all G ∈ ζ ξ G ∈ ζ ξ G inzeta_(xi)G \in \zeta_{\xi}G∈ζξ, showing that T 1 T 1 T_(1)T_{1}T1 is an loometry.
THEORSU 7. If the subset X X XXX of a motric space X X XXX pes the proper ty (HB) , tien the extension operator ⋅ ∣ ⋅ ∣ *∣\cdot \mid⋅∣ ip 0 Y → 0 Y → _(0)Yrarr_{0} \mathrm{Y} \rightarrow0Y→ Lip c X c X _(c)X_{c} \mathrm{X}cX de linear.
Proof. Let : ~ : ~ tilde(:)\tilde{:}:~ : Lip I → ζ j I → ζ j Irarrzeta_(j)\mathrm{I} \rightarrow \zeta_{j}I→ζj be the inverse of the restiliction operator r 1 = r | e r 1 = r e r_(1)=r|_(e)r_{1}=\left.r\right|_{e}r1=r|e & ⟶ ⟶ longrightarrow\longrightarrow⟶ Lip 0 I 0 I _(0)I{ }_{0} \mathrm{I}0I which, by Theorem 6, is an isoattrical isomorpicism between ζ ζ zeta\zetaζ and Lipo Y Y YYY. Then is linear and e = j ∘ 0 ~ e = j ∘ 0 ~ e=j@ tilde(0)e=j \circ \tilde{0}e=j∘0~ where j : ζ → j : ζ → j:zeta rarrj: \zeta \rightarrowj:ζ→ Lip j x j x jxj xjx, denotes the imbedding operator of ζ ζ zeta\zetaζ into Ling x x xxx.

RAITPRINCES

1 ALPSEN, E.M., EPFROSS, E. , Structure in real Banach speces , calt ac 1 ann. of Math. 96 (1972), 98-173.
2 CZIPSTR, J., GHENR, L., Extension of Functions Eatisfying a Liocchite conditions, Acta Hath, Acad. Sci. Hungar 6 (1955), 213-220
3 Tatrourr, E ., Sélections 11néaires essociées au Théorige de Fohn - Expach , J. of Functional Analysis 21 (2972), 436-452.
4 EETIMSILD, J., 14-jdec.ls, HB - subroaces and Conpact Operators, Indiana Univ. Math. J. 28 (6) (1979), 927 - 934.
5 EMTHIPID, J., Anote on M-ideals in B(X), Proc, Amer. Hat. Soc. 98 (1) (1980) , 89-92.
6 HOLIES, R.B., SCRANTON, B., WARD, J.D., Approzimation from the appe of compect operators and other x − x − x-x-x− ideals, Duke Math. J. 42 (1975), 259-26%.
7 BOLITS, R.B., Geonetric Functionol Anclssie and its Applicationg, Springer - Verlag, Now York - Heidelberg - Berlis , 1975 •
8 Mo SHANE, E.J., Thonsion of rango of functions, Bull. Amor. Math. Soc. 40 (1954) , 837-842.
9 JOEWSON, J.A., Benech Speces of Jipschitz Punctions and Vector - Velued Iipschitz Functions, Trans, knor. Meth. Soc. 248 (1970), 147-169 a pt at eccon abill
Eipschite Functions, J. Appror. Theory 29 (3) (1977) , 222-230 。
11 RESTPS, R.R., Uniqueness of Bohn - Banach Extension and Uniqus Best Approximation, Trans. Amer. Math. Boc. 45 ( 1950 ) , 238 − 255 ( 1950 ) , 238 − 255 (1950),238-255(1950), 238-255(1950),238−255.
12 OJ1, S., On the uniquaness of the norr proservine extengion of linear luuctional in the Hohu - Bancon Feooren, Proc. Acad. Science Estonian SSR 33 (4) (2984), 422-433 (Russian)
Insvitutul do Calcul
Oficiul Postal 1
C. P. 68
3400 Cluj-Hiapoca
Romenia
This note is in final form and no version of 1t is or will be submitted for publication elsawhere.
1988

Related Posts