On the metric projection and the quotient mapping

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

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

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C. Mustăţa, On the metric projection and the quotient mapping, Rev. Anal. Numér. Théor. Approx., 24 (1995) nos. 1-2, 191-199.

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Revue d’Analyse Numer. Theor. Approx.

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Publishing Romanian Academy

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https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art21

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2457-6794

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1995-Mustata-On the metric projection and the quotient mapping-Jnaat

ON THE METRIC PROJECTION AND THE QUOTIENT MAPPING

COSTICĂ MUSTĂTA

(Cluj-Napoca)

1. INTRODUCTION

For a linear space Z Z ZZZ and a nonvoid set U U UUU denote by Z U Z U Z^(U)Z^{U}ZU the linear space (with respect to the pointwise operations of addition and multiplication by real scalars) of all applications from U U UUU to Z Z ZZZ.
Let Y , X Y , X Y,XY, XY,X be two nonvoid sets such that Y ⊆ X Y ⊆ X Y sube XY \subseteq XY⊆X and let ( N Y , ‖ ⋅ ‖ Y ) N Y , ‖ ⋅ ‖ Y (N_(Y),||*||_(Y))\left(N_{Y},\|\cdot\|_{Y}\right)(NY,‖⋅‖Y) and ( N X , ‖ ⋅ ‖ X ) N X , ‖ ⋅ ‖ X (N_(X),||*||_(X))\left(N_{X},\|\cdot\|_{X}\right)(NX,‖⋅‖X) be two normed spaces contained in Z Y Z Y Z^(Y)Z^{\mathrm{Y}}ZY and Z X Z X Z^(X)Z^{X}ZX respectively. Suppose that for every F ∈ N X F ∈ N X F inN_(X)F \in N_{X}F∈NX the restriction F | Y F Y F|_(Y)\left.F\right|_{Y}F|Y of F F FFF to Y Y YYY belongs to N Y N Y N_(Y)N_{Y}NY.
DEFINITON 1. We say that the norms ‖ ⋅ ‖ Y ‖ ⋅ ‖ Y ||*||_(Y)\|\cdot\|_{Y}‖⋅‖Y and ‖ ⋅ ‖ Y ‖ ⋅ ‖ Y ||*||_(Y)\|\cdot\|_{Y}‖⋅‖Y are compatible 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} \leq\|F\|_{X} \tag{1} \end{equation*}(1)‖F|Y‖Y≤‖F‖X
for every F ∈ N X F ∈ N X F inN_(X)F \in N_{X}F∈NX.
In the following the norms ‖ ⋅ ‖ Y ‖ ⋅ ‖ Y ||*||_(Y)\|\cdot\|_{Y}‖⋅‖Y and ‖ ⋅ ‖ X ‖ ⋅ ‖ X ||*||_(X)\|\cdot\|_{X}‖⋅‖X will be supposed always compatible.
A nonvoid subset K K KKK of a normed space ( X , ‖ ⋅ ‖ ) ( X , ‖ ⋅ ‖ ) (X,||*||)(X,\|\cdot\|)(X,‖⋅‖) is called a cone if:
a) u + v ∈ K u + v ∈ K u+v in Ku+v \in Ku+v∈K, and
b) λ ⋅ u ∈ K λ ⋅ u ∈ K lambda*u in K\lambda \cdot u \in Kλ⋅u∈K,
for all u , v ∈ K u , v ∈ K u,v in Ku, v \in Ku,v∈K and λ ∈ R , λ ≥ 0 λ ∈ R , λ ≥ 0 lambda in R,lambda >= 0\lambda \in R, \lambda \geq 0λ∈R,λ≥0.
Definition 2. Let K Y K Y K_(Y)K_{Y}KY and K X K X K_(X)K_{X}KX be two cones in the linear spaces N Y N Y N_(Y)N_{Y}NY and N X N X N_(X)N_{X}NX, respectively. We say that the cone K Y K Y K_(Y)K_{Y}KY has the norm preserving extension ((NPE) in short) property with respect to K X K X K_(X)K_{X}KX if F | Y ∈ K Y F Y ∈ K Y F|_(Y)inK_(Y)\left.F\right|_{Y} \in K_{Y}F|Y∈KY, for every F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX and every f ∈ K Y f ∈ K Y f inK_(Y)f \in K_{Y}f∈KY has a norm perserving exiension F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX (i.e. F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f and ‖ F ‖ X = ‖ f ‖ Y ‖ F ‖ X = ‖ f ‖ Y ||F||_(X)=||f||_(Y)\|F\|_{X}=\|f\|_{Y}‖F‖X=‖f‖Y ).

If K Y K Y K_(Y)K_{Y}KY has the (NPE)-property with respect to K X K X K_(X)K_{X}KX let

(2) E ( f ) := { F ∈ K X : F | Y = f and ‖ F ‖ X = ‖ f ‖ Y } (2) E ( f ) := F ∈ K X : F Y = f  and  ‖ F ‖ X = ‖ f ‖ Y {:(2)E(f):={F inK_(X):F|_(Y)=f" and "||F||_(X)=||f||_(Y)}:}\begin{equation*} \mathscr{E}(f):=\left\{F \in K_{X}:\left.F\right|_{Y}=f \text { and }\|F\|_{X}=\|f\|_{Y}\right\} \tag{2} \end{equation*}(2)E(f):={F∈KX:F|Y=f and ‖F‖X=‖f‖Y}
denote the set of all (NPE) extensions of the function f ∈ K Y f ∈ K Y f inK_(Y)f \in K_{Y}f∈KY. Let also
(3) M X := K X − K X (3) M X := K X − K X {:(3)M_(X):=K_(X)-K_(X):}\begin{equation*} M_{X}:=K_{X}-K_{X} \tag{3} \end{equation*}(3)MX:=KX−KX
denote the liniar subspace of N X N X N_(X)N_{X}NX generated by the cone K X K X K_(X)K_{X}KX and let
(4) Y ⊥ := { G ∈ M X : G | Y = 0 } (4) Y ⊥ := G ∈ M X : G Y = 0 {:(4)Y^(_|_):={G inM_(X):G|_(Y)=0}:}\begin{equation*} Y^{\perp}:=\left\{G \in M_{X}:\left.G\right|_{Y}=0\right\} \tag{4} \end{equation*}(4)Y⊥:={G∈MX:G|Y=0}
be the annihilator of the set Y Y YYY in the linear space M X M X M_(X)M_{X}MX (remember that we have supposed Y ⊆ X Y ⊆ X Y sube XY \subseteq XY⊆X ).
Obviously that Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is a closed linear sunspace of M X M X M_(X)M_{X}MX.
A linear subspace V V VVV of normed space U U UUU is called proximinal if for every u ∈ U u ∈ U u in Uu \in Uu∈U there exists v 0 ∈ V v 0 ∈ V v_(0)in Vv_{0} \in Vv0∈V such that
‖ u − v 0 ‖ = d ( u , V ) := inf { ‖ u − v ‖ : v ∈ V } u − v 0 = d ( u , V ) := inf { ‖ u − v ‖ : v ∈ V } ||u-v_(0)||=d(u,V):=i n f{||u-v||:v in V}\left\|u-v_{0}\right\|=d(u, V):=\inf \{\|u-v\|: v \in V\}‖u−v0‖=d(u,V):=inf{‖u−v‖:v∈V}
The set
P V ( u ) := { v ∈ V : ‖ u − v ‖ = d ( u , V ) } P V ( u ) := { v ∈ V : ‖ u − v ‖ = d ( u , V ) } P_(V)(u):={v in V:||u-v||=d(u,V)}P_{V}(u):=\{v \in V:\|u-v\|=d(u, V)\}PV(u):={v∈V:‖u−v‖=d(u,V)}
is called the set of elements of best approximation for u u uuu by elements in V V VVV.
If P v ( w ) ≠ ∅ P v ( w ) ≠ ∅ P_(v)(w)!=O/P_{v}(w) \neq \varnothingPv(w)≠∅ for all w w www in a subset W W WWW of U U UUU then the subspace V V VVV is called W W WWW-proximinal.
THEOREM 1. If the cone K Y K Y K_(Y)K_{Y}KY has the (NPE)- property with respect to the cone K X K X K_(X)K_{X}KX then:
(a) The equality
(5)
d ( F , Y ⊥ ) = ‖ F | Y ‖ Y d F , Y ⊥ = F Y Y d(F,Y^(_|_))=||F|_(Y)||_(Y)d\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\|_{Y}d(F,Y⊥)=‖F|Y‖Y
is true for all F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX.
(b) The inclusion
(6) F − E ( F | Y ) ⊆ P Y ⊥ ( F ) (6) F − E F Y ⊆ P Y ⊥ ( F ) {:(6)F-E(F|_(Y))subeP_(Y)_|_(F):}\begin{equation*} F-\mathscr{E}\left(\left.F\right|_{Y}\right) \subseteq P_{Y} \perp(F) \tag{6} \end{equation*}(6)F−E(F|Y)⊆PY⊥(F)
holds for every F ∈ K X ′ F ∈ K X ′ F inK_(X^('))F \in K_{X^{\prime}}F∈KX′
(c) If furthermore Y ⊥ ⊆ K X Y ⊥ ⊆ K X Y^(_|_)subeK_(X)Y^{\perp} \subseteq K_{X}Y⊥⊆KX, then
F − C ( F | Y ) = P Y ⊥ ( F ) F − C F Y = P Y ⊥ ( F ) F-C(F|_(Y))=P_(Y)_|_(F)F-\mathscr{C}\left(\left.F\right|_{Y}\right)=P_{Y} \perp(F)F−C(F|Y)=PY⊥(F)
for all F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX.
Proof. Let F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX. For G ∈ Y ⊥ G ∈ Y ⊥ G inY^(_|_)G \in Y^{\perp}G∈Y⊥ taking into account that the norms ‖ ⋅ ‖ Y ‖ ⋅ ‖ Y ||*||_(Y)\|\cdot\|_{Y}‖⋅‖Y were supposed compatible in the sense of Definition 1 , we have
‖ F | Y ‖ Y = ‖ F | Y − G | Y ‖ Y = ‖ ( F − G ) | Y ‖ Y ≤ ‖ F − G ‖ X F Y Y = F Y − G Y Y = ( F − G ) Y Y ≤ ‖ F − G ‖ X ||F|_(Y)||_(Y)=||F|_(Y)-G|_(Y)||_(Y)=||(F-G)|_(Y)||_(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|_{Y}\right\|_{Y} \leq\|F-G\|_{X}‖F|Y‖Y=‖F|Y−G|Y‖Y=‖(F−G)|Y‖Y≤‖F−G‖X
It follows that
‖ 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} \leq d\left(F, Y^{\perp}\right)‖F|Y‖Y≤d(F,Y⊥)
On the other hand, because F − H ∈ Y ⊥ F − H ∈ Y ⊥ F-H inY^(_|_)F-H \in Y^{\perp}F−H∈Y⊥ for H ∈ E ( F | Y ) H ∈ E F Y H inE(F|_(Y))H \in \mathscr{E}\left(\left.F\right|_{Y}\right)H∈E(F|Y), we have
‖ F | Y ‖ Y = ‖ H ‖ X = ‖ F − ( F − H ) ‖ X X ≥ inf { ‖ F − G ‖ X : G ∈ Y ⊥ } = d ( F , Y ⊥ ) F Y Y = ‖ H ‖ X = ‖ F − ( F − H ) ‖ X X ≥ inf ‖ F − G ‖ X : G ∈ Y ⊥ = d F , Y ⊥ ||F|_(Y)||_(Y)=||H||_(X)=||F-(F-H)||X_(X) >= i n f{||F-G||_(X):G inY^(_|_)}=d(F,Y^(_|_))\left\|\left.F\right|_{Y}\right\|_{Y}=\|H\|_{X}=\|F-(F-H)\| X_{X} \geq \inf \left\{\|F-G\|_{X}: G \in Y^{\perp}\right\}=d\left(F, Y^{\perp}\right)‖F|Y‖Y=‖H‖X=‖F−(F−H)‖XX≥inf{‖F−G‖X:G∈Y⊥}=d(F,Y⊥)
Combining the obtained inequalities one can write
d ( F , Y ⊥ ) ≤ ‖ F − ( F − H ) ‖ X X = ‖ F | Y ‖ Y ≤ d ( F , Y ⊥ ) d F , Y ⊥ ≤ ‖ F − ( F − H ) ‖ X X = F Y Y ≤ d F , Y ⊥ d(F,Y^(_|_)) <= ||F-(F-H)||X_(X)=||F|_(Y)||_(Y) <= d(F,Y^(_|_))d\left(F, Y^{\perp}\right) \leq\|F-(F-H)\| X_{X}=\left\|\left.F\right|_{Y}\right\|_{Y} \leq d\left(F, Y^{\perp}\right)d(F,Y⊥)≤‖F−(F−H)‖XX=‖F|Y‖Y≤d(F,Y⊥)
It follows
‖ F − ( F − H ) ‖ X = d ( F , Y ⊥ ) = ‖ F | Y ‖ , ‖ F − ( F − H ) ‖ X = d F , Y ⊥ = F Y , ||F-(F-H)||_(X)=d(F,Y^(_|_))=||F|_(Y)||,\|F-(F-H)\|_{X}=d\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\|,‖F−(F−H)‖X=d(F,Y⊥)=‖F|Y‖,
for every H ∈ E ( F | Y ) H ∈ E F Y H inE(F|_(Y))H \in \mathscr{E}\left(\left.F\right|_{Y}\right)H∈E(F|Y), proving formula (5) and inclusion (6).
In order to prove equality (7) suppose that Y ⊥ ⊆ K X Y ⊥ ⊆ K X Y^(_|_)subeK_(X)Y^{\perp} \subseteq K_{X}Y⊥⊆KX and let G G GGG be an arbitrary element of P Y ⊥ ( F ) P Y ⊥ ( F ) P_(Y)_|_(F)P_{Y} \perp(F)PY⊥(F). Since G | Y = 0 G Y = 0 G|_(Y)=0\left.G\right|_{Y}=0G|Y=0 and the norms ‖ ⋅ ‖ Y ‖ ⋅ ‖ Y ||*||_(Y)\|\cdot\|_{Y}‖⋅‖Y and ‖ ⋅ ‖ X ‖ ⋅ ‖ X ||*||_(X)\|\cdot\|_{X}‖⋅‖X are compatible, it follows that ( 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 and, by (5),
‖ F − G ‖ X = d ( F , Y ⊥ ) = ‖ F | Y ‖ Y ‖ F − G ‖ X = d F , Y ⊥ = F Y Y ||F-G||_(X)=d(F,Y^(_|_))=||F|_(Y)||_(Y)\|F-G\|_{X}=d\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\|_{Y}‖F−G‖X=d(F,Y⊥)=‖F|Y‖Y
showing that F − G F − G F-GF-GF−G is a (NPE) extension of F | Y F Y F|_(Y)\left.F\right|_{Y}F|Y. To prove that F − G ∈ E ( F | Y ) F − G ∈ E F Y F-G inE(F|_(Y))F-G \in \mathscr{E}\left(\left.F\right|_{Y}\right)F−G∈E(F|Y) it remains to show that F − G ∈ K X F − G ∈ K X F-G inK_(X)F-G \in K_{X}F−G∈KX (see(2)). But G ∈ Y ⊥ G ∈ Y ⊥ G inY^(_|_)G \in Y^{\perp}G∈Y⊥ implies − G ∈ Y ⊥ − G ∈ Y ⊥ -G inY^(_|_)-G \in Y^{\perp}−G∈Y⊥ and, by hypothesis, Y ⊥ ⊆ K X Y ⊥ ⊆ K X Y^(_|_)subeK_(X)Y^{\perp} \subseteq K_{X}Y⊥⊆KX implying − G ∈ K X − G ∈ K X -G inK_(X)-G \in K_{X}−G∈KX, and, since K X K X K_(X)K_{X}KX is acone, F − G = F + ( − G ) ∈ K X F − G = F + ( − G ) ∈ K X F-G=F+(-G)inK_(X)F-G=F+(-G) \in K_{X}F−G=F+(−G)∈KX,
showing that F − G ∈ E ( F | Y ) F − G ∈ E F Y F-G inE(F|_(Y))F-G \in \mathscr{E}\left(\left.F\right|_{Y}\right)F−G∈E(F|Y). This last relation is equivalent to G ∈ F − E ( F | Y ) G ∈ F − E F Y G in F-E(F|_(Y))G \in F-\mathscr{E}\left(\left.F\right|_{Y}\right)G∈F−E(F|Y) for every G ∈ P Y ⊥ G ∈ P Y ⊥ G inP_(Y^(_|_))G \in P_{Y^{\perp}}G∈PY⊥ (F), i.e. P Y ⊥ ( F ) ⊆ F − E ( F | Y ) P Y ⊥ ( F ) ⊆ F − E F Y P_(Y^(_|_))(F)sube F-E(F|_(Y))P_{Y^{\perp}}(F) \subseteq F-\mathscr{E}\left(\left.F\right|_{Y}\right)PY⊥(F)⊆F−E(F|Y), which together with inclusion (6), prove equality (7).
Taking K Y = N Y K Y = N Y K_(Y)=N_(Y)K_{Y}=N_{Y}KY=NY and K X = N X K X = N X K_(X)=N_(X)K_{X}=N_{X}KX=NX one obtains:
COROLLARY 1. If N Y N Y N_(Y)N_{Y}NY has the (NPE)-propety with respect to N X N X N_(X)N_{X}NX then
(a') The subspace Y ⊥ = { G ∈ N X : G | Y = 0 } Y ⊥ = G ∈ N X : G Y = 0 Y^(_|_)={G inN_(X):G|_(Y)=0}Y^{\perp}=\left\{G \in N_{X}:\left.G\right|_{Y}=0\right\}Y⊥={G∈NX:G|Y=0} is proximal in N X N X N_(X)N_{X}NX and
( ′ ) d ( F , Y ⊥ ) = ‖ F | Y ‖ Y ( ′ ) d F , Y ⊥ = F Y Y {:('")"d(F,Y^(_|_))=||F|_(Y)||_(Y):}\begin{equation*} d\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\|_{Y} \tag{$\prime$} \end{equation*}(′)d(F,Y⊥)=‖F|Y‖Y
for every F ∈ N X F ∈ N X F inN_(X)F \in N_{X}F∈NX.
(b') The equality
(7')
P Y ⊥ ( F ) = F − E ( F | Y ) P Y ⊥ ( F ) = F − E F Y P_(Y^(_|_))(F)=F-E(F|_(Y))P_{Y^{\perp}}(F)=F-\mathscr{E}\left(\left.F\right|_{Y}\right)PY⊥(F)=F−E(F|Y)
holds for every F ∈ N X F ∈ N X F inN_(X)F \in N_{X}F∈NX.
Proof. It suffices to prove equality (7'). By Theorem 1 it follows that F − E ( F | Y ) ⊆ P Y ⊥ ( F ) F − E F Y ⊆ P Y ⊥ ( F ) F-E(F|_(Y))subeP_(Y^(_|_))(F)F-\mathscr{E}\left(\left.F\right|_{Y}\right) \subseteq P_{Y^{\perp}}(F)F−E(F|Y)⊆PY⊥(F). If F ∈ N X F ∈ N X F inN_(X)F \in N_{X}F∈NX and G ∈ P Y ⊥ ( F ) G ∈ P Y ⊥ ( F ) G inP_(Y^(_|_))(F)G \in P_{Y^{\perp}}(F)G∈PY⊥(F) then
‖ F − G ‖ X = d ( F , Y ⊥ ) = ‖ F | Y ‖ Y ‖ F − G ‖ X = d F , Y ⊥ = F Y Y ||F-G||_(X)=d(F,Y^(_|_))=||F|_(Y)||_(Y)\|F-G\|_{X}=d\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\|_{Y}‖F−G‖X=d(F,Y⊥)=‖F|Y‖Y
Since ( F − G ) Y = F | Y ( F − G ) Y = F Y (F-G)_(Y)=F|_(Y)(F-G)_{Y}=\left.F\right|_{Y}(F−G)Y=F|Y it follows that F − G F − G F-GF-GF−G is a norm preserving extension F | Y F Y F|_(Y)\left.F\right|_{Y}F|Y. But K X = N X K X = N X K_(X)=N_(X)K_{X}=N_{X}KX=NX implies M X = N X M X = N X M_(X)=N_(X)M_{X}=N_{X}MX=NX so that F − G ∈ M X F − G ∈ M X F-G inM_(X)F-G \in M_{X}F−G∈MX showing that F − G ∈ C ( F | Y ) F − G ∈ C F Y F-G inC(F|_(Y))F-G \in \mathscr{C}\left(\left.F\right|_{Y}\right)F−G∈C(F|Y) or equivalently G ∈ F − C ( F | Y ) G ∈ F − C F Y G in F-C(F|_(Y))G \in F-\mathscr{C}\left(\left.F\right|_{Y}\right)G∈F−C(F|Y) for every G ∈ P Y ⊥ ( F ) G ∈ P Y ⊥ ( F ) G inP_(Y_(_|_))(F)G \in P_{Y_{\perp}}(F)G∈PY⊥(F). This proves the inclusion P Y ⊥ ( F ) ⊆ F − E ( F | Y ) P Y ⊥ ( F ) ⊆ F − E F Y P_(Y_(_|_))(F)sube F-E(F|_(Y))P_{Y_{\perp}}(F) \subseteq F-\mathscr{E}\left(\left.F\right|_{Y}\right)PY⊥(F)⊆F−E(F|Y) and equality (7').

EXAMPLES

1 ∘ 1 ∘ 1^(@)\mathbf{1}^{\circ}1∘. Let X = [ a , b ] ⊂ R X = [ a , b ] ⊂ R X=[a,b]sub RX=[a, b] \subset RX=[a,b]⊂R and Y = { a , b } Y = { a , b } Y={a,b}Y=\{a, b\}Y={a,b}. Take N X = C [ a , b ] N X = C [ a , b ] N_(X)=C[a,b]N_{X}=C[a, b]NX=C[a,b]-the space of all realvalued continuous on [ a , b ] [ a , b ] [a,b][a, b][a,b] with the sup-norm and N Y = C ( { a , b } ) N Y = C ( { a , b } ) N_(Y)=C({a,b})N_{Y}=C(\{a, b\})NY=C({a,b}).
Let K X K X K_(X)K_{X}KX be the cone
K X := { F ∈ C [ a , b ] : F ( a ) = F ( b ) ≥ 0 } K X := { F ∈ C [ a , b ] : F ( a ) = F ( b ) ≥ 0 } K_(X):={F in C[a,b]:F(a)=F(b) >= 0}K_{X}:=\{F \in C[a, b]: F(a)=F(b) \geq 0\}KX:={F∈C[a,b]:F(a)=F(b)≥0}
and K Y = K X ∩ C ( { a , b } ) K Y = K X ∩ C ( { a , b } ) K_(Y)=K_(X)nn C({a,b})K_{Y}=K_{X} \cap C(\{a, b\})KY=KX∩C({a,b}), i.e.
K Y := { f ∈ C ( { a , b } ) : f ( a ) = f ( b ) ≥ 0 } K Y := { f ∈ C ( { a , b } ) : f ( a ) = f ( b ) ≥ 0 } K_(Y):={f in C({a,b}):f(a)=f(b) >= 0}K_{Y}:=\{f \in C(\{a, b\}): f(a)=f(b) \geq 0\}KY:={f∈C({a,b}):f(a)=f(b)≥0}
Obviously that K Y K Y K_(Y)K_{Y}KY has the (NPE)-properly with respect to K X K X K_(X)K_{X}KX. If F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX then F | Y ∈ K Y F Y ∈ K Y F|_(Y)inK_(Y)\left.F\right|_{Y} \in K_{Y}F|Y∈KY and function H ( x ) = f ( a ) , x ∈ [ a , b ] H ( x ) = f ( a ) , x ∈ [ a , b ] H(x)=f(a),x in[a,b]H(x)=f(a), x \in[a, b]H(x)=f(a),x∈[a,b], is an (NPE) extension of f ∈ C ( { a , b } ) f ∈ C ( { a , b } ) f in C({a,b})f \in C(\{a, b\})f∈C({a,b}).
The space generated by th cone K X K X K_(X)K_{X}KX is
M X := K X − K X = { F ∈ C [ a , b ] : F ( a ) = F ( b ) } M X := K X − K X = { F ∈ C [ a , b ] : F ( a ) = F ( b ) } M_(X):=K_(X)-K_(X)={F in C[a,b]:F(a)=F(b)}M_{X}:=K_{X}-K_{X}=\{F \in C[a, b]: F(a)=F(b)\}MX:=KX−KX={F∈C[a,b]:F(a)=F(b)}
and the annihilator space of the set Y Y YYY in M X M X M_(X)M_{X}MX is
Y ⊥ := { G ∈ M X : G ( a ) = G ( b ) = 0 } Y ⊥ := G ∈ M X : G ( a ) = G ( b ) = 0 Y^(_|_):={G inM_(X):G(a)=G(b)=0}Y^{\perp}:=\left\{G \in M_{X}: G(a)=G(b)=0\right\}Y⊥:={G∈MX:G(a)=G(b)=0}
By Theorem 1, the subspace Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is K X K X K_(X)K_{X}KX-proximal and d ( F , Y ⊥ ) = F ( a ) d F , Y ⊥ = F ( a ) d(F,Y^(_|_))=F(a)d\left(F, Y^{\perp}\right)=F(a)d(F,Y⊥)=F(a) and F − ϕ ( F | Y ) ⊆ P Y ⊥ ( F ) F − ϕ F Y ⊆ P Y ⊥ ( F ) F-phi(F|_(Y))subeP_(Y)_|_(F)F-\mathscr{\phi}\left(\left.F\right|_{Y}\right) \subseteq P_{Y} \perp(F)F−ϕ(F|Y)⊆PY⊥(F), for each F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX. We have
E ( F | Y ) = { H ∈ K X : H ( a ) = H ( b ) = ‖ H ‖ X } = = { H ∈ K X : H ( a ) = H ( b ) and | H ( x ) | ≤ F ( a ) , for all x ∈ [ a , b ] } E F Y = H ∈ K X : H ( a ) = H ( b ) = ‖ H ‖ X = = H ∈ K X : H ( a ) = H ( b )  and  | H ( x ) | ≤ F ( a ) ,  for all  x ∈ [ a , b ] {:[E(F|_(Y))={H inK_(X):H(a)=H(b)=||H||_(X)}=],[={H inK_(X):H(a)=H(b)" and "|H(x)| <= F(a)," for all "x in[a,b]}]:}\begin{gathered} \mathscr{E}\left(\left.F\right|_{Y}\right)=\left\{H \in K_{X}: H(a)=H(b)=\|H\|_{X}\right\}= \\ =\left\{H \in K_{X}: H(a)=H(b) \text { and }|H(x)| \leq F(a), \text { for all } x \in[a, b]\right\} \end{gathered}E(F|Y)={H∈KX:H(a)=H(b)=‖H‖X}=={H∈KX:H(a)=H(b) and |H(x)|≤F(a), for all x∈[a,b]}
It follows that Y ⊥ ⊂ K X Y ⊥ ⊂ K X Y^(_|_)subK_(X)Y^{\perp} \subset K_{X}Y⊥⊂KX and therefore the equality F − E ( F | Y ) = P Y ⊥ ( F ) F − E F Y = P Y ⊥ ( F ) F-E(F|_(Y))=P_(Y)_|_(F)F-\mathscr{E}\left(\left.F\right|_{Y}\right)=P_{Y} \perp(F)F−E(F|Y)=PY⊥(F) for each F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘. Let X = [ − 2 , 2 ] ⊂ R , Y = { − 1 , 0 , 1 } X = [ − 2 , 2 ] ⊂ R , Y = { − 1 , 0 , 1 } X=[-2,2]sub R,Y={-1,0,1}X=[-2,2] \subset R, Y=\{-1,0,1\}X=[−2,2]⊂R,Y={−1,0,1} and let N X := Lip 0 [ − 2 , 2 ] = { F : [ − 2 , 2 ] → R : F N X := Lip 0 [ − 2 , 2 ] = { F : [ − 2 , 2 ] → R : F N_(X):=Lip_(0)[-2,2]={F:[-2,2]rarr R:FN_{X}:=\operatorname{Lip}_{0}[-2,2]=\{F:[-2,2] \rightarrow R: FNX:=Lip0[−2,2]={F:[−2,2]→R:F is Lipschitz on [ − 2 , 2 ] [ − 2 , 2 ] [-2,2][-2,2][−2,2] and F ( 0 ) = 0 } F ( 0 ) = 0 } F(0)=0}F(0)=0\}F(0)=0} equipped with the Lipschitz norm
‖ F ‖ X = sup { | F ( x ) − F ( y ) | / | x − y | : x , y ∈ [ − 2 , 2 ] , x ≠ y } ‖ F ‖ X = sup { | F ( x ) − F ( y ) | / | x − y | : x , y ∈ [ − 2 , 2 ] , x ≠ y } ||F||_(X)=s u p{|F(x)-F(y)|//|x-y|:x,y in[-2,2],x!=y}\|F\|_{X}=\sup \{|F(x)-F(y)| /|x-y|: x, y \in[-2,2], x \neq y\}‖F‖X=sup{|F(x)−F(y)|/|x−y|:x,y∈[−2,2],x≠y}
For N Y N Y N_(Y)N_{Y}NY take
N Y := Lip 0 { − 1 , 0 , 1 } = { f : { − 1 , 0 , 1 } → R : f ( 0 ) = 0 } N Y := Lip 0 { − 1 , 0 , 1 } = { f : { − 1 , 0 , 1 } → R : f ( 0 ) = 0 } N_(Y):=Lip_(0){-1,0,1}={f:{-1,0,1}rarr R:f(0)=0}N_{Y}:=\operatorname{Lip}_{0}\{-1,0,1\}=\{f:\{-1,0,1\} \rightarrow R: f(0)=0\}NY:=Lip0{−1,0,1}={f:{−1,0,1}→R:f(0)=0}
equipped with the norm
‖ f ‖ Y = max { | f ( − 1 ) | , | f ( 1 ) | } . ‖ f ‖ Y = max { | f ( − 1 ) | , | f ( 1 ) | } . ||f||_(Y)=max{|f(-1)|,|f(1)|}.\|f\|_{Y}=\max \{|f(-1)|,|f(1)|\} .‖f‖Y=max{|f(−1)|,|f(1)|}.
For K X K X K_(X)K_{X}KX and K Y K Y K_(Y)K_{Y}KY take
K X := { F ∈ Lip 0 [ − 2 , 2 ] : F is convex on [ − 2 , 2 ] } K X := { f ∈ Lip 0 { − 1 , 0 , 1 } : f is convex on { − 1 , 0 , 1 } } K X := F ∈ Lip 0 [ − 2 , 2 ] : F  is convex on  [ − 2 , 2 ] K X := f ∈ Lip 0 { − 1 , 0 , 1 } : f  is convex on  { − 1 , 0 , 1 } {:[K_(X):={F inLip_(0)[-2,2]:F" is convex on "[-2,2]}],[K_(X):={f inLip_(0){-1,0,1}:f" is convex on "{-1,0,1}}]:}\begin{gathered} K_{X}:=\left\{F \in \operatorname{Lip}_{0}[-2,2]: F \text { is convex on }[-2,2]\right\} \\ K_{X}:=\left\{f \in \operatorname{Lip}_{0}\{-1,0,1\}: f \text { is convex on }\{-1,0,1\}\right\} \end{gathered}KX:={F∈Lip0[−2,2]:F is convex on [−2,2]}KX:={f∈Lip0{−1,0,1}:f is convex on {−1,0,1}}
By definition F ∈ Lip 0 [ − 2 , 2 ] F ∈ Lip 0 [ − 2 , 2 ] F inLip_(0)[-2,2]F \in \operatorname{Lip}_{0}[-2,2]F∈Lip0[−2,2] is in K X K X K_(X)K_{X}KX if
F ( λ x + ( 1 − λ ) y ) ≤ λ F ( x ) + ( 1 − λ ) F ( y ) F ( λ x + ( 1 − λ ) y ) ≤ λ F ( x ) + ( 1 − λ ) F ( y ) F(lambda x+(1-lambda)y) <= lambda F(x)+(1-lambda)F(y)F(\lambda x+(1-\lambda) y) \leq \lambda F(x)+(1-\lambda) F(y)F(λx+(1−λ)y)≤λF(x)+(1−λ)F(y)
for every x , y ∈ [ − 2 , 2 ] x , y ∈ [ − 2 , 2 ] x,y in[-2,2]x, y \in[-2,2]x,y∈[−2,2] and every λ ∈ [ 0 , 1 ] λ ∈ [ 0 , 1 ] lambda in[0,1]\lambda \in[0,1]λ∈[0,1] and f f fff is in K Y K Y K_(Y)K_{Y}KY if the divided difference [ − 1 , 0 , 1 ; f ] [ − 1 , 0 , 1 ; f ] [-1,0,1;f][-1,0,1 ; f][−1,0,1;f] is nonnegative.
Obviously that K Y K Y K_(Y)K_{Y}KY has the (NPE)-property with respect to K X K X K_(X)K_{X}KX : for F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX the restriction F | Y F Y F|_(Y)\left.F\right|_{Y}F|Y is in K Y K Y K_(Y)K_{Y}KY and F ( x ) = min y ∈ { − 1 , 0 , 1 } [ f ( y ) + ‖ f ‖ Y | x − y | ] , x ∈ [ − 2 , 2 ] F ( x ) = min y ∈ { − 1 , 0 , 1 }   f ( y ) + ‖ f ‖ Y | x − y | , x ∈ [ − 2 , 2 ] F(x)=min_(y in{-1,0,1})[f(y)+||f||_(Y)|x-y|],x in[-2,2]F(x)=\min _{y \in\{-1,0,1\}}\left[f(y)+\|f\|_{Y}|x-y|\right], x \in[-2,2]F(x)=miny∈{−1,0,1}[f(y)+‖f‖Y|x−y|],x∈[−2,2] is a (NPE) extension in K X K X K_(X)K_{X}KX of f ∈ K Y f ∈ K Y f inK_(Y)f \in K_{Y}f∈KY.
Let
F ( x ) = { − x , x ∈ [ − 2 , 0 ] 1 2 x , x ∈ ( 0 , − 2 ] F ( x ) = − x ,      x ∈ [ − 2 , 0 ] 1 2 x ,      x ∈ ( 0 , − 2 ] F(x)={[-x",",x in[-2","0]],[(1)/(2)x",",x in(0","-2]]:}F(x)= \begin{cases}-x, & x \in[-2,0] \\ \frac{1}{2} x, & x \in(0,-2]\end{cases}F(x)={−x,x∈[−2,0]12x,x∈(0,−2]
and
G ( x ) = { − x − 1 , x ∈ [ − 2 , − 1 ) 0 , x ∈ [ − 1 , 1 ] 1 2 x − 1 2 , x ∈ [ 1 , 1 2 ] G ( x ) = − x − 1 ,      x ∈ [ − 2 , − 1 ) 0 ,      x ∈ [ − 1 , 1 ] 1 2 x − 1 2 ,      x ∈ 1 , 1 2 G(x)={[-x-1",",x in[-2","-1)],[0",",x in[-1","1]],[(1)/(2)x-(1)/(2)",",x in[1,(1)/(2)]]:}G(x)= \begin{cases}-x-1, & x \in[-2,-1) \\ 0, & x \in[-1,1] \\ \frac{1}{2} x-\frac{1}{2}, & x \in\left[1, \frac{1}{2}\right]\end{cases}G(x)={−x−1,x∈[−2,−1)0,x∈[−1,1]12x−12,x∈[1,12]
It follows that G ∈ Y ⊥ G ∈ Y ⊥ G inY^(_|_)G \in Y^{\perp}G∈Y⊥ (in fact G ∈ K X ⊆ K X − K X G ∈ K X ⊆ K X − K X G inK_(X)subeK_(X)-K_(X)G \in K_{X} \subseteq K_{X}-K_{X}G∈KX⊆KX−KX ),
F ( x ) − G ( x ) = { 1 , x ∈ [ − 2 , − 1 ] − x , x ∈ ( − 1 , 0 ] 1 2 x , x ∈ ( 0 , 1 ] 1 2 , x ∈ ( 1 , 2 ] F ( x ) − G ( x ) = 1 ,      x ∈ [ − 2 , − 1 ] − x ,      x ∈ ( − 1 , 0 ] 1 2 x ,      x ∈ ( 0 , 1 ] 1 2 ,      x ∈ ( 1 , 2 ] F(x)-G(x)={[1",",x in[-2","-1]],[-x",",x in(-1","0]],[(1)/(2)x",",x in(0","1]],[(1)/(2)",",x in(1","2]]:}F(x)-G(x)= \begin{cases}1, & x \in[-2,-1] \\ -x, & x \in(-1,0] \\ \frac{1}{2} x, & x \in(0,1] \\ \frac{1}{2}, & x \in(1,2]\end{cases}F(x)−G(x)={1,x∈[−2,−1]−x,x∈(−1,0]12x,x∈(0,1]12,x∈(1,2]
and
‖ F − G ‖ X = ‖ F | Y ‖ Y = d ( F , Y ⊥ ) , ( F − G ) | Y = F | Y , ‖ F − G ‖ X = F Y Y = d F , Y ⊥ , ( F − G ) Y = F Y , ||F-G||_(X)=||F|_(Y)||_(Y)=d(F,Y^(_|_)),(F-G)|_(Y)=F|_(Y),\|F-G\|_{X}=\left\|\left.F\right|_{Y}\right\|_{Y}=d\left(F, Y^{\perp}\right),\left.(F-G)\right|_{Y}=\left.F\right|_{Y},‖F−G‖X=‖F|Y‖Y=d(F,Y⊥),(F−G)|Y=F|Y,
implying that F − G F − G F-GF-GF−G is a (NPE) extension of F | Y F Y F|_(Y)\left.F\right|_{Y}F|Y. But F − G F − G F-GF-GF−G is not convex function on [ − 2 , 2 ] [ − 2 , 2 ] [-2,2][-2,2][−2,2] showing that, in general, F − E ( F | Y ) F − E F Y F-E(F|_(Y))F-\mathscr{E}\left(\left.F\right|_{Y}\right)F−E(F|Y) can be strictly contained in P Y ⊥ ( F ) P Y ⊥ ( F ) P_(Y^(_|_))(F)P_{Y^{\perp}}(F)PY⊥(F). Therefore equality (7) is not true without any supplementary hypotheses on the space Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥.
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘. Let X , Y , N X ′ N Y X , Y , N X ′ N Y X,Y,N_(X^('))N_(Y)X, Y, N_{X^{\prime}} N_{Y}X,Y,NX′NY be as in Example 2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ and
Y ⊥ = { G ∈ N X : G | Y = 0 } Y ⊥ = G ∈ N X : G Y = 0 Y^(_|_)={G inN_(X):G|_(Y)=0}Y^{\perp}=\left\{G \in N_{X}:\left.G\right|_{Y}=0\right\}Y⊥={G∈NX:G|Y=0}
By Mc Shane's theorem (see [9]), the space N Y N Y N_(Y)N_{Y}NY has the (NPE)-property with respect to N X N X N_(X)N_{X}NX. By Corollary the subspace Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is proximinal in N X N X N_(X)N_{X}NX and every element
G G GGG of best approximation of a function F ∈ N X F ∈ N X F inN_(X)F \in N_{X}F∈NX by elements in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ has the form
G = F − H G = F − H G=F-HG=F-HG=F−H for a function H ∈ E ( F | Y ) H ∈ E F Y H inE(F|_(Y))H \in \mathscr{E}\left(\left.F\right|_{Y}\right)H∈E(F|Y) Also d ( F , Y ⊥ ) = ‖ F | Y ‖ Y d F , Y ⊥ = F Y Y d(F,Y^(_|_))=||F|_(Y)||_(Y)d\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\|_{Y}d(F,Y⊥)=‖F|Y‖Y and P Y ⊥ ( F ) = F − E ( F | Y ) P Y ⊥ ( F ) = F − E F Y P_(Y^(_|_))(F)=F-E(F|_(Y))P_{Y^{\perp}}(F)=F-\mathscr{E}\left(\left.F\right|_{Y}\right)PY⊥(F)=F−E(F|Y).
Other examples to which Corollary 1 applies are given by Hahn-Banach extension theorem, by Tietze extension theorem, by Helly extension Theorem (see [7], [8], [12], [16]).

2. THE QUOTIENT MAPPING

Consider the quotient subspace M X / Y ⊥ M X / Y ⊥ M_(X)//Y^(_|_)M_{X} / Y^{\perp}MX/Y⊥ with respect to its subspace Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥, defined by
(8) M X / Y ⊥ = { F + Y ⊥ : F ∈ M X } . (8) M X / Y ⊥ = F + Y ⊥ : F ∈ M X . {:(8)M_(X)//Y^(_|_)={F+Y^(_|_):F inM_(X)}.:}\begin{equation*} M_{X} / Y^{\perp}=\left\{F+Y^{\perp}: F \in M_{X}\right\} . \tag{8} \end{equation*}(8)MX/Y⊥={F+Y⊥:F∈MX}.
Since the subspace Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is closed in M X M X M_(X)M_{X}MX it follows that
(9)
‖ F + Y ⊥ ‖ = d ( F , Y ⊥ ) , F ∈ M X F + Y ⊥ = d F , Y ⊥ , F ∈ M X ||F+Y^(_|_)||=d(F,Y^(_|_)),F inM_(X)\left\|F+Y^{\perp}\right\|=d\left(F, Y^{\perp}\right), F \in M_{X}‖F+Y⊥‖=d(F,Y⊥),F∈MX
is a norm on M X / Y ⊥ M X / Y ⊥ M_(X)//Y^(_|_)M_{X} / Y^{\perp}MX/Y⊥
Let
(10) K X / Y ⊥ := { F + Y ⊥ : F ∈ K X } (10) K X / Y ⊥ := F + Y ⊥ : F ∈ K X {:(10)K_(X)//Y^(_|_):={F+Y^(_|_):F inK_(X)}:}\begin{equation*} K_{X} / Y^{\perp}:=\left\{F+Y^{\perp}: F \in K_{X}\right\} \tag{10} \end{equation*}(10)KX/Y⊥:={F+Y⊥:F∈KX}
and let
(11) Ker P Y ⊥ ∣ K X := { F ∈ K X : 0 ∈ P Y ⊥ ( F ) } (11) Ker P Y ⊥ ∣ K X := F ∈ K X : 0 ∈ P Y ⊥ ( F ) {:(11)KerP_(Y^(_|_))∣K_(X):={F inK_(X):0inP_(Y^(_|_))(F)}:}\begin{equation*} \operatorname{Ker} P_{Y^{\perp}} \mid K_{X}:=\left\{F \in K_{X}: 0 \in P_{Y^{\perp}}(F)\right\} \tag{11} \end{equation*}(11)KerPY⊥∣KX:={F∈KX:0∈PY⊥(F)}
be the kernel of the restriction of the metric projection P Y ⊥ P Y ⊥ P_(Y^(_|_))P_{Y^{\perp}}PY⊥ to K X K X K_(X)K_{X}KX.
Obviously that
Ker P Y ⊥ | K X = { F ∈ K X : ‖ F ‖ X = d ( F , Y ⊥ ) } = { F ∈ K X : ‖ F ‖ X = ‖ F ˙ | Y ‖ Y } . Ker P Y ⊥ K X = F ∈ K X : ‖ F ‖ X = d F , Y ⊥ = F ∈ K X : ‖ F ‖ X = F ˙ Y Y . KerP_(Y^(_|_))|_(K_(X))={F inK_(X):||F||_(X)=d(F,Y^(_|_))}={F inK_(X):||F||_(X)=||(F^(˙))|_(Y)||_(Y)}.\left.\operatorname{Ker} P_{Y^{\perp}}\right|_{K_{X}}=\left\{F \in K_{X}:\|F\|_{X}=d\left(F, Y^{\perp}\right)\right\}=\left\{F \in K_{X}:\|F\|_{X}=\left\|\left.\dot{F}\right|_{Y}\right\|_{Y}\right\} .KerPY⊥|KX={F∈KX:‖F‖X=d(F,Y⊥)}={F∈KX:‖F‖X=‖F˙|Y‖Y}.
The application
(12) Q K X : K X → K X / Y ⊥ , Q ( f ) = F + Y ⊥ , F ∈ K X , (12) Q K X : K X → K X / Y ⊥ , Q ( f ) = F + Y ⊥ , F ∈ K X , {:(12)Q_(K_(X)):K_(X)rarrK_(X)//Y^(_|_)","quad Q(f)=F+Y^(_|_)","quad F inK_(X)",":}\begin{equation*} Q_{K_{X}}: K_{X} \rightarrow K_{X} / Y^{\perp}, \quad Q(f)=F+Y^{\perp}, \quad F \in K_{X}, \tag{12} \end{equation*}(12)QKX:KX→KX/Y⊥,Q(f)=F+Y⊥,F∈KX,
is called the quotient mapping of the cone K X K X K_(X)K_{X}KX onto the cone K X / Y ⊥ K X / Y ⊥ K_(X)//Y^(_|_)K_{X} / Y^{\perp}KX/Y⊥. THEOREM 2. If the cone K Y K Y K_(Y)K_{Y}KY has the (NPE)-property with respect to the cone K X K X K_(X)K_{X}KX then
1 0 . K X ⊆ Y ⊥ + Ker P Y ⊥ ∣ K X = { G + H : G ∈ Y ⊥ , G ∈ Ker P Y ⊥ ∣ K X } 1 0 . K X ⊆ Y ⊥ + Ker P Y ⊥ ∣ K X = G + H : G ∈ Y ⊥ , G ∈ Ker P Y ⊥ ∣ K X 1^(0).K_(X)subeY^(_|_)+KerP_(Y^(_|_))∣K_(X)={G+H:G inY^(_|_),G in KerP_(Y^(_|_))∣K_(X)}\mathbf{1}^{0} . K_{X} \subseteq Y^{\perp}+\operatorname{Ker} P_{Y^{\perp}} \mid K_{X}=\left\{G+H: G \in Y^{\perp}, G \in \operatorname{Ker} P_{Y^{\perp}} \mid K_{X}\right\}10.KX⊆Y⊥+KerPY⊥∣KX={G+H:G∈Y⊥,G∈KerPY⊥∣KX}
2 ∘ ⋅ Q ( Ker P Y ⊥ | K X ) = K X / Y ⊥ ; 2 ∘ ⋅ Q Ker P Y ⊥ K X = K X / Y ⊥ ; 2^(@)*Q( KerP_(Y^(_|_))|_(K_(X)))=K_(X)//Y^(_|_);2^{\circ} \cdot Q\left(\left.\operatorname{Ker} P_{Y^{\perp}}\right|_{K_{X}}\right)=K_{X} / Y^{\perp} ;2∘⋅Q(KerPY⊥|KX)=KX/Y⊥;
30. F − ( Q | Ker P Y ⊥ | | X ) − 1 ( F + Y ⊥ ) ⊆ P Y ⊥ | K X ( F ) F − Q Ker P Y ⊥ | | X − 1 F + Y ⊥ ⊆ P Y ⊥ K X ( F ) F-(Q|_(KerP_(Y^(_|_))||_(X)))^(-1)(F+Y^(_|_))subeP_(Y^(_|_))|_(K_(X))(F)F-\left.\left(\left.Q\right|_{\operatorname{Ker} P_{Y^{\perp}}| |_{X}}\right)^{-1}\left(F+Y^{\perp}\right) \subseteq P_{Y^{\perp}}\right|_{K_{X}}(F)F−(Q|KerPY⊥||X)−1(F+Y⊥)⊆PY⊥|KX(F), for every F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX.
Proof. 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘. By Theorem 1. (b), each F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX there has an element of best approximation G ∈ Y ⊥ G ∈ Y ⊥ G inY^(_|_)G \in Y^{\perp}G∈Y⊥ and G = F − H G = F − H G=F-HG=F-HG=F−H for a function H ∈ E ( F | Y ) H ∈ E F Y H inE(F|_(Y))H \in \mathscr{E}\left(\left.F\right|_{Y}\right)H∈E(F|Y). It follows that ‖ H ‖ X = ‖ F | Y ‖ Y = d ( H , Y ⊥ ) ‖ H ‖ X = F Y Y = d H , Y ⊥ ||H||_(X)=||F|_(Y)||_(Y)=d(H,Y^(_|_))\|H\|_{X}=\left\|\left.F\right|_{Y}\right\|_{Y}=d\left(H, Y^{\perp}\right)‖H‖X=‖F|Y‖Y=d(H,Y⊥) implying H ∈ Ker P Y ⊥ | K X H ∈ Ker P Y ⊥ K X H in KerP_(Y^(_|_))|_(K_(X))\left.H \in \operatorname{Ker} P_{Y^{\perp}}\right|_{K_{X}}H∈KerPY⊥|KX; and then F = G + Y ⊥ F = G + Y ⊥ F=G+Y^(_|_)F=G+Y^{\perp}F=G+Y⊥, H ∈ Ker P Y ⊥ ∣ K X H ∈ Ker P Y ⊥ ∣ K X H in KerP_(Y^(_|_))∣K_(X)H \in \operatorname{Ker} P_{Y^{\perp}} \mid K_{X}H∈KerPY⊥∣KX.
2 ∘ 2 ∘ 2^(@)\mathbf{2}^{\boldsymbol{\circ}}2∘. Let F + Y ⊥ ∈ K X / Y ⊥ F + Y ⊥ ∈ K X / Y ⊥ F+Y^(_|_)inK_(X)//Y^(_|_)F+Y^{\perp} \in K_{X} / Y^{\perp}F+Y⊥∈KX/Y⊥. Since F | Y ∈ K Y F Y ∈ K Y F|_(Y)inK_(Y)\left.F\right|_{Y} \in K_{Y}F|Y∈KY and K Y K Y K_(Y)K_{Y}KY has the (NPE)-property with respect to K X K X K_(X)K_{X}KX it follows that for every H ∈ E ( F | Y ) ⊆ Ker P Y ⊥ | K X H ∈ E F Y ⊆ Ker P Y ⊥ K X H inE(F|_(Y))sube KerP_(Y^(_|_))|_(K_(X))\left.H \in \mathscr{E}\left(\left.F\right|_{Y}\right) \subseteq \operatorname{Ker} P_{Y^{\perp}}\right|_{K_{X}}H∈E(F|Y)⊆KerPY⊥|KX we have
Q ( H ) = H + Y ⊥ = F + Y ⊥ Q ( H ) = H + Y ⊥ = F + Y ⊥ Q(H)=H+Y^(_|_)=F+Y^(_|_)Q(H)=H+Y^{\perp}=F+Y^{\perp}Q(H)=H+Y⊥=F+Y⊥
because F − H ∈ Y ⊥ F − H ∈ Y ⊥ F-H inY^(_|_)F-H \in Y^{\perp}F−H∈Y⊥. Therefore Q Ker P Y ⊥ | K X Q Ker P Y ⊥ K X Q_(KerP_(Y^(_|_)))|_(K_(X))\left.Q_{\operatorname{Ker} P_{Y^{\perp}}}\right|_{K_{X}}QKerPY⊥|KX is a surjection.
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘. By 2 ∘ 2 ∘ 2^(@)2^{\circ}2∘, the application
(13) E = ( Q Ker P Y ⊥ ∣ K X ) − 1 : K X / Y ⊥ → 2 Ker P Y ⊥ ∣ K X (13) E = Q Ker P Y ⊥ ∣ K X − 1 : K X / Y ⊥ → 2 Ker P Y ⊥ ∣ K X {:(13)E=(Q_(KerP_(Y^(_|_))∣K_(X)))^(-1):K_(X)//Y^(_|_)rarr2^(KerP_(Y^(_|_))∣K_(X)):}\begin{equation*} E=\left(Q_{\operatorname{Ker} P_{Y^{\perp}} \mid K_{X}}\right)^{-1}: K_{X} / Y^{\perp} \rightarrow 2^{\operatorname{Ker} P_{Y^{\perp}} \mid K_{X}} \tag{13} \end{equation*}(13)E=(QKerPY⊥∣KX)−1:KX/Y⊥→2KerPY⊥∣KX
given by
(14) E ( F + Y ⊥ ) = E ( F | Y ) ⊂ Ker P Y ⊥ | K X (14) E F + Y ⊥ = E F Y ⊂ Ker P Y ⊥ K X {:(14)E(F+Y^(_|_))=E(F|_(Y))sub KerP_(Y^(_|_))|_(K_(X)):}\begin{equation*} E\left(F+Y^{\perp}\right)=\left.\mathscr{E}\left(\left.F\right|_{Y}\right) \subset \operatorname{Ker} P_{Y^{\perp}}\right|_{K_{X}} \tag{14} \end{equation*}(14)E(F+Y⊥)=E(F|Y)⊂KerPY⊥|KX
is well defined.
But then, by Theorem 1. (b), it follows that
F − E ( F | Y ) ⊆ P Y ⊥ | K X ( F ) . F − E F Y ⊆ P Y ⊥ K X ( F ) . F-E(F|_(Y))subeP_(Y^(_|_))|_(K_(X))(F).F-\left.\mathscr{E}\left(\left.F\right|_{Y}\right) \subseteq P_{Y^{\perp}}\right|_{K_{X}}(F) .F−E(F|Y)⊆PY⊥|KX(F).
Theorem 2 is proved.
Using Theorem 2 one can obtain some relations between the properties of the selections associated to the metric projection P Y ⊥ | K X P Y ⊥ K X P_(Y^(_|_))|_(K_(X))\left.P_{Y^{\perp}}\right|_{K_{X}}PY⊥|KX and the properties of the selections associated to the application E E EEE defined by (13) and (14). (see [3], [12], [13], [16]).

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Received 15 IX 1994
Academia Română
Institutul de Calcul
"Tiberiu Popoviciu"
P.O. Box 68
3400 Cluj-Napoca 1
România
    1. Banach, S., Wstep to teorii funkji rzeczywistych, Warsawa-Wroclaw, 1951.
    2. Cobzaş, S., Mustăţa, C., Norm Preserving Extension of Convex Lipschitz Functions, J.A.T. 34, (1978), 236-244.
1995

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