A characterization of Chebyshevian subspaces of Y-type

Costica Mustata
(original), Approximation Theory, best approximation, paper

Abstract

Authors

Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

Keywords

Paper coordinates

C.Mustăţa, A characterization of Chebyshevian subspaces of \(Y^\perp\)-type, Anal. Numér. Théor. Approx., 6 (1977) 1, 51-56.

PDF

https://ictp.acad.ro/jnaat/journal/article/view/1977-vol6-no1-art8/Mustata

About this paper

Journal

Revue d’Analyse Numer. Theor. Approximation

Publisher Name

Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/1977-vol6-no1-art8/

Print ISSN

2457-6794

Online ISSN

2501-059X

MR 58 # 29722

google scholar link

[1] Banach, Stefan, Wstęp do teorii funkcji rzeczywistych. (Polish) [Introduction to the theory of real functions] Monografie Matematyczne. Tom XVII.] Polskie Towarzystwo Matematyczne, Warszawa-Wrocław, 1951. iv+224 pp., MR0043161.
[2] Czipszer, J., Gehér, L., Extension of functions satisfying a Lipschitz condition. Acta Math. Acad. Sci. Hungar. 6 (1955), 213-220, MR0071493, https://doi.org/10.1007/bf02021278
[3] Kolumban, I., On the uniqueness of the extension of linear functionals. (Russian) Mathematica (Cluj) 4 (27) 1962 267-270, MR0164223.
[4] Mustăţa, Costică, On certain Čebyšev subspaces of the normed space of Lipschitzian functions. (Romanian) Rev. Anal. Numer. Teoria Aproximaţiei 2 (1973), 81-87, MR0387920.
[5] Mustăţa, Costică, A monotonicity property of the operator of best approximation in the space of Lipschitzian functions. (Romanian) Rev. Anal. Numer. Teoria Aproximaţiei 3 (1974), no. 2, 153-160 (1975), MR0387921.
[6] Nachbin, Leopoldo, A theorem of the Hahn-Banach type for linear transformations. Trans. Amer. Math. Soc. 68, (1950). 28-46, MR0032932, https://doi.org/10.1090/s0002-9947-1950-0032932-3
[7] Pantelidis, Georgios, Approximationstheorie für metrische lineare Räume. (German) Math. Ann. 184 1969 30-48, MR0262754, https://doi.org/10.1007/bf01350613
[8] Phelps, R. R., Uniqueness of Hahn-Banach extensions and unique best approximation. Trans. Amer. Math. Soc. 95 1960 238-255, MR0113125, https://doi.org/10.1090/s0002-9947-1960-0113125-4
[9] Singer, Ivan, Cea mai bună aproximare în spaţii vectoriale normate prin elemente din subspaţii vectoriale. (Romanian) [Best approximation in normed vector spaces by elements of vector subspaces] Editura Academiei Republicii Socialiste România, Bucharest 1967 386 pp., MR0235368.

Paper (preprint) in HTML form

1977-Mustata-A characterization of Chebyshevian subspaces of -Mathematica

A CHARACTERISATION OF CHEBYSHEVIAN SUBSPACE OF Y ⊥ Y ⊥ Y^(_|_)\mathrm{Y}^{\perp}Y⊥ - TYPE

by
COSTICA MUSTATA
(Cluj-Napoca)
  1. Let be given a real linear space Z Z ZZZ. For any nonvoid set E E EEE we denote by Z E Z E Z^(E)Z^{E}ZE the linear space of all functions from E E EEE to Z Z ZZZ with the operations of addition and multiplication by real scalars defined pointwisely.
Consider now two nonvoid sets X , Y X , Y X,YX, YX,Y such that Y ⊆ X Y ⊆ X Y sube XY \subseteq XY⊆X and two normed linear subspace M X M X M_(X)M_{X}MX and M Y M Y M_(Y)M_{Y}MY of Z X Z X Z^(X)Z^{X}ZX, respectively of Z Y Z Y Z^(Y)Z^{Y}ZY, such that f | Y ∈ M Y f Y ∈ M Y f|_(Y)inM_(Y)\left.f\right|_{Y} \in M_{Y}f|Y∈MY for all f ∈ M X f ∈ M X f inM_(X)f \in M_{X}f∈MX, where f | Y f Y f|_(Y)\left.f\right|_{Y}f|Y denotes the restriction of f f fff to Y Y YYY. Denote by ‖ ‖ X ‖ X ||||_(X):}\left\|\|_{X}\right.‖‖X and ‖ ‖ Y ‖ ‖ Y ||||_(Y)\| \|_{Y}‖‖Y the norms on M X M X M_(X)M_{X}MX, respectively M Y M Y M_(Y)M_{Y}MY.
Definition 1. We say that the norm ‖ ‖ Y ‖ Y ||||_(Y):}\left\|\|_{Y}\right.‖‖Y is compatible with the norm ‖ ‖ X ‖ X ||||_(X):}\left\|\|_{X}\right.‖‖X if
(1) ‖ f | Y ‖ Y ⩽ ‖ f ‖ X , (1) f Y Y ⩽ ‖ f ‖ X , {:(1)||f|_(Y)||_(Y) <= ||f||_(X)",":}\begin{equation*} \left\|\left.f\right|_{Y}\right\|_{Y} \leqslant\|f\|_{X}, \tag{1} \end{equation*}(1)‖f|Y‖Y⩽‖f‖X,
for all f ∈ M X f ∈ M X f inM_(X)f \in M_{X}f∈MX. patible.
In the sequel, the norms ‖ ‖ X ‖ X ||||_(X):}\left\|\|_{X}\right.‖‖X and ‖ ‖ Y ‖ ‖ Y ||||_(Y)\| \|_{Y}‖‖Y will considered always com-
Let K X ⊆ M X K X ⊆ M X K_(X)subeM_(X)K_{X} \subseteq M_{X}KX⊆MX and K Y ⊆ M Y K Y ⊆ M Y K_(Y)subeM_(Y)K_{Y} \subseteq M_{Y}KY⊆MY be two convex cones with the vertex in the origin of M X M X M_(X)M_{X}MX, respectively M Y M Y M_(Y)M_{Y}MY such that f | X ∈ K Y f X ∈ K Y f|_(X)inK_(Y)\left.f\right|_{X} \in K_{Y}f|X∈KY, for all f ∈ K X f ∈ K X f inK_(X)f \in K_{X}f∈KX.
Definition 2. We say that K Y K Y K_(Y)K_{Y}KY is a P P PPP-cone if for all f ∈ K Y f ∈ K Y f inK_(Y)f \in K_{Y}f∈KY there exists F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX such that
  1. f = F | Y f = F Y f=F|_(Y)f=\left.F\right|_{Y}f=F|Y,
  2. ‖ f ‖ Y = ‖ F ‖ X ‖ f ‖ Y = ‖ F ‖ X ||f||_(Y)=||F||_(X)\|f\|_{Y}=\|F\|_{X}‖f‖Y=‖F‖X.
If further, the function F F FFF with the properties 1) and 2) is unique, K Y K Y K_(Y)K_{Y}KY is called P U P U PUP UPU-cone. The function F F FFF is called an extension of f f fff.

2. Let

(2)
X K = K X − K X X K = K X − K X X_(K)=K_(X)-K_(X)X_{K}=K_{X}-K_{X}XK=KX−KX
be the linear subspace of M X M X M_(X)M_{X}MX, generated by the cone K X K X K_(X)K_{X}KX and
(3) Y X K ⊥ = { g : g ∈ X K , g | Y = θ Y } (3) Y X K ⊥ = g : g ∈ X K , g Y = θ Y {:(3)Y_(X_(K))^(_|_)={g:g inX_(K),g|_(Y)=theta_(Y)}:}\begin{equation*} Y_{X_{K}}^{\perp}=\left\{g: g \in X_{K},\left.g\right|_{Y}=\theta_{Y}\right\} \tag{3} \end{equation*}(3)YXK⊥={g:g∈XK,g|Y=θY}
where θ Y θ Y theta_(Y)\theta_{Y}θY denotes the zero function in M Y M Y M_(Y)M_{Y}MY, i.e. θ Y ( y ) = 0 θ Y ( y ) = 0 theta_(Y)(y)=0\theta_{Y}(y)=0θY(y)=0, for all y ∈ X y ∈ X y in Xy \in Xy∈X.
Definition 3. We say that the subspace Y X K ⊥ Y X K ⊥ Y_(X_(K))^(_|_)Y_{X_{K}}^{\perp}YXK⊥ is K K K K K_(K)K_{K}KK - proximinal if for all f ∈ K X f ∈ K X f inK_(X)f \in K_{X}f∈KX there exists an element g 0 ∈ Y X K ⊥ g 0 ∈ Y X K ⊥ g_(0)inY_(X_(K))^(_|_)g_{0} \in Y_{X_{K}}^{\perp}g0∈YXK⊥ such that
(4) ‖ f − g 0 ‖ X = d ( f , Y X K ⊥ ) = inf { ‖ f − g ‖ X : g ∈ Y X K ⊥ } . (4) f − g 0 X = d f , Y X K ⊥ = inf ‖ f − g ‖ X : g ∈ Y X K ⊥ . {:(4)||f-g_(0)||_(X)=d(f,Y_(X_(K))^(_|_))=i n f{||f-g||_(X):g inY_(X_(K))^(_|_)}.:}\begin{equation*} \left\|f-g_{0}\right\|_{X}=d\left(f, Y_{X_{K}}^{\perp}\right)=\inf \left\{\|f-g\|_{X}: g \in Y_{X_{K}}^{\perp}\right\} . \tag{4} \end{equation*}(4)‖f−g0‖X=d(f,YXK⊥)=inf{‖f−g‖X:g∈YXK⊥}.
If further, for all f ∈ K X f ∈ K X f inK_(X)f \in K_{X}f∈KX there exists a unique g 0 ∈ Y X K ⊥ g 0 ∈ Y X K ⊥ g_(0)inY_(X_(K))^(_|_)g_{0} \in Y_{X_{K}}^{\perp}g0∈YXK⊥ such that the equality (4) holds, then Y X K ⊥ Y X K ⊥ Y_(X_(K))^(_|_)Y_{X_{K}}^{\perp}YXK⊥ is called K X K X K_(X)K_{X}KX-Chebyshveian. An element g 0 ∈ Y X K ⊥ g 0 ∈ Y X K ⊥ g_(0)inY_(X_(K))^(_|_)g_{0} \in Y_{X_{K}}^{\perp}g0∈YXK⊥ such that ‖ f − g 0 ‖ X = d ( f , Y X K ⊥ ) f − g 0 X = d f , Y X K ⊥ ||f-g_(0)||_(X)=d(f,Y_(X_(K))^(_|_))\left\|f-g_{0}\right\|_{X}=d\left(f, Y_{X_{K}}^{\perp}\right)‖f−g0‖X=d(f,YXK⊥) is called an element of best approximation of f f fff by elements of Y X K ⊥ Y X K ⊥ Y_(X_(K))^(_|_)Y_{X_{K}}^{\perp}YXK⊥.
3. The following two theorems show that the best approximation properties of the subspace Y X K ⊥ Y X K ⊥ Y_(X_(K))^(_|_)Y_{X_{K}}^{\perp}YXK⊥ in M X M X M_(X)M_{X}MX are connected with the extension properties of K Y K Y K_(Y)K_{Y}KY.
THEOREM 1. If K Y K Y K_(Y)K_{Y}KY is a P P PPP-cone then:
(a) for all f ∈ K X f ∈ K X f inK_(X)f \in K_{X}f∈KX, the following equality holds
(5) ‖ f | Y ‖ = d ( f , Y X K ⊥ ) (5) f Y = d f , Y X K ⊥ {:(5)||f|_(Y)||=d(f,Y_(X_(K))^(_|_)):}\begin{equation*} \left\|\left.f\right|_{Y}\right\|=d\left(f, Y_{X_{K}}^{\perp}\right) \tag{5} \end{equation*}(5)‖f|Y‖=d(f,YXK⊥)
(b) for every f ∈ K X f ∈ K X f inK_(X)f \in K_{X}f∈KX, the elements of best approximation of f f fff by elements of Y X K ⊥ Y X K ⊥ Y_(X_(K))^(_|_)Y_{X_{K}}^{\perp}YXK⊥ are exactly the elements of the form f − F f − F f-Ff-Ff−F, where F F FFF is an extension of f ^ | Y f ^ Y ( hat(f))|_(Y)\left.\hat{f}\right|_{Y}f^|Y.
Proof. (a) For g ∈ Y X K ⊥ g ∈ Y X K ⊥ g inY_(X_(K))^(_|_)g \in Y_{X_{K}}^{\perp}g∈YXK⊥ we have:
‖ f | Y ‖ Y = ‖ f | Y − g | Y ‖ Y = ‖ ( f − g ) | X ‖ Y ⩽ ‖ f − g ‖ X f Y Y = f Y − g Y Y = ( f − g ) X Y ⩽ ‖ f − g ‖ X ||f|_(Y)||_(Y)=||f|_(Y)-g|_(Y)||_(Y)=||(f-g)|_(X)||_(Y) <= ||f-g||_(X)\left\|\left.f\right|_{Y}\right\|_{Y}=\left\|\left.f\right|_{Y}-\left.g\right|_{Y}\right\|_{Y}=\left\|\left.(f-g)\right|_{X}\right\|_{Y} \leqslant\|f-g\|_{X}‖f|Y‖Y=‖f|Y−g|Y‖Y=‖(f−g)|X‖Y⩽‖f−g‖X
such that ‖ f | Y ‖ Y ⩽ d ( f , Y X K ⊥ ) f Y Y ⩽ d f , Y X K ⊥ ||f|_(Y)||_(Y) <= d(f,Y_(X_(K))^(_|_))\left\|\left.f\right|_{Y}\right\|_{Y} \leqslant d\left(f, Y_{X_{K}}^{\perp}\right)‖f|Y‖Y⩽d(f,YXK⊥).
On the other hand,
‖ f | Y ‖ Y = ‖ f − ( f − F ) ‖ X ⩾ inf { ‖ f − g ‖ X : g ∈ Y X K ⊥ } = d ( f , Y X K ⊥ ) , f Y Y = ‖ f − ( f − F ) ‖ X ⩾ inf ‖ f − g ‖ X : g ∈ Y X K ⊥ = d f , Y X K ⊥ , ||f|_(Y)||_(Y)=||f-(f-F)||_(X) >= i n f{||f-g||_(X):g inY_(X_(K))^(_|_)}=d(f,Y_(X_(K))^(_|_)),\left\|\left.f\right|_{Y}\right\|_{Y}=\|f-(f-F)\|_{X} \geqslant \inf \left\{\|f-g\|_{X}: g \in Y_{X_{K}}^{\perp}\right\}=d\left(f, Y_{X_{K}}^{\perp}\right),‖f|Y‖Y=‖f−(f−F)‖X⩾inf{‖f−g‖X:g∈YXK⊥}=d(f,YXK⊥),
where F F FFF is an extension of f | Y f Y f|_(Y)\left.f\right|_{Y}f|Y to X X XXX. Therefore, the equality (5) holds.
(b) If f ∈ K X f ∈ K X f inK_(X)f \in K_{X}f∈KX and g ∈ Y X K ⊥ g ∈ Y X K ⊥ g inY_(X_(K))^(_|_)g \in Y_{X_{K}}^{\perp}g∈YXK⊥ is an element of best approximation of f f fff, then by (5), ‖ f − g ‖ X = d ( f , Y X K ⊥ ) = ‖ f | Y ‖ Y ‖ f − g ‖ X = d f , Y X K ⊥ = f Y Y ||f-g||_(X)=d(f,Y_(X_(K))^(_|_))=||f|_(Y)||_(Y)\|f-g\|_{X}=d\left(f, Y_{X_{K}}^{\perp}\right)=\left\|\left.f\right|_{Y}\right\|_{Y}‖f−g‖X=d(f,YXK⊥)=‖f|Y‖Y and ( f − g ) | Y = f | Y ( f − g ) Y = f Y (f-g)|_(Y)=f|_(Y)\left.(f-g)\right|_{Y}=\left.f\right|_{Y}(f−g)|Y=f|Y. It follows
that f − g f − g f-gf-gf−g is an extension of f | Y f Y f|_(Y)\left.f\right|_{Y}f|Y to X X XXX. The fact that f − F f − F f-Ff-Ff−F is a best approximation of f f fff by elements of Y X K ⊥ Y X K ⊥ Y_(X_(K))^(_|_)Y_{X_{K}}^{\perp}YXK⊥, for every extension F F FFF of f | Y f Y f|_(Y)\left.f\right|_{Y}f|Y to X X XXX, follows by the equalities:
d ( f , Y X K ⊥ ) = ‖ f | Y ‖ Y = ‖ f − ( f − F ) ‖ X d f , Y X K ⊥ = f Y Y = ‖ f − ( f − F ) ‖ X d(f,Y_(X_(K))^(_|_))=||f|_(Y)||_(Y)=||f-(f-F)||_(X)d\left(f, Y_{X_{K}}^{\perp}\right)=\left\|\left.f\right|_{Y}\right\|_{Y}=\|f-(f-F)\|_{X}d(f,YXK⊥)=‖f|Y‖Y=‖f−(f−F)‖X
THEOREM 2. (a) If K Y K Y K_(Y)K_{Y}KY is a P P PPP-cone, then Y X ¯ K ⊥ Y X ¯ K ⊥ Y_( bar(X)_(K))^(_|_)Y_{\bar{X}_{K}}^{\perp}YX¯K⊥ is K X K X K_(X)K_{X}KX-proximinal;
(b) If K Y K Y K_(Y)K_{Y}KY is a P P PPP-cone, then Y X K ⊥ Y X K ⊥ Y_(X_(K))^(_|_)Y_{X_{K}}^{\perp}YXK⊥ is K X K X K_(X)K_{X}KX-Chebyshevian if and only if K Y K Y K_(Y)K_{Y}KY is a P U P U PUP UPU-cone.
Proof. The theorem follows from theorem 1 (b).
∴ ∴ :.quad\therefore \quad∴ If K Y = M Y K Y = M Y K_(Y)=M_(Y)K_{Y}=M_{Y}KY=MY and K Y K Y K_(Y)K_{Y}KY is P P PPP - cone, respectively P U P U PUP UPU - cone, then M Y M Y M_(Y)M_{Y}MY is called P P PPP - space, respectively P U − P U − PU-P U-PU− space.
Let us denote by Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥, the following subspace of M X M X M_(X)M_{X}MX :
(6)
Y ⊥ = { f : f ∈ M X , f | Y = θ Y } Y ⊥ = f : f ∈ M X , f Y = θ Y Y^(_|_)={f:f inM_(X),f|_(Y)=theta_(Y)}Y^{\perp}=\left\{f: f \in M_{X},\left.f\right|_{Y}=\theta_{Y}\right\}Y⊥={f:f∈MX,f|Y=θY}
Then, the theorems 1 and 2 become:
THEOREM 3. If M M MMM is P P PPP - space, then:
(a) for all f ∈ M X f ∈ M X f inM_(X)f \in M_{X}f∈MX, the following equality holds:
(7)
‖ f | Y ‖ Y = d ( f , Y ⊥ ) f Y Y = d ( f , Y ⊥ ) ||f|_(Y)||_(Y)=d(f,Y _|_)\left\|\left.f\right|_{Y}\right\|_{Y}=d(f, Y \perp)‖f|Y‖Y=d(f,Y⊥)
(b) for every f ∈ M X f ∈ M X f inM_(X)f \in M_{X}f∈MX, the elements of best approximation of f f fff by elements of Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ are exactly the elements of the form f − F f − F f-Ff-Ff−F, where F F FFF is an extension of f | Y f Y f|_(Y)\left.f\right|_{Y}f|Y to X X XXX.
THEOREM 4. (a) If M Y M Y M_(Y)M_{Y}MY is a P P PPP - space, then Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is proximinal;
(b) If M Y M Y M_(Y)M_{Y}MY is a P P PPP - space, then Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is Chebyshevian if and only if M Y M Y M_(Y)M_{Y}MY is a P U 2 P U 2 PU^(2)P U^{2}PU2 - space.
For the definition of proximinal and Chebyshevian sets see [9].
4. We shall give some particular cases of the above theorems.
I. If X X XXX is a normed linear space, Y Y YYY a linear subspace of X , X ∗ X , X ∗ X,X^(**)X, X^{*}X,X∗ the conjugate space of X , Y ∗ X , Y ∗ X,Y^(**)X, Y^{*}X,Y∗ the conjugate space of Y Y YYY, then by the HahnBanach theorem, Y ∗ Y ∗ Y^(**)Y^{*}Y∗ is a P P PPP-space. In this case, theorem 3 ( a ) 3 ( a ) 3(a)3(a)3(a) and theorem 4 (b) were proved by R. R. PHELPS [8].
II. For a metric space ( X , d X , d X,dX, dX,d ), a subset Y Y YYY of X X XXX and a fixed element x 0 x 0 x_(0)x_{0}x0 of Y Y YYY, let
(8) Iip 0 X = { f : f : X → R , sup x ≠ y x , y ∈ X | f ( x ) − f ( y ) | d ( x , y ) < ∞ , f ( x 0 ) = 0 } (8) Iip 0 X = f : f : X → R , sup x ≠ y x , y ∈ X   | f ( x ) − f ( y ) | d ( x , y ) < ∞ , f x 0 = 0 {:(8)Iip_(0)X={f:f:X rarrR,s u p_({:[x!=y],[x","y in X]:})(|f(x)-f(y)|)/(d(x,y)) < oo,f(x_(0))=0}:}\begin{equation*} \operatorname{Iip}_{0} X=\left\{f: f: X \rightarrow \mathbb{R}, \sup _{\substack{x \neq y \\ x, y \in X}} \frac{|f(x)-f(y)|}{d(x, y)}<\infty, f\left(x_{0}\right)=0\right\} \tag{8} \end{equation*}(8)Iip0X={f:f:X→R,supx≠yx,y∈X|f(x)−f(y)|d(x,y)<∞,f(x0)=0}
(9) Lip 0 Y = { h : h : Y → R , sup x ≠ y x , y ∈ Y | h ( x ) − h ( y ) | d ( x , y ) < ∞ , h ( x 0 ) = 0 } (9) Lip 0 Y = h : h : Y → R , sup x ≠ y x , y ∈ Y   | h ( x ) − h ( y ) | d ( x , y ) < ∞ , h x 0 = 0 {:(9)Lip_(0)Y={h:h:Y rarrR,s u p_({:[x!=y],[x","y in Y]:})(|h(x)-h(y)|)/(d(x,y)) < oo,h(x_(0))=0}:}\begin{equation*} \operatorname{Lip}_{0} Y=\left\{h: h: Y \rightarrow \mathbf{R}, \sup _{\substack{x \neq y \\ x, y \in Y}} \frac{|h(x)-h(y)|}{d(x, y)}<\infty, h\left(x_{0}\right)=0\right\} \tag{9} \end{equation*}(9)Lip0Y={h:h:Y→R,supx≠yx,y∈Y|h(x)−h(y)|d(x,y)<∞,h(x0)=0}
be the linear space of Lipschitz functions on X X XXX, respectively Y Y YYY, which vanish on x 0 x 0 x_(0)x_{0}x0, with the norms
(10)
(11)
‖ f ‖ X = sup { | f ( x ) − f ( y ) | / d ( x , y ) : x ≠ y , x , y ∈ X } ‖ h ‖ Y = sup { | h ( x ) − h ( y ) | / d ( x , y ) : x ≠ y , x , y ∈ Y } ‖ f ‖ X = sup { | f ( x ) − f ( y ) | / d ( x , y ) : x ≠ y , x , y ∈ X } ‖ h ‖ Y = sup { | h ( x ) − h ( y ) | / d ( x , y ) : x ≠ y , x , y ∈ Y } {:[||f||_(X)=s u p{|f(x)-f(y)|//d(x","y):x!=y","x","y in X}],[||h||_(Y)=s u p{|h(x)-h(y)|//d(x","y):x!=y","x","y in Y}]:}\begin{aligned} & \|f\|_{X}=\sup \{|f(x)-f(y)| / d(x, y): x \neq y, x, y \in X\} \\ & \|h\|_{Y}=\sup \{|h(x)-h(y)| / d(x, y): x \neq y, x, y \in Y\} \end{aligned}‖f‖X=sup{|f(x)−f(y)|/d(x,y):x≠y,x,y∈X}‖h‖Y=sup{|h(x)−h(y)|/d(x,y):x≠y,x,y∈Y}
By a theorem of S. BANACH [1], rediscovered by J. CZIPSER and L. GÉHER [2], the space Lip 0 Y Lip 0 Y Lip_(0)Y\operatorname{Lip}_{0} YLip0Y is a P P PPP-space with respect to Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X. In this case, theorems 3 and 4 were proved in [5].
III. A topological space is called extremally disconnected if the closure of every open set is open. If Ω Ω Omega\OmegaΩ is a compact Hausdorff space, denote by C ( Ω ) C ( Ω ) C(Omega)C(\Omega)C(Ω) the Banach space of all continuous real functions defined on Ω Ω Omega\OmegaΩ with the sup-norm.
Let Ω Ω Omega\OmegaΩ be an extremally disconnected compact Hausdorff space, X X XXX a Banach space, Y Y YYY a subspace of X X XXX. By a theorem of L . L . L_(.)\mathrm{L}_{.}L.nachbin [6] L ( Y , C ( Ω ) ) L ( Y , C ( Ω ) ) L(Y,C(Omega))L(Y, C(\Omega))L(Y,C(Ω)) is a P P PPP-space in L ( X , C ( Ω ) ) L ( X , C ( Ω ) ) L(X,C(Omega))L(X, C(\Omega))L(X,C(Ω)), so that theorems 3 and 4 can be applied. Here L ( E , F ) L ( E , F ) L(E,F)L(E, F)L(E,F) denotes the space of all continuous linear operators between the Banach spaces E E EEE and F F FFF.
IV. Let ( X , d X , d X,dX, dX,d ) be a metric linear space, d d ddd being a invariant metric for translation, i.e. d ( x , y ) = d ( x − y , θ ^ ) d ( x , y ) = d ( x − y , θ ^ ) d(x,y)=d(x-y, hat(theta))d(x, y)=d(x-y, \hat{\theta})d(x,y)=d(x−y,θ^). Let
(12) S X ∘ = { f : f : X → R , sup { | f ( x ) | / d ( x , θ ) : x ≠ θ , x ∈ X } < ∞ f ( θ ) = 0 , f ( x + y ) ⩽ f ( x ) + f ( y ) , x , y ∈ X } (12) S X ∘ = { f : f : X → R , sup { | f ( x ) | / d ( x , θ ) : x ≠ θ , x ∈ X } < ∞ f ( θ ) = 0 , f ( x + y ) ⩽ f ( x ) + f ( y ) , x , y ∈ X } {:[(12)S_(X)^(@)={f:f:X rarrR","s u p{|f(x)|//d(x","theta):x!=theta","x in X} < oo],[f(theta)=0","f(x+y) <= f(x)+f(y)","x","y in X}]:}\begin{gather*} S_{X}^{\circ}=\{f: f: X \rightarrow \mathbf{R}, \sup \{|f(x)| / d(x, \theta): x \neq \theta, x \in X\}<\infty \tag{12}\\ f(\theta)=0, f(x+y) \leqslant f(x)+f(y), x, y \in X\} \end{gather*}(12)SX∘={f:f:X→R,sup{|f(x)|/d(x,θ):x≠θ,x∈X}<∞f(θ)=0,f(x+y)⩽f(x)+f(y),x,y∈X}
be the cone defined by g. PANTELIDIS [7].
For a subspace Y Y YYY of X X XXX, the cone S Y ∘ S Y ∘ S_(Y)^(@)S_{Y}^{\circ}SY∘ is defined in a similar way.
It was proved in [5], that S X ˙ ∘ S X ˙ ∘ S_(X^(˙))^(@)S_{\dot{X}}^{\circ}SX˙∘ is a convex cone in Lip 0 X , S Y ˙ ∘ Lip 0 X , S Y ˙ ∘ Lip_(0)X,S_(Y^(˙))^(@)\operatorname{Lip}_{0} X, S_{\dot{Y}}^{\circ}Lip0X,SY˙∘ is a convex cone in Lip 0 Y Lip 0 Y Lip_(0)Y\operatorname{Lip}_{0} YLip0Y and S Y ∘ S Y ∘ S_(Y)^(@)S_{\mathrm{Y}}^{\circ}SY∘ is a P P PPP-cone.
Let
(13)
X S = S X ∘ − S X ∘ , X S = S X ∘ − S X ∘ , X_(S)=S_(X)^(@)-S_(X)^(@),X_{S}=S_{X}^{\circ}-S_{X}^{\circ},XS=SX∘−SX∘,
be the linear space generated by the cone S X ∘ S X ∘ S_(X)^(@)S_{X}^{\circ}SX∘. In this case, theorem 1 and theorem 2 were proved in [5].
V . If X X XXX is a normed linear space, Y Y YYY a nonvoid convex subset of X X XXX such that θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y, put
(14) C X = { f : f ∈ Lip o X , f is convex } (15) C Y = { h : h ∈ Lip o Y . h is convex } . (14) C X = f : f ∈ Lip o X , f  is convex  (15) C Y = h : h ∈ Lip o Y . h  is convex  . {:[(14)C_(X)={f:f inLip_(o)X,f" is convex "}],[(15)C_(Y)={h:h inLip_(o)Y.h" is convex "}.]:}\begin{align*} & C_{X}=\left\{f: f \in \operatorname{Lip}_{o} X, f \text { is convex }\right\} \tag{14}\\ & C_{Y}=\left\{h: h \in \operatorname{Lip}_{o} Y . h \text { is convex }\right\} . \tag{15} \end{align*}(14)CX={f:f∈LipoX,f is convex }(15)CY={h:h∈LipoY.h is convex }.
Then C Y C Y C_(Y)C_{Y}CY is a P P PPP-cone and theorem 1 and theorem 2 can be applied.
5. In this section we intend to study the relation between the extremal elements of the unit ball of M Y M Y M_(Y)M_{Y}MY and the faces of the unit ball of M X M X M_(X)M_{X}MX (the notation are as in section 1.).
If ( E , ‖ ‖ E , ‖ ‖ E,||||E,\| \|E,‖‖ ) is a normed space, denote by B E B E B_(E)B_{E}BE and S E S E S_(E)S_{E}SE the unit ball, respectively the unit sphere of E E EEE, i.e.
(16) B E = { x ∈ E : ‖ x ‖ ⩽ 1 } S E = { x ∈ E : ‖ x ‖ = 1 } (16) B E = { x ∈ E : ‖ x ‖ ⩽ 1 } S E = { x ∈ E : ‖ x ‖ = 1 } {:[(16)B_(E)={x in E:||x|| <= 1}],[S_(E)={x in E:||x||=1}]:}\begin{align*} & B_{E}=\{x \in E:\|x\| \leqslant 1\} \tag{16}\\ & S_{E}=\{x \in E:\|x\|=1\} \end{align*}(16)BE={x∈E:‖x‖⩽1}SE={x∈E:‖x‖=1}
An extremal element of a convex set C C CCC in a linear space E E EEE is an element x ∈ C x ∈ C x in Cx \in Cx∈C such that λ x 1 + ( 1 − λ ) x 2 = x λ x 1 + ( 1 − λ ) x 2 = x lambdax_(1)+(1-lambda)x_(2)=x\lambda x_{1}+(1-\lambda) x_{2}=xλx1+(1−λ)x2=x for x 1 , x 2 ∈ C x 1 , x 2 ∈ C x_(1),x_(2)in Cx_{1}, x_{2} \in Cx1,x2∈C and λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) implies x 1 = x = x 2 x 1 = x = x 2 x_(1)=x=x_(2)x_{1}=x=x_{2}x1=x=x2.
A face of the unit ball B E B E B_(E)B_{E}BE is a convex subset F F FFF of S E S E S_(E)S_{E}SE such that λ x 1 + ( 1 − λ ) x 2 ∈ F λ x 1 + ( 1 − λ ) x 2 ∈ F lambdax_(1)+(1-lambda)x_(2)in F\lambda x_{1}+(1-\lambda) x_{2} \in Fλx1+(1−λ)x2∈F for x 1 , x 2 ∈ B E x 1 , x 2 ∈ B E x_(1),x_(2)inB_(E)x_{1}, x_{2} \in B_{E}x1,x2∈BE and λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) implies that x 1 , x 2 ∈ F x 1 , x 2 ∈ F x_(1),x_(2)in Fx_{1}, x_{2} \in Fx1,x2∈F. Obviously, a face which contain exactly one element is an extremal element of B E B E B_(E)B_{E}BE.
For h ∈ M Y h ∈ M Y h inM_(Y)h \in M_{Y}h∈MY, denote by
(17) P Y ( h ) = { f : f ∈ M X , f | Y = h , ‖ f ‖ X = ‖ h ‖ Y } , (17) P Y ( h ) = f : f ∈ M X , f Y = h , ‖ f ‖ X = ‖ h ‖ Y , {:(17)P_(Y)(h)={f:f inM_(X),f|_(Y)=h,||f||_(X)=||h||_(Y)}",":}\begin{equation*} P_{Y}(h)=\left\{f: f \in M_{X},\left.f\right|_{Y}=h,\|f\|_{X}=\|h\|_{Y}\right\}, \tag{17} \end{equation*}(17)PY(h)={f:f∈MX,f|Y=h,‖f‖X=‖h‖Y},
the set of all extension of h h hhh.
Then P Y ( h ) P Y ( h ) P_(Y)(h)P_{Y}(h)PY(h) is a nonvoid, convex, bounded and closed subset of M X M X M_(X)M_{X}MX.
THEOREM 5. An element h ∈ B M Y h ∈ B M Y h inB_(M_(Y))h \in B_{M_{Y}}h∈BMY is an extremal element of B M Y B M Y B_(M_(Y))B_{M_{Y}}BMY if and only if P Y ( h ) P Y ( h ) P_(Y)(h)P_{Y}(h)PY(h) is a face of B M X B M X B_(M_(X))B_{M_{X}}BMX.
Proof. Suppose h h hhh is an extremal element of B M Y B M Y B_(M_(Y))B_{M_{Y}}BMY. Let λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) and f 1 , f 2 ∈ B M X f 1 , f 2 ∈ B M X f_(1),f_(2)inB_(M_(X))f_{1}, f_{2} \in B_{M_{X}}f1,f2∈BMX be such that λ f 1 + ( 1 − λ ) f 2 ∈ P Y ( h ) λ f 1 + ( 1 − λ ) f 2 ∈ P Y ( h ) lambdaf_(1)+(1-lambda)f_(2)inP_(Y)(h)\lambda f_{1}+(1-\lambda) f_{2} \in P_{Y}(h)λf1+(1−λ)f2∈PY(h). Then λ f 1 | Y + ( 1 − λ ) f 2 | Y = h λ f 1 Y + ( 1 − λ ) f 2 Y = h lambdaf_(1)|_(Y)+(1-lambda)f_(2)|_(Y)=h\left.\lambda f_{1}\right|_{Y}+\left.(1-\lambda) f_{2}\right|_{Y} =hλf1|Y+(1−λ)f2|Y=h, and since h h hhh is an extremal element of B M Y B M Y B_(M_(Y))B_{M_{Y}}BMY, it follows that f 1 | Y = f 2 | Y == h f 1 Y = f 2 Y == h f_(1)|_(Y)=f_(2)|_(Y)==h\left.f_{1}\right|_{Y}=\left.f_{2}\right|_{Y}= =hf1|Y=f2|Y==h, so that ‖ f 1 | Y ‖ Y = ‖ f 2 | Y ‖ Y = ‖ h ‖ Y = 1 f 1 Y Y = f 2 Y Y = ‖ h ‖ Y = 1 ||f_(1)|_(Y)||_(Y)=||f_(2)|_(Y)||_(Y)=||h||_(Y)=1\left\|\left.f_{1}\right|_{Y}\right\|_{Y}=\left\|\left.f_{2}\right|_{Y}\right\|_{Y}=\|h\|_{Y}=1‖f1|Y‖Y=‖f2|Y‖Y=‖h‖Y=1. Since the norms ‖ ‖ X ‖ X ||||_(X):}\left\|\|_{X}\right.‖‖X and ‖ ‖ Y ‖ ‖ Y ||||_(Y)\| \|_{Y}‖‖Y are supposed compatible (see definition 1.) it follows that ‖ f 1 ‖ X = ‖ f 2 ‖ X = 1 f 1 X = f 2 X = 1 ||f_(1)||_(X)=||f_(2)||_(X)=1\left\|f_{1}\right\|_{X}=\left\|f_{2}\right\|_{X}=1‖f1‖X=‖f2‖X=1. We proved that f 1 , f 2 ∈ P Y ( h ) f 1 , f 2 ∈ P Y ( h ) f_(1),f_(2)inP_(Y)(h)f_{1}, f_{2} \in P_{Y}(h)f1,f2∈PY(h) which shows that P Y ( h ) P Y ( h ) P_(Y)(h)P_{Y}(h)PY(h) is a face of B M X B M X B_(M_(X))B_{M_{X}}BMX.
Conversely, suppose h h hhh is not an extremal element of B M Y B M Y B_(M_(Y))B_{M_{Y}}BMY. Then there exist two elements h 1 , h 2 ∈ B M Y , h 1 ≠ h , h 2 ≠ h h 1 , h 2 ∈ B M Y , h 1 ≠ h , h 2 ≠ h h_(1),h_(2)inB_(M_(Y)),h_(1)!=h,h_(2)!=hh_{1}, h_{2} \in B_{M_{Y}}, h_{1} \neq h, h_{2} \neq hh1,h2∈BMY,h1≠h,h2≠h and λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) such that λ h 1 + ( 1 − λ ) h 2 = h λ h 1 + ( 1 − λ ) h 2 = h lambdah_(1)+(1-lambda)h_(2)=h\lambda h_{1}+(1-\lambda) h_{2}=hλh1+(1−λ)h2=h. Let f 1 ′ ∈ P Y ( h 1 ) f 1 ′ ∈ P Y h 1 f_(1)^(')inP_(Y)(h_(1))f_{1}^{\prime} \in P_{Y}\left(h_{1}\right)f1′∈PY(h1) and f 2 ′ ∈ P Y ( h 2 ) f 2 ′ ∈ P Y h 2 f_(2)^(')inP_(Y)(h_(2))f_{2}^{\prime} \in P_{Y}\left(h_{2}\right)f2′∈PY(h2). Then λ f 1 ′ | Y + + ( 1 − λ ) f 2 ′ | Y = h λ f 1 ′ Y + + ( 1 − λ ) f 2 ′ Y = h lambdaf_(1)^(')|_(Y)++(1-lambda)f_(2)^(')|_(Y)=h\left.\lambda f_{1}^{\prime}\right|_{Y}+ +\left.(1-\lambda) f_{2}^{\prime}\right|_{Y}=hλf1′|Y++(1−λ)f2′|Y=h and 1 = ‖ λ f 1 ′ | Y + ( 1 − λ ) f 2 ′ | Y ‖ Y ⩽ ‖ λ f 1 ′ + ( 1 − λ ) f 2 ′ ‖ X ⩽ 1 1 = λ f 1 ′ Y + ( 1 − λ ) f 2 ′ Y Y ⩽ λ f 1 ′ + ( 1 − λ ) f 2 ′ X ⩽ 1 1=|| lambdaf_(1)^(')|_(Y)+(1-lambda)f_(2)^(')|_(Y)||_(Y) <= ||lambdaf_(1)^(')+(1-lambda)f_(2)^(')||_(X) <= 11=\left\|\left.\lambda f_{1}^{\prime}\right|_{Y}+\left.(1-\lambda) f_{2}^{\prime}\right|_{Y}\right\|_{Y} \leqslant\left\|\lambda f_{1}^{\prime}+(1-\lambda) f_{2}^{\prime}\right\|_{X} \leqslant 11=‖λf1′|Y+(1−λ)f2′|Y‖Y⩽‖λf1′+(1−λ)f2′‖X⩽1, so that λ f 1 ′ + ( 1 − λ ) f 2 ′ ∈ P Y ( h ) λ f 1 ′ + ( 1 − λ ) f 2 ′ ∈ P Y ( h ) lambdaf_(1)^(')+(1-lambda)f_(2)^(')inP_(Y)(h)\lambda f_{1}^{\prime}+(1-\lambda) f_{2}^{\prime} \in P_{Y}(h)λf1′+(1−λ)f2′∈PY(h). But f 1 ′ f 1 ′ f_(1)^(')f_{1}^{\prime}f1′ and f 2 ′ f 2 ′ f_(2)^(')f_{2}^{\prime}f2′ do not belong to P Y ( h ) P Y ( h ) P_(Y)(h)P_{Y}(h)PY(h) since f 1 ′ | Y ≠ h f 1 ′ Y ≠ h f_(1)^(')|_(Y)!=h\left.f_{1}^{\prime}\right|_{Y} \neq hf1′|Y≠h and f 2 ′ | Y ≠ h f 2 ′ Y ≠ h f_(2)^(')|_(Y)!=h\left.f_{2}^{\prime}\right|_{Y} \neq hf2′|Y≠h, so that P Y ( h ) P Y ( h ) P_(Y)(h)P_{Y}(h)PY(h) is not a face of B M X B M X B_(M_(X))B_{M_{X}}BMX.
Suppose now, Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X and Lip 0 Y Lip 0 Y Lip_(0)Y\operatorname{Lip}_{0} YLip0Y be as in the case II. from section 4. If h ∈ Lip 0 Y h ∈ Lip 0 Y h inLip_(0)Yh \in \operatorname{Lip}_{0} Yh∈Lip0Y, then the functions
(18) f 1 ( x ) = inf { h ( y ) + ‖ h ‖ Y d ( x , y ) : y ∈ Y } , x ∈ X f 2 ( x ) = sup { h ( y ) − ‖ h ‖ Y d ( x , y ) : y ∈ Y } , x ∈ X (18) f 1 ( x ) = inf h ( y ) + ‖ h ‖ Y d ( x , y ) : y ∈ Y , x ∈ X f 2 ( x ) = sup h ( y ) − ‖ h ‖ Y d ( x , y ) : y ∈ Y , x ∈ X {:[(18)f_(1)(x)=i n f{h(y)+||h||_(Y)d(x,y):y in Y}","x in X],[f_(2)(x)=s u p{h(y)-||h||_(Y)d(x,y):y in Y}","x in X]:}\begin{align*} & f_{1}(x)=\inf \left\{h(y)+\|h\|_{Y} d(x, y): y \in Y\right\}, x \in X \tag{18}\\ & f_{2}(x)=\sup \left\{h(y)-\|h\|_{Y} d(x, y): y \in Y\right\}, x \in X \end{align*}(18)f1(x)=inf{h(y)+‖h‖Yd(x,y):y∈Y},x∈Xf2(x)=sup{h(y)−‖h‖Yd(x,y):y∈Y},x∈X
are extensions of h h hhh (see [4]) and further, they are extremal elements of the set P Y ( h ) P Y ( h ) P_(Y)(h)P_{Y}(h)PY(h).
Indeed, one can prove that
(19) f 2 ( x ) ⩽ f ( x ) ⩽ f 1 ( x ) , x ∈ X (19) f 2 ( x ) ⩽ f ( x ) ⩽ f 1 ( x ) , x ∈ X {:(19)f_(2)(x) <= f(x) <= f_(1)(x)","quad x in X:}\begin{equation*} f_{2}(x) \leqslant f(x) \leqslant f_{1}(x), \quad x \in X \tag{19} \end{equation*}(19)f2(x)⩽f(x)⩽f1(x),x∈X
for all f ∈ P ( h ) Y f ∈ P ( h ) Y f in P(h)_(Y)f \in P(h)_{Y}f∈P(h)Y (see [5]). If φ , ψ ∈ P Y ( h ) φ , ψ ∈ P Y ( h ) varphi,psi inP_(Y)(h)\varphi, \psi \in P_{Y}(h)φ,ψ∈PY(h) and λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) are such that λ φ + ( 1 − λ ) ψ = f 1 λ φ + ( 1 − λ ) ψ = f 1 lambda varphi+(1-lambda)psi=f_(1)\lambda \varphi+(1-\lambda) \psi=f_{1}λφ+(1−λ)ψ=f1, then
(20) 0 ⩽ λ ( f 1 − φ ) = ( 1 − λ ) ( ψ − f 1 ) (20) 0 ⩽ λ f 1 − φ = ( 1 − λ ) ψ − f 1 {:(20)0 <= lambda(f_(1)-varphi)=(1-lambda)(psi-f_(1)):}\begin{equation*} 0 \leqslant \lambda\left(f_{1}-\varphi\right)=(1-\lambda)\left(\psi-f_{1}\right) \tag{20} \end{equation*}(20)0⩽λ(f1−φ)=(1−λ)(ψ−f1)
and by (19) it follows φ = ψ = f 1 φ = ψ = f 1 varphi=psi=f_(1)\varphi=\psi=f_{1}φ=ψ=f1, so that f 1 f 1 f_(1)f_{1}f1 is an extremal element of P Y ( h ) P Y ( h ) P_(Y)(h)P_{Y}(h)PY(h). In a similar way one can show that f 2 f 2 f_(2)f_{2}f2 is an extremal element of P Y ( h ) P Y ( h ) P_(Y)(h)P_{Y}(h)PY(h).
Since, by theorem 5 , h 5 , h 5,h5, h5,h is an extremal element of B Lip 0 Y B Lip  0 Y B_("Lip "_(0)Y)B_{\text {Lip }{ }_{0} Y}BLip 0Y if and only if P Y ( h ) P Y ( h ) P_(Y)(h)P_{Y}(h)PY(h) is a face of B Lip , X B Lip  , X B_("Lip ",X)B_{\text {Lip }, X}BLip ,X, and an extremal element of a face of the unit ball of a normed linear space is an extremal element of the ball; it follows:
If h h hhh is an extremal element of the unit ball of Lip 0 Y Lip 0 Y Lip_(0)Y\operatorname{Lip}_{0} YLip0Y, then the functions f 1 , f 2 f 1 , f 2 f_(1),f_(2)f_{1}, f_{2}f1,f2 defined by the formulae (18) are extremal elements of the unit ball of Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X.

REFERENCES

[1] Banach, S., Wstep to teovii funkji rzeczywistych, Warszawa-Wroclaw, 1951.
[2] Czipser, J., Géher. L., Extension of function satisfying a Lipschitz condition, Acta Math. Acad. Sci. Hungar., 6, 213-220, (1955).
[3] Kolumbán, I., Ob edinstvenosti prodolzenia lineinyh funktionalov, Mathematica (Cluj), 4 (27), 267-270, (1962).
[4] Mustăfa, C., Asupra unor subspafii cebîseviene din spajiul normat al functiilor lipschitziene, Rev. Anal. Num. Teor. Aprox., 2, 1, 81-87, (1973).
[5] - , O proprietate de monotonie a operatorului de cea mai bună aproximare in spativl functiilor lipschitziene, Rev. Anal. Num. Teor. Aprox., 3, 2, 153-160, (1974).
[6] Nach bin, L., A theorem of the Hahn-Banach type for linear transformation, Trans. Anter. Math. Soc., 68, 28-46, (1950).
[7] Pantelidis, G., Approximationstheorie für metrische lineave Räume, Math. Ann., 184, 30-48, (1969).
[8] Phelps, R. R., Uniqueness of Hahn-Banach extension and unique best approximation, Trans. Amer. Math. Soc., 25, 238-255, (1960).
[9] Singer, I., Cea mai bunä aproximare in spafii vectoriale normate prin elemente din subspafii vectoriale, E.d. Acad. R.S.R., Bucureşti, 1967.
Received 16. II. 1977.
Universitatea ,,Babes-Bolyai"
Cluj-Napoca
Institutul de matematică
1977

Related Posts