On the uniqueness of the extension of continuous p-seminorms

Costica Mustata
(original), Approximation Theory, paper

Original title (in Romanian)

Asupra unicității prelungirii p-seminormelor continue

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

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C. Mustăţa, On the uniqueness of the extension of continuous p-seminorms, Rev. Anal. Numer. Teoria Approximatiei 2 (1973) no. 2, 173-177 (MR 53 # 8759) (in Romanian).

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Rev. Anal. Numer. Theoria Approximatiei

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Romanian Academy

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https://ictp.acad.ro/ranta-ro-72-74/journal/article/view/32

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MR 53 # 8759

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[1] Czipser, J. si Geher, L., Extension of functions satisfying a Lipschitz condition, Acta Math. Acad. Sci. Hungar 6, 213-220, 1955
[2] Kolumban I., Ob edinstvenosti prodoljenia linein]h functionalov, Mathematica, Cluj, 4 (27), 267-270, 1962
[3] Phelps, R.P., Uniquencess of Hahn-Banach extension and unique best approximation, Trans. Amer. Math. Soc. 95, 238-255, 1960.

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1973-Mustata-RANTA-On-the-uniqueness-of-the-extension-of-continuous-p-seminorms

ASUPRA UNICITATTII PRELUNGIRII p p ppp-SEMINORMELOR CONTINUE

de COSTICA MUSTATA(Cluj)

  1. Fie X X XXX un spaţiu liniar real și p ∈ ( 0 , 1 ] p ∈ ( 0 , 1 ] p in(0,1]p \in(0,1]p∈(0,1]. O funcţională ‖ ‖ ‖ : X → R ‖ ‖ ‖ : X → R ||||||:X rarr R\|\|\|: X \rightarrow R‖‖‖:X→R se numeşte o p p ppp-normă pe X X XXX dacă ea verifică axiomele:
p 1 ) ‖ | x | ‖ ≧ 0 , ‖ x ‖ = 0 ⇔ x = 0 , x ∈ X p 2 ) ‖ | x + y ‖ p ≦ ‖ | x | ‖ p + ‖ | y ∣ ‖ p , x , y ∈ X p 3 ) ‖ λ x | ‖ = | λ | ⋅ ‖ x | ‖ , x ∈ X , λ ∈ R p 1 ‖ | x | ‖ ≧ 0 , ‖ x ‖ = 0 ⇔ x = 0 ,      x ∈ X p 2 x + y p ≦ x p + y ∣ p ,      x , y ∈ X p 3 ‖ λ x | ‖ = | λ | ⋅ ‖ x | ‖ ,      x ∈ X , λ ∈ R {:[{:p_(1))quad|||x||| >= 0","||x||=0<=>x=0",",x in X],[{:p_(2))quad|||x+y||^(p) <= |||x|||^(p)+|||y∣||^(p)",",x","y in X],[{:p_(3))quad||lambda x|||=|lambda|*||x|||",",x in X","lambda in R]:}\begin{array}{ll} \left.p_{1}\right) \quad\||x|\| \geqq 0,\|x\|=0 \Leftrightarrow x=0, & x \in X \\ \left.p_{2}\right) \quad\left\|\left|x+y\left\|^{p} \leqq\right\|\right| x\left|\left\|^{p}+\right\|\right| y \mid\right\|^{p}, & x, y \in X \\ \left.p_{3}\right) \quad\|\lambda x|\|=|\lambda| \cdot\| x|\|, & x \in X, \lambda \in R \end{array}p1)‖|x|‖≧0,‖x‖=0⇔x=0,x∈Xp2)‖|x+y‖p≦‖|x|‖p+‖|y∣‖p,x,y∈Xp3)‖λx|‖=|λ|⋅‖x|‖,x∈X,λ∈R
Spatiul liniar real X X XXX, înzestrat cu p p ppp-norma ||| ||| îl numim spaţiu p p ppp-normat şi îl notăm ( X , ‖ | ‖ ∣ ‖ X , ‖ | ‖ ∣ ‖ X,|||||∣||X,\||\|\mid\|X,‖|‖∣‖ ).
In lucrarea [4], w. RUESS defineşte conul convex al p p ppp-seminormelor continue definite pe spaţiul p p ppp-normat ( X , ‖ ‖ ‖ ) ( X , ‖ ‖ ‖ ) (X,||||||)(X,\| \| \|)(X,‖‖‖) :
(1) C X p τ = { h ∣ h : X → R + , ∗ ) ∀ x , y ∈ X , ∀ λ ∈ R , h ( x + y ) ≦ h ( x ) + h ( y ) şi h ( λ x ) = | λ | p h ( x ) } (1) C X p τ = h ∣ h : X → R + , ∗ ∀ x , y ∈ X , ∀ λ ∈ R , h ( x + y ) ≦ h ( x ) + h ( y )  şi  h ( λ x ) = | λ | p h ( x ) {:[(1)C_(X)^(p)tau={h∣h:X rarrR^(+),^(**))quad AA x","y in X","quad AA lambda in R","quad h(x+y) <= ],[{:h(x)+h(y)quad" şi "h(lambda x)=|lambda|^(p)h(x)}]:}\begin{gather*} C_{X}^{p} \tau=\left\{h \mid h: X \rightarrow R^{+},{ }^{*}\right) \quad \forall x, y \in X, \quad \forall \lambda \in R, \quad h(x+y) \leqq \tag{1}\\ \left.h(x)+h(y) \quad \text { şi } h(\lambda x)=|\lambda|^{p} h(x)\right\} \end{gather*}(1)CXpτ={h∣h:X→R+,∗)∀x,y∈X,∀λ∈R,h(x+y)≦h(x)+h(y) şi h(λx)=|λ|ph(x)}
şi spațiul generat de C X p C X p C_(X)^(p)C_{X}^{p}CXp, anume
(2) X τ p = C X p τ − C X p τ . (2) X τ p = C X p τ − C X p τ . {:(2)X_(tau)^(p)=C_(X)^(p)tau-C_(X)^(p)tau.:}\begin{equation*} X_{\tau}^{p}=C_{X}^{p} \tau-C_{X}^{p} \tau . \tag{2} \end{equation*}(2)Xτp=CXpτ−CXpτ.
Vom nota cu X # X # X^(#)X^{\#}X# spatiul liniar real al functionalelor lipschitziene definite pe spatiul p p ppp-normat ( X , | | | | | | X , | | | | | | X,||||||X,||||| |X,|||||| ) (vezi [2]):
(3) ∀ f ∈ X # , ∃ M ≧ 0 , ∀ x , y ∈ X , | f ( x ) − f ( y ) | ≦ M ⋅ | ‖ x − y ‖ | ∀ f ∈ X # , ∃ M ≧ 0 , ∀ x , y ∈ X , | f ( x ) − f ( y ) | ≦ M ⋅ | ‖ x − y ‖ | AA f inX^(#),EE M >= 0,AA x,y in X,|f(x)-f(y)| <= M*|||x-y|||\forall f \in X^{\#}, \exists M \geqq 0, \forall x, y \in X,|f(x)-f(y)| \leqq M \cdot|\|x-y\||∀f∈X#,∃M≧0,∀x,y∈X,|f(x)−f(y)|≦M⋅|‖x−y‖|
Cu Φ Φ Phi\PhiΦ vom nota funcţionala nulă pe ( X , | | | | | | X , | | | | | | X,||||||X,||||| |X,|||||| ).
C X p τ C X p τ C_(X)^(p_(tau))C_{X}^{p_{\tau}}CXpτ este un con convex din spaţiul liniar X # X # X^(#)X^{\#}X#. Intr-adevăr dacă h ∈ C γ τ p p τ h ∈ C γ τ p p τ h inC_(gamma_(tau)^(p))^(p_(tau))h \in C_{\gamma_{\tau}^{p}}^{p_{\tau}}h∈Cγτppτ atunci pentru orice x 1 , x 2 ∈ X x 1 , x 2 ∈ X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2∈X avem | h ( x 1 ) − h ( x 2 ) | ≦ h ( x 1 − x 2 ) h x 1 − h x 2 ≦ h x 1 − x 2 |h(x_(1))-h(x_(2))| <= h(x_(1)-x_(2))\left|h\left(x_{1}\right)-h\left(x_{2}\right)\right| \leqq h\left(x_{1}-x_{2}\right)|h(x1)−h(x2)|≦h(x1−x2) de unde, pentru x 1 ≠ x 2 x 1 ≠ x 2 x_(1)!=x_(2)x_{1} \neq x_{2}x1≠x2
(4) | h ( x 1 ) − h ( x 2 ) | ‖ x 1 − x 2 ‖ ∣ ≦ h ( x 1 − x 2 ) ‖ ∣ x 1 − x 2 ‖ ‖ ≦ sup x ≠ y x , y ∈ X h ( x − y ) ‖ x − y ‖ ‖ (4) h x 1 − h x 2 x 1 − x 2 ∣ ≦ h x 1 − x 2 ∣ x 1 − x 2 ‖ ≦ sup x ≠ y x , y ∈ X   h ( x − y ) ‖ x − y ‖ ‖ {:(4)(|h(x_(1))-h(x_(2))|)/(||x_(1)-x_(2)||∣) <= (h(x_(1)-x_(2)))/(||∣x_(1)-x_(2)||||) <= s u p_({:[x!=y],[x","y in X]:})(h(x-y))/(||x-y||||):}\begin{equation*} \frac{\left|h\left(x_{1}\right)-h\left(x_{2}\right)\right|}{\left\|x_{1}-x_{2}\right\| \mid} \leqq \frac{h\left(x_{1}-x_{2}\right)}{\left\|\mid x_{1}-x_{2}\right\| \|} \leqq \sup _{\substack{x \neq y \\ x, y \in X}} \frac{h(x-y)}{\|x-y\| \|} \tag{4} \end{equation*}(4)|h(x1)−h(x2)|‖x1−x2‖∣≦h(x1−x2)‖∣x1−x2‖‖≦supx≠yx,y∈Xh(x−y)‖x−y‖‖
Pe de altă parte h ∈ C X p h ∈ C X p h inC_(X)^(p)h \in C_{X}^{p}h∈CXp dacă şi numai dacă sup x ∈ X − { θ } h ( x ) ‖ x ‖ < ∞ sup x ∈ X − { θ }   h ( x ) ‖ x ‖ < ∞ s u p_(x in X-{theta})(h(x))/(||x||) < oo\sup _{x \in X-\{\theta\}} \frac{h(x)}{\|x\|}<\inftysupx∈X−{θ}h(x)‖x‖<∞ (vezi [4] pag. 16, Obs. 2.6) și avînd în vedere (4) rezultă că h ∈ X # h ∈ X # h inX^(#)h \in X^{\#}h∈X#.
Evident X τ p X τ p X_(tau)^(p)X_{\tau}^{p}Xτp este un subspaţiu liniar al lui X # X # X^(#)X^{\#}X#.
Pe| X τ p X τ p X_(tau)^(p)X_{\tau}^{p}Xτp se definesc următoarele norme (vezi Definitia 2.7 şi Lema 2.8 din [4]) :
(5) ‖ ‖ X 1 : X τ p → R + (5) ‖ X 1 : X τ p → R + {:(5)||||_(X)^(1):X_(tau)^(p)rarrR^(+):}:}\begin{equation*} \left\|\|_{X}^{1}: X_{\tau}^{p} \rightarrow R^{+}\right. \tag{5} \end{equation*}(5)‖‖X1:Xτp→R+
∀ f ∈ X τ p , ‖ f ‖ X 1 = sup ‖ x ‖ ‖ ⩽ 1 | f ( x ) | , ∀ f ∈ X τ p , ‖ f ‖ X 1 = sup ‖ x ‖ ‖ ⩽ 1   | f ( x ) | , AA f inX_(tau)^(p),||f||_(X)^(1)=s u p_(||x|||| <= 1)|f(x)|,\forall f \in X_{\tau}^{p},\|f\|_{X}^{1}=\sup _{\|x\| \| \leqslant 1}|f(x)|,∀f∈Xτp,‖f‖X1=sup‖x‖‖⩽1|f(x)|,
(6)
‖ ‖ X : X τ p → R + ‖ X : X τ p → R + ||||_(X):X_(tau)^(p)rarrR^(+):}\left\|\|_{X}: X_{\tau}^{p} \rightarrow R^{+}\right.‖‖X:Xτp→R+
∀ f ∈ X τ p , ‖ f ‖ X = q ( B 1 ( 0 , 1 ) ∩ ∩ ∫ X p τ − B 1 ( 0 , 1 ) ∩ c X p τ ) ( f ) ∀ f ∈ X τ p , ‖ f ‖ X = q B 1 ( 0 , 1 ) ∩ ∩ ∫ X p   τ − B 1 ( 0 , 1 ) ∩ c X p τ ( f ) AA f inX_(tau)^(p),quad||f||_(X)=q_((B_(1)(0,1)nn nnint_(X)^(p)tau-B_(1)(0,1)nnc_(X)^(p)tau))(f)\forall f \in X_{\tau}^{p}, \quad\|f\|_{X}=q_{\left(B_{1}(0,1) \cap \cap \int_{X}^{p} \tau-B_{1}(0,1) \cap c_{X}^{p} \tau\right)}(f)∀f∈Xτp,‖f‖X=q(B1(0,1)∩∩∫Xpτ−B1(0,1)∩cXpτ)(f)
unde q q qqq este funcţionala lui Minkowski ataşată mulțimii B 1 ( 0 , 1 ) ∩ C X p τ B 1 ( 0 , 1 ) ∩ C X p τ B_(1)(0,1)nnC_(X)^(p)tauB_{1}(0,1) \cap C_{X}^{p} \tauB1(0,1)∩CXpτ - − B 1 ( 0 , 1 ) ∩ C X p τ − B 1 ( 0 , 1 ) ∩ C X p τ -B_(1)(0,1)nnC_(X)^(p_(tau))-B_{1}(0,1) \cap C_{X}^{p_{\tau}}−B1(0,1)∩CXpτ iar B 1 ( 0 , 1 ) = { f ∈ X τ p , ‖ f ‖ X 1 ≦ 1 } B 1 ( 0 , 1 ) = f ∈ X τ p , ‖ f ‖ X 1 ≦ 1 B_(1)(0,1)={f inX_(tau)^(p),||f||_(X)^(1) <= 1}B_{1}(0,1)=\left\{f \in X_{\tau}^{p},\|f\|_{X}^{1} \leqq 1\right\}B1(0,1)={f∈Xτp,‖f‖X1≦1}.
Conform Lemei 2.8 din [4], ( X τ p , ‖ ‖ X X τ p , ‖ ‖ X X_(tau)^(p),||||_(X)X_{\tau}^{p},\| \|_{X}Xτp,‖‖X ) este un spațiu Banach şi pentru orice f ∈ X τ p , ‖ f ‖ X 1 ≦ ‖ f ‖ X f ∈ X τ p , ‖ f ‖ X 1 ≦ ‖ f ‖ X f inX_(tau)^(p),||f||_(X)^(1) <= ||f||_(X)f \in X_{\tau}^{p},\|f\|_{X}^{1} \leqq\|f\|_{X}f∈Xτp,‖f‖X1≦‖f‖X. Dacă h h hhh este chiar din C X p τ C X p τ C_(X)^(p)tauC_{X}^{p} \tauCXpτ atunci ‖ h ‖ X = ‖ h ‖ X 1 ‖ h ‖ X = ‖ h ‖ X 1 ||h||_(X)=||h||_(X)^(1)\|h\|_{X}=\|h\|_{X}^{1}‖h‖X=‖h‖X1.
‥ Fie Y Y YYY un subspatiu liniar al lui ( X , | | | | | | ) X , | | | ) (X,||||||):}\left(X,\left|\left|\left||| |)\right.\right.\right.\right.(X,||||||). Pentru h ∈ C X p τ h ∈ C X p τ h inC_(X)^(p_(tau))h \in C_{X}^{p_{\tau}}h∈CXpτ vom nota cu h | Y h Y h|_(Y)\left.h\right|_{Y}h|Y restrictia lui h h hhh pe subspatiul Y Y YYY.
teorema 1. (Teorema 2.9 din [4]). Fie Y Y YYY un subspatiu liniar al lui ( X X XXX, III III) si h ∈ C X p τ h ∈ C X p τ h inC_(X)^(p_(tau))h \in C_{X}^{p_{\tau}}h∈CXpτ. Atunci functionala
(7)
H : X → R + H : X → R + H:X rarrR^(+)H: X \rightarrow R^{+}H:X→R+
∀ x ∈ X H ( x ) = inf y ∈ Y { h | Y ( y ) + ‖ h | Y ‖ Y ⋅ ‖ ∣ x − y ‖ ‖ } ∀ x ∈ X H ( x ) = inf y ∈ Y   h Y ( y ) + h Y Y ⋅ ‖ ∣ x − y ‖ ‖ AA x in X quad H(x)=i n f_(y in Y){h|_(Y)(y)+||h|_(Y)||_(Y)*||∣x-y||||}\forall x \in X \quad H(x)=\inf _{y \in Y}\left\{\left.h\right|_{Y}(y)+\left\|\left.h\right|_{Y}\right\|_{Y} \cdot\|\mid x-y\| \|\right\}∀x∈XH(x)=infy∈Y{h|Y(y)+‖h|Y‖Y⋅‖∣x−y‖‖}
verifică proprietățile:
(8) H ∈ C X p τ ; H | Y = h | Y ; ‖ H ‖ X = ‖ h | Y ‖ Y . (8) H ∈ C X p τ ; H Y = h Y ; ‖ H ‖ X = h Y Y . {:(8)H inC_(X)^(p)tau;H|_(Y)=h|_(Y);||H||_(X)=||h|_(Y)||_(Y).:}\begin{equation*} H \in C_{X}^{p} \tau ;\left.H\right|_{Y}=\left.h\right|_{Y} ;\|H\|_{X}=\left\|\left.h\right|_{Y}\right\|_{Y} . \tag{8} \end{equation*}(8)H∈CXpτ;H|Y=h|Y;‖H‖X=‖h|Y‖Y.
Functionala H H HHH din teorema 1 se numeşte o prelungire a restrictiei lui h ∈ C X p τ h ∈ C X p τ h inC_(X)^(p_(tau))h \in C_{X}^{p_{\tau}}h∈CXpτ pe Y Y YYY, de pe Y Y YYY pe X X XXX cu păstrarea normei de pe Y Y YYY.
2. În general, prelungirea H H HHH, cu proprietățile (8) nu este unică. În cele ce urmează vom găsi o condiţie necesată şi suficientă pentru unicitatea unei astfel de prelungiri. Pentru alte tipuri de functionale, condiţii pentru unicitatea prelungirii se pot găsi în lucrările [1], [2], [3].
Definifia 1. Fie ( X , ‖ ‖ X , ‖ ‖ X,||||X,\| \|X,‖‖ ) un spatiu liniar normat, V V VVV o submultime a sa nevidă şi Y Y YYY un subspațiu liniar al lui X X XXX. Vom zice că subspatiul Y Y YYY este V V VVV-cebîsevian dacă dîndu-se v ∈ V v ∈ V v in Vv \in Vv∈V există un singur element y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y astfel ca
(9)
‖ v − y 0 ‖ = inf y ∈ Y ‖ v − y ‖ = d ( v , Y ) . v − y 0 = inf y ∈ Y   ‖ v − y ‖ = d ( v , Y ) . ||v-y_(0)||=i n f_(y in Y)||v-y||=d(v,Y).\left\|v-y_{0}\right\|=\inf _{y \in Y}\|v-y\|=d(v, Y) .‖v−y0‖=infy∈Y‖v−y‖=d(v,Y).
Fie Y Y YYY un subspatiu liniar al lui ( X , ‖ ‖ ‖ X , ‖ ‖ ‖ X,||||||X,\| \| \|X,‖‖‖ ). Vom nota
(10) Y X τ p ⊥ = { f ∈ X τ p , f ( y ) = 0 pentru toti y ∈ Y } . (10) Y X τ p ⊥ = f ∈ X τ p , f ( y ) = 0  pentru toti  y ∈ Y . {:(10)Y_(X_(tau)^(p))^(_|_)={f inX_(tau)^(p),f(y)=0" pentru toti "y in Y}.:}\begin{equation*} Y_{X_{\tau}^{p}}^{\perp}=\left\{f \in X_{\tau}^{p}, f(y)=0 \text { pentru toti } y \in Y\right\} . \tag{10} \end{equation*}(10)YXτp⊥={f∈Xτp,f(y)=0 pentru toti y∈Y}.
Evident Y X τ p ⊥ Y X τ p ⊥ Y_(X_(tau)^(p))^(_|_)Y_{X_{\tau}^{p}}^{\perp}YXτp⊥ este un subspatiu al lui X τ p X τ p X_(tau)^(p)X_{\tau}^{p}Xτp.
Lema 1. Fie Y Y YYY un subspatiu liniar al lui ( X , ‖ ‖ ‖ ) ( X , ‖ ‖ ‖ ) (X,||||||)(X,\| \| \|)(X,‖‖‖) şi h ∈ C X p τ h ∈ C X p τ h inC_(X)^(p)tauh \in C_{X}^{p} \tauh∈CXpτ. Atunci are loc următoavea egalitate:
(11)
‖ h | Y ‖ Y = d ( h , Y X p ⊥ ) . h Y Y = d h , Y X p ⊥ . ||h|_(Y)||_(Y)=d(h,Y_(X^(p))^(_|_)).\left\|\left.h\right|_{Y}\right\|_{Y}=d\left(h, Y_{X^{p}}^{\perp}\right) .‖h|Y‖Y=d(h,YXp⊥).
Demonstratie. Conform teoremer 1 , dacă h ∈ C X p h ∈ C X p h inC_(X)^(p)h \in C_{X}^{p}h∈CXp, pentru h | Y h Y h|_(Y)\left.h\right|_{Y}h|Y există H ∈ C X p τ H ∈ C X p τ H inC_(X)^(p)tauH \in C_{X}^{p} \tauH∈CXpτ cu proprietățile (8). Conform Lemei 2.8 (d) din [4] avem:
‖ h | Y ‖ Y = ‖ h | Y ‖ Y 1 = sup ‖ y ‖ ‖ 1 < 1 y ∈ Y h ( y ) = sup ‖ y 1 ‖ ‖ 1 < 1 y ∈ Y | h ( y ) | = = sup | ( h − g ) ( y ) | ≦ sup | ( h − g ) ( x ) | = h Y Y = h Y Y 1 = sup ‖ y ‖ ‖ 1 < 1 y ∈ Y   h ( y ) = sup y 1 ‖ 1 < 1 y ∈ Y   | h ( y ) | = = sup | ( h − g ) ( y ) | ≦ sup | ( h − g ) ( x ) | = {:[||h|_(Y)||_(Y)=||h|_(Y)||_(Y)^(1)=s u p_({:[||y||||_(1) < 1],[y in Y]:})h(y)=s u p_({:[||y_(1)||||_(1) < 1],[y in Y]:})|h(y)|=],[=s u p|(h-g)(y)| <= s u p|(h-g)(x)|=]:}\begin{aligned} & \left\|\left.h\right|_{Y}\right\|_{Y}=\left\|\left.h\right|_{Y}\right\|_{Y}^{1}=\sup _{\substack{\|y\| \|_{1}<1 \\ y \in Y}} h(y)=\sup _{\substack{\left\|y_{1}\right\| \|_{1}<1 \\ y \in Y}}|h(y)|= \\ & =\sup |(h-g)(y)| \leqq \sup |(h-g)(x)|= \end{aligned}‖h|Y‖Y=‖h|Y‖Y1=sup‖y‖‖1<1y∈Yh(y)=sup‖y1‖‖1<1y∈Y|h(y)|==sup|(h−g)(y)|≦sup|(h−g)(x)|=
= ‖ h − g ‖ X 1 ≦ ‖ h − g ‖ X . = ‖ h − g ‖ X 1 ≦ ‖ h − g ‖ X . {:=||h-g||_(X)^(1) <= ||h-g||_(X).:}\begin{aligned} & =\|h-g\|_{X}^{1} \leqq\|h-g\|_{X} . \end{aligned}=‖h−g‖X1≦‖h−g‖X.
De aici rezultă că
‖ h | Y ‖ Y ≦ inf ε ∈ Y ⊥ ‖ h − g ‖ X p = d ( h , Y X τ p ⊥ ) h Y Y ≦ inf ε ∈ Y ⊥   ‖ h − g ‖ X p = d h , Y X τ p ⊥ ||h|_(Y)||_(Y) <= i n f_(epsi inY^(_|_))||h-g||_(X)^(p)=d(h,Y_(X_(tau)^(p))^(_|_))\left\|\left.h\right|_{Y}\right\|_{Y} \leqq \inf _{\varepsilon \in Y^{\perp}}\|h-g\|_{X}^{p}=d\left(h, Y_{X_{\tau}^{p}}^{\perp}\right)‖h|Y‖Y≦infε∈Y⊥‖h−g‖Xp=d(h,YXτp⊥)
Invers, conform teoremei 1 avem:
‖ h | Y ‖ Y = ‖ H ‖ X = ‖ h − ( h − H ) ‖ X ≧ d ( h , Y X τ p ⊥ ) . h Y Y = ‖ H ‖ X = ‖ h − ( h − H ) ‖ X ≧ d h , Y X τ p ⊥ . ||h|_(Y)||_(Y)=||H||_(X)=||h-(h-H)||_(X) >= d(h,Y_(X_(tau)^(p))^(_|_)).\left\|\left.h\right|_{Y}\right\|_{Y}=\|H\|_{X}=\|h-(h-H)\|_{X} \geqq d\left(h, Y_{X_{\tau}^{p}}^{\perp}\right) .‖h|Y‖Y=‖H‖X=‖h−(h−H)‖X≧d(h,YXτp⊥).
Deci
‖ h | Y ‖ Y = d ( h , Y X τ p ⊥ ) . h Y Y = d h , Y X τ p ⊥ . ||h|_(Y)||_(Y)=d(h,Y_(X_(tau)^(p))^(_|_)).\left\|\left.h\right|_{Y}\right\|_{Y}=d\left(h, Y_{X_{\tau}^{p}}^{\perp}\right) .‖h|Y‖Y=d(h,YXτp⊥).
teorema 2. Fie Y Y YYY un subspatiu liniar al lui ( X , ‖ ‖ X , ‖ ‖ X,||||X,\| \|X,‖‖ |||) şi h ∈ C X p h ∈ C X p h inC_(X)^(p)h \in C_{X}^{p}h∈CXp. Următoavele două afirmații sînt echivalente:
a) Oricare ar fi h ∈ C X p τ , h | Y h ∈ C X p τ , h Y h inC_(X)^(p)tau,h|_(Y)h \in C_{X}^{p} \tau,\left.h\right|_{Y}h∈CXpτ,h|Y are o prelungire H H HHH, cave verifică proprietătile (8) unică.
b) Y X τ p ⊥ Y X τ p ⊥ Y_(X_(tau)^(p))^(_|_)Y_{X_{\tau}^{p}}^{\perp}YXτp⊥ este C X p τ − C X p τ − C_(X)^(p)tau-C_{X}^{p} \tau-CXpτ− cebîşevian.
Demonstratie. a) ⇒ ⇒ =>\Rightarrow⇒ b). Mai întîi observăm că pentru orice h ∈ C X p τ h ∈ C X p τ h inC_(X)^(p)tauh \in C_{X}^{p} \tauh∈CXpτ există un element g 0 ∈ Y X τ p ⊥ g 0 ∈ Y X τ p ⊥ g_(0)inY_(X_(tau)^(p))^(_|_)g_{0} \in Y_{X_{\tau}^{p}}^{\perp}g0∈YXτp⊥ astfel ca ‖ h − g 0 ‖ X = d ( h , Y X τ p ⊥ ) h − g 0 X = d h , Y X τ p ⊥ ||h-g_(0)||_(X)=d(h,Y_(X_(tau)^(p))^(_|_))\left\|h-g_{0}\right\|_{X}=d\left(h, Y_{X_{\tau}^{p}}^{\perp}\right)‖h−g0‖X=d(h,YXτp⊥). Intr-adevăr, conform TEOREMEI 1 și LEMEI 1 , h | Y 1 , h Y 1,h|_(Y)1,\left.h\right|_{Y}1,h|Y are o prelungire H ∈ C X p τ H ∈ C X p τ H inC_(X)^(p)tauH \in C_{X}^{p} \tauH∈CXpτ astfel ca
‖ h | Y ‖ Y = d ( h , Y X τ p ⊥ ) = ‖ h − ( h − H ) ‖ X . h Y Y = d h , Y X τ p ⊥ = ‖ h − ( h − H ) ‖ X . ||h|_(Y)||_(Y)=d(h,Y_(X_(tau)^(p))^(_|_))=||h-(h-H)||_(X).\left\|\left.h\right|_{Y}\right\|_{Y}=d\left(h, Y_{X_{\tau}^{p}}^{\perp}\right)=\|h-(h-H)\|_{X} .‖h|Y‖Y=d(h,YXτp⊥)=‖h−(h−H)‖X.
Deci g 0 = h − H g 0 = h − H g_(0)=h-Hg_{0}=h-Hg0=h−H.
Să presupunem acum că Y X ⊥ Y X ⊥ Y_(X)^(_|_)Y_{X}^{\perp}YX⊥ nu este C X p C X p C_(X)^(p)C_{X}^{p}CXp - cevişevian; atunci există h ∈ C X p τ h ∈ C X p τ h inC_(X)^(p)tauh \in C_{X}^{p} \tauh∈CXpτ şi există g 1 , g 2 din Y X τ p τ , g 1 ≠ g 2 g 1 , g 2 din Y X τ p τ , g 1 ≠ g 2 g_(1),g_(2)dinY_(X_(tau)^(p))^(tau),g_(1)!=g_(2)g_{1}, g_{2} \operatorname{din} Y_{X_{\tau}^{p}}^{\tau}, g_{1} \neq g_{2}g1,g2dinYXτpτ,g1≠g2 astfel ca
(12) ‖ h − g 1 ‖ X = ‖ h − g 2 ‖ X = d ( h , Y X τ p ⊥ ) = ‖ h | Y ‖ Y . (12) h − g 1 X = h − g 2 X = d h , Y X τ p ⊥ = h Y Y . {:(12)||h-g_(1)||_(X)=||h-g_(2)||_(X)=d(h,Y_(X_(tau)^(p))^(_|_))=||h|_(Y)||_(Y).:}\begin{equation*} \left\|h-g_{1}\right\|_{X}=\left\|h-g_{2}\right\|_{X}=d\left(h, Y_{X_{\tau}^{p}}^{\perp}\right)=\left\|\left.h\right|_{Y}\right\|_{Y} . \tag{12} \end{equation*}(12)‖h−g1‖X=‖h−g2‖X=d(h,YXτp⊥)=‖h|Y‖Y.
Dar atunci, avînd în vedere şi egalitățile (12) rezultă că h − g 1 h − g 1 h-g_(1)h-g_{1}h−g1 şi h − g 2 h − g 2 h-g_(2)h-g_{2}h−g2 sînt două prelungiri diferite ale lui h | Y h Y h|_(Y)\left.h\right|_{Y}h|Y.
b) ⇒ ⇒ =>\Rightarrow⇒ a). Să presupunem că există h ∈ C X p h ∈ C X p h inC_(X)^(p)h \in C_{X}^{p}h∈CXp astfel ca h | Y h Y h|_(Y)\left.h\right|_{Y}h|Y să aibă prelungirile H 1 , H 2 ∈ C X p τ , H 1 ≠ H 2 cu H 1 , H 2 ∈ C X p τ , H 1 ≠ H 2 cu H_(1),H_(2)inC_(X)^(p)tau,H_(1)!=H_(2)cuH_{1}, H_{2} \in C_{X}^{p} \tau, H_{1} \neq H_{2} \mathrm{cu}H1,H2∈CXpτ,H1≠H2cu proprietățile (8). Atunci din mema 1 rezultă că:
‖ H 1 ‖ X = ‖ H 1 − ( H 1 − H 2 ) ‖ X = ‖ h | Y ‖ Y = ‖ H 1 | Y ‖ Y = d ( H 1 , Y X τ p ⊥ ) . H 1 X = H 1 − H 1 − H 2 X = h Y Y = H 1 Y Y = d H 1 , Y X τ p ⊥ . ||H_(1)||_(X)=||H_(1)-(H_(1)-H_(2))||_(X)=||h|_(Y)||_(Y)=||H_(1)|_(Y)||_(Y)=d(H_(1),Y_(X_(tau)^(p))^(_|_)).\left\|H_{1}\right\|_{X}=\left\|H_{1}-\left(H_{1}-H_{2}\right)\right\|_{X}=\left\|\left.h\right|_{Y}\right\|_{Y}=\left\|\left.H_{1}\right|_{Y}\right\|_{Y}=d\left(H_{1}, Y_{X_{\tau}^{p}}^{\perp}\right) .‖H1‖X=‖H1−(H1−H2)‖X=‖h|Y‖Y=‖H1|Y‖Y=d(H1,YXτp⊥).
Dar aceasta înseamnă că pentru H 1 H 1 H_(1)H_{1}H1 există două elemente din Y x τ p ⊥ Y x τ p ⊥ Y_(x_(tau)^(p))^(_|_)Y_{x_{\tau}^{p}}^{\perp}Yxτp⊥ pentru care are loc (9) şi anume Φ Φ Phi\PhiΦ şi H 1 − H 2 ≠ Φ H 1 − H 2 ≠ Φ H_(1)-H_(2)!=PhiH_{1}-H_{2} \neq \PhiH1−H2≠Φ, deci Y X ⊥ τ p Y X ⊥ τ p Y_(X)^(_|_)_(tau)^(p)Y_{X}^{\perp}{ }_{\tau}^{p}YX⊥τp nu este C X p C X p C_(X)^(p)C_{X}^{p}CXp r-cebîşevian.

SUR L'UNICITÉ DU PROLONGEMENT DES p p ppp-SÉMINORMES CONTINUES

RÉSUME

Soit ( X , ‖ ‖ ‖ X , ‖ ‖ ‖ X,||||||X,\| \| \|X,‖‖‖ ) un espace p p ppp-normé réel ( p ∈ ( 0 , 1 ] p ∈ ( 0 , 1 ] p in(0,1]p \in(0,1]p∈(0,1] ), Y Y YYY un sousespace de ( X , ‖ ∣ ‖ ) ( X , ‖ ∣ ‖ ) (X,||∣||)(X,\|\mid\|)(X,‖∣‖) et soit C X p C X p C_(X)^(p)C_{X}^{p}CXp r le cône des p p ppp-séminormes continues sur ( X , ‖ ‖ ‖ ) ( X , ‖ ‖ ‖ ) (X,||||||)(X,\| \| \|)(X,‖‖‖). Soit h ∈ C X p τ h ∈ C X p τ h inC_(X)^(p)tauh \in C_{X}^{p} \tauh∈CXpτ et H H HHH un prolongement de h | Y h Y h|_(Y)\left.h\right|_{Y}h|Y de Y Y YYY sur X X XXX qui conserve 1a norme de Y Y YYY. On montre que H H HHH est un prolongement unique si et seulement si ∣ Y C ⊥ X p τ − C X p τ ∣ Y C ⊥ X p τ − C X p τ ∣Y_(C)^(_|_)_(X)^(p)tau-C_(X)^(p)tau\mid Y_{C}^{\perp}{ }_{X}^{p} \tau-C_{X}^{p} \tau∣YC⊥Xpτ−CXpτ lest un sousespace de Tchébycheff pour les éléments de C X p τ C X p τ C_(X)^(p)tauC_{X}^{p} \tauCXpτ.

BIBLIOGRAFIE

[1] Kolumban I., Ob edinstvenosti prodoljenia lineinîh functionalov, Mathematica, vol. 4
[2] (24), 2, 1962), 267-270. schitziene, "Revista de analiză numerică şi teoria aproximației", vol. 2, fasc. 1, (1973), "81-87.
[3] Phelps R. R., Uniqueness of Hahn-Banach extension and unique best approximction, Trans. Amer. Math. Soc., 95, (1960), 238-255.
[4] Ruess W., Ein Dualkegel für p-konvexe topologische lineare Räume, Gesellschaft für Mathematik und Datenverarbeltung, Bonn, Nr. 60, (1973).
Institutul de calcul din Clui
al Academiei Republicii Socialiste
Románia
Primit la 28. V. 1973.
    • R + R + R+\mathrm{R}+R+ reprezintă mulțimea numerelor reale nenegative.
1973

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