Extension of bilinear operators and best approximation in 2-normed space

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

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Stefan Cobzas
Babes-Bolyai University

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

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Şt. Cobzaş, C. Mustăţa, Extension of bilinear operators and best approximation in 2-normed spaces, Rev. Anal. Numér. Théor. Approx., 25 (1996) nos. 1-2, 63-75.

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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/1996-vol25-nos1-2-art7

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2457-6794

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2501-059X

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[1] I. Beg, M. Iqbal, Extension of linear 2-operators, Mathematica Montesnigri 2 (1993), 1-10.
[2] S. Cobzas, C. Mustata, Extension of bilinear functionals and best approximation in 2-normed spaces (to appear).
[3] N. Dunford, J.T. Schwartz, Linear Operators Part I: General theory, Interscience Publishers, New York, 1958.
[4] I. Franic, An extension theorem for bounded linear 2-functionals and applications, Math. Japonica 40(1994), 79-85.
[5] S. Gahler, 2-Metrische Raume and ihre Topologische Struktur, Math. Nachr. 26 (1963/64), 115-148.
[6] S. Gahler, Linear 2-Normierte Raume, Math. Nachr. 28 (1965), 1-45.
[7] S. Gahler, Uber 2-Banach-Raume, Math. Nachr. 42 (1969),, 335-347.
[8] T.L. Hayden, The extension of bilinear funcitonals, Pacific, J. Math. 22 (1967), 99-108.
[9] R.B. Holmes, On the continuity of best approximaiton operators, inSymposium on Infinite Dimensional Topology (R.D. Anderson Editor), Annals of Math. Studies, Princeton Univ. Press, Princeton 1972, 137-158.
[10] R.B. Holmes, Geometric Functional Analysis and its Application, Springer-Verlag, Berlin 1975.
[11] K. Iseki, Mathematics on 2-normed spaces, Bull. Korean, Math. Soc., 13 (1976), 127-136.
[12] L.Lindenstrauss, Extension of compact operators, Mem. Amer. Math.Soc. 48 (1964)
[13] S. Mabizela, On bounded 2-linear fucntionals, Math. Japonica, 35 (1990), 51-55.
[14] K. Nachbin, A theorem of Hahn-Banach type for linear transformation, Trans. Amer. Math. Soc., 68 (1950), 28-46.
[15] L. Nachnin, Some problems in extending and lifting continuous linear transformations, in Proc.International symposium on Linear Spaces, Jerusalem, 1960.
[16] R.R. Phelps, Uniquencess of Hahn-Banach extension and unique best approximaiton, Trans. Amer. Math. Soc., 95 (1960), 238-255.
[17] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Publishing House of the Romanian Academy and Springer Verlag, Bucharest and Berlin, 1970.
[18] A.G. White Jr., 2-Banach spaces, Math. Nachr., 42 (1969), 43-60.
[19] A.G. White and Yeol Je Cho, Linear mappings on linear 2-normed spaces, Bull. Korean Math. Soc. 21 (1984), 1-6.

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1996-Mustata-Extension of bilinear operators and best approximation-Jnaat

EXTENSION OF BILINEAR OPERATORS AND BEST APPROXIMATION IN 2-NORMED SPACES

S. COBZAŞ and C. MUSTÃȚA(Cluj-Napoca)

1. INTRODUCTION

In 1965 S. Gähler [6] defined 2-normed spaces and studied their basic properties. Since then the field has considerably grown, the research being directed to obtain a theory similar to that of normed spaces. A key result in developing such a theory is a Hahn-Banach type theorem for bilinear functional on 2-normed spaces. But, as remarked S. Gähler [7, p.345], a general Hahn-Banach theorem doesn't hold in this setting. Some extension theorems for bounded bilinear functionals defined on subspaces of the form Z × [ b ] Z × [ b ] Z xx[b]Z \times[b]Z×[b] to X × [ b ] ( X − a 2 X × [ b ] ( X − a 2 X xx[b](X-a2X \times[b](X-a 2X×[b](X−a2-normed space, Z Z ZZZ a subspace of X X XXX and [ b ] [ b ] [b][b][b] - the subspace of X X XXX spanned by b ∈ X ∖ { 0 } b ∈ X ∖ { 0 } b in X\\{0}b \in X \backslash\{0\}b∈X∖{0} ) were proved by A. G. White [18], S. Mabizela [13] and I. Franić [4]. In [2] it was shown that all these results follow directly from the classical Hahn-Banach theorem for linear functionals on seminormed spaces.
Inspired by some results of L. Nachbin [14], [15] and of J. Lindenstrauss [12], I. Beg and M. Iqbal [1] proved some extension theorems for bounded or compact bilinear operators defined also on subspaces of the form Z × [ b ] Z × [ b ] Z xx[b]Z \times[b]Z×[b]. In this paper we shall show that, again, all these results follow directly from the corresponding results for linear operators on normed spaces. The key tool will be a result relating the bilinear operators from Z × [ b ] Z × [ b ] Z xx[b]Z \times[b]Z×[b] to a semi-normed space ( Y , q Y , q Y,qY, qY,q ) and the linear operators from Z Z ZZZ to Y Y YYY (Proposition 3.2 below). The extension results are applied to obtain some duality results for best approximation in spaces of bounded bilinear operators.

2. BOUNDED LINEAR OPERATORS ON SEMINORMED SPACES

Let ( X , p ) ( X , p ) (X,p)(X, p)(X,p) and ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) be two seminormed spaces. It is well known that a linear operator A : X → Y A : X → Y A:X rarr YA: X \rightarrow YA:X→Y is continuous if and only if it is bounded (or Lipschitz), i.e. there exists a number L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 such that
q ( A x ) ≤ L ⋅ p ( x ) , for all x ∈ X q ( A x ) ≤ L ⋅ p ( x ) ,  for all  x ∈ X q(Ax) <= L*p(x)," for all "x in Xq(A x) \leq L \cdot p(x), \text { for all } x \in Xq(Ax)≤L⋅p(x), for all x∈X
A number L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 verifying (2.1) is called a Lipschitz constant for A A AAA. For a bounded linear operator A : X → Y A : X → Y A:X rarr YA: X \rightarrow YA:X→Y define by

(2.2)

‖ A ‖ = sup { q ( A x ) : x ∈ X , p ( x ) ≤ 1 } , ‖ A ‖ = sup { q ( A x ) : x ∈ X , p ( x ) ≤ 1 } , ||A||=s u p{q(Ax):x in X,quad p(x) <= 1},\|A\|=\sup \{q(A x): x \in X, \quad p(x) \leq 1\},‖A‖=sup{q(Ax):x∈X,p(x)≤1},
the norm of the operator A A AAA. The following results are well known in the case of normed spaces (see e.g. [3]). Since their proofs can be transposed with slight and obvious modifications to the case of seminormed spaces we shall omit them. Denote by L ( X , Y ) L ( X , Y ) L(X,Y)L(X, Y)L(X,Y) the space of all bounded linear operators from X X XXX to Y Y YYY.
Proposition 2.1 Let ( X , p ) ( X , p ) (X,p)(X, p)(X,p) and ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) be seminormed spaces. Then the following assertions hold:
1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ If A ∈ L ( X , Y ) A ∈ L ( X , Y ) A in L(X,Y)A \in L(X, Y)A∈L(X,Y) then the number ‖ A ‖ ‖ A ‖ ||A||\|A\|‖A‖, defined by (2.2) is the smallest Lipschitz constant for A A AAA, i.e.
‖ A ‖ = min { L ≥ 0 : L is a Lipschitz constant for A } . ‖ A ‖ = min { L ≥ 0 : L  is a Lipschitz constant for  A } .  ||A||=min{L >= 0:L" is a Lipschitz constant for "A}". "\|A\|=\min \{L \geq 0: L \text { is a Lipschitz constant for } A\} \text {. }‖A‖=min{L≥0:L is a Lipschitz constant for A}. 
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ The application ‖ ⋅ ‖ : L ( X , Y ) → [ 0 , ∞ ) ‖ ⋅ ‖ : L ( X , Y ) → [ 0 , ∞ ) ||*||:L(X,Y)rarr[0,oo)\|\cdot\|: L(X, Y) \rightarrow[0, \infty)‖⋅‖:L(X,Y)→[0,∞) is a seminorm on L ( X , Y ) L ( X , Y ) L(X,Y)L(X, Y)L(X,Y) which is a norm if and only if q q qqq is a norm on Y Y YYY.
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘ The seminormed space ( L ( X , Y ) , ‖ ⋅ ‖ ) ( L ( X , Y ) , ‖ ⋅ ‖ ) (L(X,Y),||*||)(L(X, Y),\|\cdot\|)(L(X,Y),‖⋅‖) is complete if (and only if when q ≠ 0 q ≠ 0 q!=0q \neq 0q≠0 ) the seminormed space ( Y , q Y , q Y,qY, qY,q ) is complete.

3. BOUNDED BILINEAR OPERATORS ON 2-NORMED SPACES

Let X X XXX be a real vector space of dimension at least 2 . An application ‖ ⋅ ‖ : , X × X → [ 0 , ∞ ) ‖ ⋅ ‖ : , X × X → [ 0 , ∞ ) ||*||:,X xx X rarr[0,oo)\|\cdot\|:, X \times X \rightarrow[0, \infty)‖⋅‖:,X×X→[0,∞) is called a 2 -norm on X X XXX if
BN 1) ‖ x , y ‖ = 0 ‖ x , y ‖ = 0 ||x,y||=0\|x, y\|=0‖x,y‖=0 if and only if x , y x , y x,yx, yx,y are linearly dependent,
BN 2) ‖ x , y ‖ = ‖ y , x ‖ ‖ x , y ‖ = ‖ y , x ‖ ||x,y||=||y,x||\|x, y\|=\|y, x\|‖x,y‖=‖y,x‖,
BN 3) ‖ λ x , y ‖ = | λ | ‖ x , y ‖ ‖ λ x , y ‖ = | λ | ‖ x , y ‖ ||lambda x,y||=|lambda|||x,y||\|\lambda x, y\|=|\lambda|\|x, y\|‖λx,y‖=|λ|‖x,y‖,
BN 4) ‖ x + y , z ‖ ≤ ‖ x , z ‖ + ‖ y , x ‖ ‖ x + y , z ‖ ≤ ‖ x , z ‖ + ‖ y , x ‖ ||x+y,z|| <= ||x,z||+||y,x||\|x+y, z\| \leq\|x, z\|+\|y, x\|‖x+y,z‖≤‖x,z‖+‖y,x‖,
for all x , y , z ∈ X x , y , z ∈ X x,y,z in Xx, y, z \in Xx,y,z∈X and all λ ∈ R λ ∈ R lambda inR\lambda \in \mathbf{R}λ∈R (see [6]).
A 2 -normed space is a real linear space equipped with a 2 -norm ‖ ⋅ , ⋅ ‖ ‖ ⋅ , ⋅ ‖ ||*,*||\|\cdot, \cdot\|‖⋅,⋅‖. If ( X , ‖ ⋅ ‖ ( X , ‖ ⋅ ‖ (X,||*||(X,\|\cdot\|(X,‖⋅‖, ) i s a 2 ) i s a 2 )isa2) is a 2)isa2-normed space and b ∈ X b ∈ X b in Xb \in Xb∈X then the functional p b : X → [ 0 , ∞ ) p b : X → [ 0 , ∞ ) p_(b):X rarr[0,oo)p_{b}: X \rightarrow[0, \infty)pb:X→[0,∞),
defined by p b ( x ) = ‖ x , b ‖ , x ∈ X p b ( x ) = ‖ x , b ‖ , x ∈ X p_(b)(x)=||x,b||,x in Xp_{b}(x)=\|x, b\|, x \in Xpb(x)=‖x,b‖,x∈X, is a seminorm on X X XXX. The locally convex topology generated by the family P = { p b : b ∈ X } P = p b : b ∈ X P={p_(b):b in X}P=\left\{p_{b}: b \in X\right\}P={pb:b∈X} of seminoms is called the natural topology of X X XXX induced by the 2-norm ‖ ⋅ ⋅ ‖ ‖ ⋅ ⋅ ‖ ||**||\|\cdot \cdot\|‖⋅⋅‖, (see [6]).
Let ( X , ‖ ⋅ ‖ ( X , ‖ ⋅ ‖ (X,||*||(X,\|\cdot\|(X,‖⋅‖, ) b e a 2 − n o m e d s p a c e a n d X 1 , X 2 ) b e a 2 − n o m e d s p a c e a n d X 1 , X 2 )bea2-nomedspaceandX_(1),X_(2)) be a 2-nomed space and X_{1}, X_{2})bea2−nomedspaceandX1,X2 subspaces of X X XXX. A bilinear operator is an application T T TTT from X 1 × X 2 X 1 × X 2 X_(1)xxX_(2)X_{1} \times X_{2}X1×X2 to a seminormed space ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) such that: (BL) T ( ⋅ , y ) : X 1 → Y T ( ⋅ , y ) : X 1 → Y T(*,y):X_(1)rarr YT(\cdot, y): X_{1} \rightarrow YT(⋅,y):X1→Y and T ( x , ⋅ ) : X 2 → Y T ( x , ⋅ ) : X 2 → Y T(x,*):X_(2)rarr YT(x, \cdot): X_{2} \rightarrow YT(x,⋅):X2→Y are linear operators, for all x ∈ X 1 x ∈ X 1 x inX_(1)x \in X_{1}x∈X1 and all y ∈ X 2 y ∈ X 2 y inX_(2)y \in X_{2}y∈X2.
A bilinear operator T : X 1 × X 2 → Y T : X 1 × X 2 → Y T:X_(1)xxX_(2)rarr YT: X_{1} \times X_{2} \rightarrow YT:X1×X2→Y is called bounded if there exists a number L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 such that
(3.1) q ( T ( x , y ) ) ≤ L ‖ x , y ‖ , for all ( x , y ) ∈ X 1 × X 2 . (3.1) q ( T ( x , y ) ) ≤ L ‖ x , y ‖ ,  for all  ( x , y ) ∈ X 1 × X 2 . {:(3.1)q(T(x","y)) <= L||x","y||","" for all "(x","y)inX_(1)xxX_(2).:}\begin{equation*} q(T(x, y)) \leq L\|x, y\|, \text { for all }(x, y) \in X_{1} \times X_{2} . \tag{3.1} \end{equation*}(3.1)q(T(x,y))≤L‖x,y‖, for all (x,y)∈X1×X2.
A number L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 verifying (3.1) is called a Lipschitz constant for T T TTT. A bilinear functional is a bilinear operator F : X 1 × X 2 → R F : X 1 × X 2 → R F:X_(1)xxX_(2)rarrRF: X_{1} \times X_{2} \rightarrow \mathrm{R}F:X1×X2→R. As it was shown by A. G. White [18] in the case of bilinear functionals, and by I. Beg and M. Iqbal [1] in general, the boundedness of a bilinear operator can be characterized by a kind of a continuity condition, called 2 -continuity by S . Gähler [7]. It turns that the 2 -continuity of a bilinear operator at ( 0,0 ) implies its 2-continuity on the whole X 1 × X 2 X 1 × X 2 X_(1)xxX_(2)X_{1} \times X_{2}X1×X2. A typical example of a (nonlinear) functional which is continuous on X × X X × X X xx XX \times XX×X is the 2-norm ‖ ⋅ ⋅ ⋅ ‖ ‖ ⋅ ⋅ ⋅ ‖ ||***||\|\cdot \cdot \cdot\|‖⋅⋅⋅‖. As remarked S S SSS. Gähler [7] this notion of 2-continuity is different from the continuity with respect to the natural product topology on X × X X × X X xx XX \times XX×X.
For a bounded bilinear operator T : X 1 × X 2 → Y T : X 1 × X 2 → Y T:X_(1)xxX_(2)rarr YT: X_{1} \times X_{2} \rightarrow YT:X1×X2→Y define
(3.2) v ( T ) = sup { q ( T ( x , y ) ) : ( x , y ) ∈ X 1 × X 2 , ‖ x , y ‖ ≤ 1 } (3.2) v ( T ) = sup q ( T ( x , y ) ) : ( x , y ) ∈ X 1 × X 2 , ‖ x , y ‖ ≤ 1 {:(3.2)v(T)=s u p{q(T(x,y)):(x,y)inX_(1)xxX_(2),||x,y|| <= 1}:}\begin{equation*} v(T)=\sup \left\{q(T(x, y)):(x, y) \in X_{1} \times X_{2},\|x, y\| \leq 1\right\} \tag{3.2} \end{equation*}(3.2)v(T)=sup{q(T(x,y)):(x,y)∈X1×X2,‖x,y‖≤1}
and denote by L 2 ( X 1 × X 2 , Y ) L 2 X 1 × X 2 , Y L_(2)(X_(1)xxX_(2),Y)L_{2}\left(X_{1} \times X_{2}, Y\right)L2(X1×X2,Y) the linear space of all bounded bilinear operators from X 1 × X 2 X 1 × X 2 X_(1)xxX_(2)X_{1} \times X_{2}X1×X2 to Y Y YYY. As in the case of linear operators on normed spaces one can easily prove:
Proposition 3.1 Let ( X , ‖ ⋅ ‖ ( X , ‖ ⋅ ‖ (X,||*||(X,\|\cdot\|(X,‖⋅‖, ) b e a 2 ) b e a 2 )bea2) be a 2)bea2-normed space X 1 , X 2 X 1 , X 2 X_(1),X_(2)X_{1}, X_{2}X1,X2 subspaces of X X XXX and ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) a seminormed space. Then the following assertions hold:
1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ If T ∈ L 2 ( X 1 × X 2 , Y ) T ∈ L 2 X 1 × X 2 , Y T inL_(2)(X_(1)xxX_(2),Y)T \in L_{2}\left(X_{1} \times X_{2}, Y\right)T∈L2(X1×X2,Y) then v ( T ) v ( T ) v(T)v(T)v(T) is the smallest Lipschits constant for T T TTT, i.e.
(3.3) v ( T ) = min { L ≥ 0 : L is a Lipschitz constant for T } . (3.3) v ( T ) = min { L ≥ 0 : L  is a Lipschitz constant for  T } . {:(3.3)v(T)=min{L >= 0:L" is a Lipschitz constant for "T}.:}\begin{equation*} v(T)=\min \{L \geq 0: L \text { is a Lipschitz constant for } T\} . \tag{3.3} \end{equation*}(3.3)v(T)=min{L≥0:L is a Lipschitz constant for T}.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ The application v v vvv : L 2 ( X 1 > X 2 , Y ) → [ 0 , ∞ ) L 2 X 1 > X 2 , Y → [ 0 , ∞ ) L_(2)(X_(1) > X_(2),Y)rarr[0,oo)L_{2}\left(X_{1}>X_{2}, Y\right) \rightarrow[0, \infty)L2(X1>X2,Y)→[0,∞) is a seminorm on L 2 ( X 1 × X 2 , Y ) L 2 X 1 × X 2 , Y L_(2)(X_(1)xxX_(2),Y)L_{2}\left(X_{1} \times X_{2}, Y\right)L2(X1×X2,Y) which is a norm if and only if q q qqq is a norm on Y Y YYY.
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘ The seminormed space ( L 2 ( X 1 × X 2 , Y ) , v L 2 X 1 × X 2 , Y , v L_(2)(X_(1)xxX_(2),Y),vL_{2}\left(X_{1} \times X_{2}, Y\right), vL2(X1×X2,Y),v ) is complete if the seminormed space ( Y , q Y , q Y,qY, qY,q ) is complete.
Remark. The completeness of L 2 ( X × X , Y ) L 2 ( X × X , Y ) L_(2)(X xx X,Y)L_{2}(X \times X, Y)L2(X×X,Y) for the case of a Banach space ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) was proved by A. G. White Jr. [18].
For an element b ∈ X ∖ { 0 } b ∈ X ∖ { 0 } b in X\\{0}b \in X \backslash\{0\}b∈X∖{0} denote by [ b ] [ b ] [b][b][b] the subspace of X X XXX spanned by b b bbb (i.e. [ b ] = R ⋅ b [ b ] = R ⋅ b [b]=R*b[b]=\mathbf{R} \cdot b[b]=R⋅b ). If Z Z ZZZ is a subspace of X X XXX let p b p b p_(b)p_{b}pb denote the seminorm p b ( z ) = ‖ z , b ‖ p b ( z ) = ‖ z , b ‖ p_(b)(z)=||z,b||p_{b}(z)=\|z, b\|pb(z)=‖z,b‖, z ∈ Z z ∈ Z z in Zz \in Zz∈Z. The bilinear operators from Z × [ b ] Z × [ b ] Z xx[b]Z \times[b]Z×[b] to a seminormed space ( Y , q Y , q Y,qY, qY,q ) and the linear operators between the seminormed spaces ( Z , p b ) Z , p b (Z,p_(b))\left(Z, p_{b}\right)(Z,pb) and ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) are related as in the following proposition. Here ‖ A ‖ ‖ A ‖ ||A||\|A\|‖A‖ and ν ( T ) ν ( T ) nu(T)\nu(T)ν(T) denote the norms of a linear operator A A AAA (cf. (2.2)) and respectively of a bilinear operator T T TTT (cf. (3.2)).
Proposition 3.21 ∘ 3.21 ∘ 3.21^(@)3.21^{\circ}3.21∘ If T : Z × [ b ] → Y T : Z × [ b ] → Y T:Z xx[b]rarr YT: Z \times[b] \rightarrow YT:Z×[b]→Y is a bounded bilinear operator then the operator A : ( Z , p b ) → ( Y , q ) A : Z , p b → ( Y , q ) A:(Z,p_(b))rarr(Y,q)A:\left(Z, p_{\mathrm{b}}\right) \rightarrow(Y, q)A:(Z,pb)→(Y,q) defined by A z = T ( z , b ) , z ∈ Z A z = T ( z , b ) , z ∈ Z Az=T(z,b),z in ZA z=T(z, b), z \in ZAz=T(z,b),z∈Z, is a continuous linear operator and
(3.4)
‖ A ‖ = v ( T ) . ‖ A ‖ = v ( T ) . ||A||=v(T).\|A\|=v(T) .‖A‖=v(T).
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ Conversely, if A : ( Z , p b ) → ( Y , q ) A : Z , p b → ( Y , q ) A:(Z,p_(b))rarr(Y,q)A:\left(Z, p_{b}\right) \rightarrow(Y, q)A:(Z,pb)→(Y,q) is a continuous linear operator, then the operator T : Z × [ b ] → Y T : Z × [ b ] → Y T:Z xx[b]rarr YT: Z \times[b] \rightarrow YT:Z×[b]→Y, defined by T ( z , α b ) = α ⋅ A z T ( z , α b ) = α ⋅ A z T(z,alpha b)=alpha*AzT(z, \alpha b)=\alpha \cdot A zT(z,αb)=α⋅Az, for z ∈ Z z ∈ Z z in Zz \in Zz∈Z and α ∈ R α ∈ R alpha inR\alpha \in \mathbf{R}α∈R, is a bounded bilinear operator and
(3.5) ν ( T ) = ‖ A ‖ (3.5) ν ( T ) = ‖ A ‖ {:(3.5)nu(T)=||A||:}\begin{equation*} \nu(T)=\|A\| \tag{3.5} \end{equation*}(3.5)ν(T)=‖A‖
Proof: 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ If T : Z × [ b ] → Y T : Z × [ b ] → Y T:Z xx[b]rarr YT: Z \times[b] \rightarrow YT:Z×[b]→Y is a bilinear operator, it is immediate that the operator A : Z → Y A : Z → Y A:Z rarr YA: Z \rightarrow YA:Z→Y defined by A z = T ( z , b ) , z ∈ Z A z = T ( z , b ) , z ∈ Z Az=T(z,b),z in ZA z=T(z, b), z \in ZAz=T(z,b),z∈Z, is linear. Since
q ( A z ) = q ( T ( z , b ) ) ≤ v ( T ) ‖ z , b ‖ = v ( T ) ⋅ p b ( z ) , q ( A z ) = q ( T ( z , b ) ) ≤ v ( T ) ‖ z , b ‖ = v ( T ) ⋅ p b ( z ) , q(Az)=q(T(z,b)) <= v(T)||z,b||=v(T)*p_(b)(z),q(A z)=q(T(z, b)) \leq v(T)\|z, b\|=v(T) \cdot p_{b}(z),q(Az)=q(T(z,b))≤v(T)‖z,b‖=v(T)⋅pb(z),
for all z ∈ Z z ∈ Z z in Zz \in Zz∈Z, it follows that A A AAA is continuous and ‖ A ‖ ≤ ν ( T ) ‖ A ‖ ≤ ν ( T ) ||A|| <= nu(T)\|A\| \leq \nu(T)‖A‖≤ν(T)
But
q ( T ( z , α b ) ) = q ( T ( α z , b ) ) = q ( A ( α z ) ) ≤ ‖ A ‖ ⋅ p b ( α z ) = = ‖ A ‖ ⋅ ‖ α z , b ‖ = ‖ A ‖ ⋅ ‖ z , α b ‖ q ( T ( z , α b ) ) = q ( T ( α z , b ) ) = q ( A ( α z ) ) ≤ ‖ A ‖ ⋅ p b ( α z ) = = ‖ A ‖ ⋅ ‖ α z , b ‖ = ‖ A ‖ ⋅ ‖ z , α b ‖ {:[q(T(z","alpha b))=q(T(alpha z","b))=q(A(alpha z)) <= ||A||*p_(b)(alpha z)=],[=||A||*||alpha z","b||=||A||*||z","alpha b||]:}\begin{aligned} q(T(z, \alpha b))= & q(T(\alpha z, b))=q(A(\alpha z)) \leq\|A\| \cdot p_{b}(\alpha z)= \\ & =\|A\| \cdot\|\alpha z, b\|=\|A\| \cdot\|z, \alpha b\| \end{aligned}q(T(z,αb))=q(T(αz,b))=q(A(αz))≤‖A‖⋅pb(αz)==‖A‖⋅‖αz,b‖=‖A‖⋅‖z,αb‖
for all z ∈ Z z ∈ Z z in Zz \in Zz∈Z and all α ∈ R α ∈ R alpha inR\alpha \in \mathbf{R}α∈R implying v ( T ) ≤ ‖ A ‖ v ( T ) ≤ ‖ A ‖ v(T) <= ||A||v(T) \leq\|A\|v(T)≤‖A‖ and ‖ A ‖ = v ( T ) ‖ A ‖ = v ( T ) ||A||=v(T)\|A\|=v(T)‖A‖=v(T).
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ Suppose now that A : ( Z , p b ) → ( Y , q ) A : Z , p b → ( Y , q ) A:(Z,p_(b))rarr(Y,q)A:\left(Z, p_{b}\right) \rightarrow(Y, q)A:(Z,pb)→(Y,q) is a continuous linear operator and let T : Z × [ b ] → Y T : Z × [ b ] → Y T:Z xx[b]rarr YT: Z \times[b] \rightarrow YT:Z×[b]→Y be defined by T ( z , α b ) = α ⋅ A z T ( z , α b ) = α ⋅ A z T(z,alpha b)=alpha*AzT(z, \alpha b)=\alpha \cdot A zT(z,αb)=α⋅Az, for z ∈ Z z ∈ Z z in Zz \in Zz∈Z and α ∈ R α ∈ R alpha inR\alpha \in \mathbf{R}α∈R. It is obvious that T T TTT is a bilinear operator and from
q ( T ( z , α b ) ) = q ( α ⋅ A z ) = q ( A ( α z ) ) ≤ ‖ A ‖ ⋅ p b ( α z ) = ‖ A ‖ ⋅ ‖ α z , b ‖ = ‖ A ‖ ⋅ ‖ z , α b ‖ q ( T ( z , α b ) ) = q ( α ⋅ A z ) = q ( A ( α z ) ) ≤ ‖ A ‖ ⋅ p b ( α z ) = ‖ A ‖ ⋅ ‖ α z , b ‖ = ‖ A ‖ ⋅ ‖ z , α b ‖ q(T(z,alpha b))=q(alpha*Az)=q(A(alpha z)) <= ||A||*p_(b)(alpha z)=||A||*||alpha z,b||=||A||*||z,alpha b||q(T(z, \alpha b))=q(\alpha \cdot A z)=q(A(\alpha z)) \leq\|A\| \cdot p_{b}(\alpha z)=\|A\| \cdot\|\alpha z, b\|=\|A\| \cdot\|z, \alpha b\|q(T(z,αb))=q(α⋅Az)=q(A(αz))≤‖A‖⋅pb(αz)=‖A‖⋅‖αz,b‖=‖A‖⋅‖z,αb‖
we get v ( T ) ≤ ‖ A ‖ v ( T ) ≤ ‖ A ‖ v(T) <= ||A||v(T) \leq\|A\|v(T)≤‖A‖.
The equalities A ( α z ) = T ( z , α b ) , ‖ α z , b ‖ = ‖ z , α b ‖ A ( α z ) = T ( z , α b ) , ‖ α z , b ‖ = ‖ z , α b ‖ A(alpha z)=T(z,alpha b),||alpha z,b||=||z,alpha b||A(\alpha z)=T(z, \alpha b),\|\alpha z, b\|=\|z, \alpha b\|A(αz)=T(z,αb),‖αz,b‖=‖z,αb‖, and the definitions of the norms ‖ A ‖ ‖ A ‖ ||A||\|A\|‖A‖ and v ( T ) v ( T ) v(T)v(T)v(T) (relations (2.2) and (3.2) respectively) imply
‖ A ‖ = sup { q ( A z ) : z ∈ Z , p b ( z ) ≤ 1 } = sup { q ( A z ) : z ∈ Z , ‖ z , b ‖ ≤ 1 } ≤ sup { q ( A ( α z ) ) : z ∈ Z , α ∈ R , ‖ α z , b ‖ ≤ 1 } = sup { q ( T ( z , α b ) ) : z ∈ Z , α ∈ R , ‖ z , α b ‖ ≤ 1 } = v ( T ) ‖ A ‖ = sup q ( A z ) : z ∈ Z , p b ( z ) ≤ 1 = sup { q ( A z ) : z ∈ Z , ‖ z , b ‖ ≤ 1 } ≤ sup { q ( A ( α z ) ) : z ∈ Z , α ∈ R , ‖ α z , b ‖ ≤ 1 } = sup { q ( T ( z , α b ) ) : z ∈ Z , α ∈ R , ‖ z , α b ‖ ≤ 1 } = v ( T ) {:[||A||= s u p{q(Az):z in Z,p_(b)(z) <= 1}=s u p{q(Az):z in Z","||z","b|| <= 1}],[ <= s u p{q(A(alpha z)):z in Z","alpha inR","||alpha z","b|| <= 1}],[=s u p{q(T(z","alpha b)):z in Z","alpha inR","||z","alpha b|| <= 1}=v(T)]:}\begin{aligned} \|A\|= & \sup \left\{q(A z): z \in Z, p_{b}(z) \leq 1\right\}=\sup \{q(A z): z \in Z,\|z, b\| \leq 1\} \\ & \leq \sup \{q(A(\alpha z)): z \in Z, \alpha \in \mathbf{R},\|\alpha z, b\| \leq 1\} \\ & =\sup \{q(T(z, \alpha b)): z \in Z, \alpha \in \mathbf{R},\|z, \alpha b\| \leq 1\}=v(T) \end{aligned}‖A‖=sup{q(Az):z∈Z,pb(z)≤1}=sup{q(Az):z∈Z,‖z,b‖≤1}≤sup{q(A(αz)):z∈Z,α∈R,‖αz,b‖≤1}=sup{q(T(z,αb)):z∈Z,α∈R,‖z,αb‖≤1}=v(T)
showing that v ( T ) = ‖ A ‖ v ( T ) = ‖ A ‖ v(T)=||A||v(T)=\|A\|v(T)=‖A‖

4. NORM PRESERVING EXTENSIONS OF BILINEAR OPERATORS

Let ( X , ‖ ‖ ( X , ‖ ‖ (X,||||(X,\|\|(X,‖‖, ) b e a 2 − n o r m e d s p a c e a n d X 1 , X 2 ) b e a 2 − n o r m e d s p a c e a n d X 1 , X 2 )bea2-normedspaceandX_(1),X_(2)) be a 2-normed space and X_{1}, X_{2})bea2−normedspaceandX1,X2 linear subspaces of X X XXX. A normpreserving extension of a bounded bilinear operator T T TTT from X 1 × X 2 X 1 × X 2 X_(1)xxX_(2)X_{1} \times X_{2}X1×X2 to a semmormed space ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) is a bounded bilinear operator T ~ : X ~ 1 × X ~ 2 → Y T ~ : X ~ 1 × X ~ 2 → Y widetilde(T): widetilde(X)_(1)xx widetilde(X)_(2)rarr Y\widetilde{T}: \widetilde{X}_{1} \times \widetilde{X}_{2} \rightarrow YT~:X~1×X~2→Y, (where X ~ 1 X ~ 1 widetilde(X)_(1)\widetilde{X}_{1}X~1 and X ~ 2 X ~ 2 widetilde(X)_(2)\widetilde{X}_{2}X~2 are linear subspaces of X X XXX containing X 1 X 1 X_(1)X_{1}X1 respectively X 2 X 2 X_(2)X_{2}X2 ), such that
i) T ~ ( x 1 , x 2 ) = T ( x 1 , x 2 ) T ~ x 1 , x 2 = T x 1 , x 2 widetilde(T)(x_(1),x_(2))=T(x_(1),x_(2))\widetilde{T}\left(x_{1}, x_{2}\right)=T\left(x_{1}, x_{2}\right)T~(x1,x2)=T(x1,x2), for all ( x 1 , x 2 ) ∈ X 1 × X 2 x 1 , x 2 ∈ X 1 × X 2 (x_(1),x_(2))inX_(1)xxX_(2)\left(x_{1}, x_{2}\right) \in X_{1} \times X_{2}(x1,x2)∈X1×X2 and
ii) v ( T ~ ) = v ( T ) v ( T ~ ) = v ( T ) v( widetilde(T))=v(T)v(\widetilde{T})=v(T)v(T~)=v(T).
For two seminormed spaces ( X , p X , p X,pX, pX,p ) and ( Y , q Y , q Y,qY, qY,q ) a norm-preserving extension of a bounded linear operator A A AAA, defined on a subspace Z Z ZZZ of X X XXX and taking values in Y Y YYY, is a bounded linear operator A ~ A ~ widetilde(A)\widetilde{A}A~ defined on a subspace Z ~ Z ~ widetilde(Z)\widetilde{Z}Z~ of X , Z ⊆ Z ~ X , Z ⊆ Z ~ X,Z sube widetilde(Z)X, Z \subseteq \widetilde{Z}X,Z⊆Z~, and taking values in Y Y YYY, such that
i) A ~ z = A z A ~ z = A z widetilde(A)z=Az\widetilde{A} z=A zA~z=Az, for all z ∈ Z z ∈ Z z in Zz \in Zz∈Z, and
ii) ‖ A ~ ‖ = ‖ A ‖ ‖ A ~ ‖ = ‖ A ‖ || widetilde(A)||=||A||\|\widetilde{A}\|=\|A\|‖A~‖=‖A‖.
A normed space ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) is said to have the extension property if for any normed space ( X , p X , p X,pX, pX,p ), every continuous linear operator A A AAA, defined on a subspace Z Z ZZZ of X X XXX and taking values in Y Y YYY, has a norm-preserving extension A : X → Y A : X → Y A:X rarr YA: X \rightarrow YA:X→Y. A normed space ( Y , q Y , q Y,qY, qY,q ) is said to have the binary intersection property if every family of mutually intersecting closed convex balls in Y Y YYY has a nonvoid intersection. By a famous result of L. Nachbin [14] (see also [15]) a normed space ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) has the extension property if and only if it has the binary intersection property. The binary intersection property, and the extension property can be defined in a similar way for seminormed spaces, yielding a seminormed version of Nachbin's result.
I. Beg and M. Iqbal [1] transposed the sufficiency part of Nachbin's theorem to bilinear operators defined on subspaces of the form Z × [ b ] Z × [ b ] Z xx[b]Z \times[b]Z×[b]. We shall show that this result is an immediate consequence of Nachbin's result and of Proposition 3.2.
First we prove the following result:
Proposition 4.1 Let ( X , ‖ , ⋅ ‖ X , ‖ , ⋅ ‖ X,||,*||X,\|, \cdot\|X,‖,⋅‖ ) be a 2 -normed space ( Y , q Y , q Y,qY, qY,q ) a seminormed space. Let Z Z ZZZ and Z ~ Z ~ widetilde(Z)\widetilde{Z}Z~ be subspaces of X X XXX such that Z ⊆ Z ~ Z ⊆ Z ~ Z sube widetilde(Z)Z \subseteq \widetilde{Z}Z⊆Z~ and let b ∈ X ∖ { 0 } b ∈ X ∖ { 0 } b in X\\{0}b \in X \backslash\{0\}b∈X∖{0}. Suppose that the operators T : Z × [ b ] → Y , T ~ : Z ~ × [ b ] → Y T : Z × [ b ] → Y , T ~ : Z ~ × [ b ] → Y T:Z xx[b]rarr Y, widetilde(T): widetilde(Z)xx[b]rarr YT: Z \times[b] \rightarrow Y, \widetilde{T}: \widetilde{Z} \times[b] \rightarrow YT:Z×[b]→Y,T~:Z~×[b]→Y and A : Z → Y A : Z → Y A:Z rarr YA: Z \rightarrow YA:Z→Y respectively A ~ : Z ~ → Y A ~ : Z ~ → Y widetilde(A): widetilde(Z)rarr Y\widetilde{A}: \widetilde{Z} \rightarrow YA~:Z~→Y, are related as in Proposition 3.2.
Then T ~ T ~ widetilde(T)\widetilde{T}T~ is a norm-preserving extension of T T TTT if and only if the corresponding linear operator A ~ A ~ widetilde(A)\widetilde{A}A~ is a norm-preserving extension of A A AAA.
Proof. The proof is an immediate consequence of the relations T ( z , α b ) = α A z T ( z , α b ) = α A z T(z,alpha b)=alpha AzT(z, \alpha b)=\alpha A zT(z,αb)=αAz, ( z , α ) ∈ Z × R , v ( T ) = ‖ A ‖ ( z , α ) ∈ Z × R , v ( T ) = ‖ A ‖ (z,alpha)in Z xxR,v(T)=||A||(z, \alpha) \in Z \times \mathbf{R}, v(T)=\|A\|(z,α)∈Z×R,v(T)=‖A‖ and T ~ ( z ~ , α b ) = α ⋅ A ~ z ~ , ( z ~ , α ) ∈ Z ~ × R , v ( T ~ ) = ‖ A ~ ‖ T ~ ( z ~ , α b ) = α ⋅ A ~ z ~ , ( z ~ , α ) ∈ Z ~ × R , v ( T ~ ) = ‖ A ~ ‖ widetilde(T)( widetilde(z),alpha b)=alpha* widetilde(A) widetilde(z),( widetilde(z),alpha)in widetilde(Z)xxR,v( widetilde(T))=|| widetilde(A)||\widetilde{T}(\widetilde{z}, \alpha b)=\alpha \cdot \widetilde{A} \widetilde{z},(\widetilde{z}, \alpha) \in \widetilde{Z} \times \mathbf{R}, v(\widetilde{T})=\|\widetilde{A}\|T~(z~,αb)=α⋅A~z~,(z~,α)∈Z~×R,v(T~)=‖A~‖, relating the operators T T TTT and A A AAA, respectively T ~ T ~ widetilde(T)\widetilde{T}T~ and A ~ A ~ widetilde(A)\widetilde{A}A~
Let us agree to say that a seminormed space ( Y , q Y , q Y,qY, qY,q ) has the restricted extension property for bilinear operators if for any 2 -normed space ( X , ‖ X , ‖ X,||X,\|X,‖, ‖ ) e v e r y ‖ ) e v e r y ||)every\| ) every‖)every bounded bilinear operator T : Z × [ b ] → Y ( Z T : Z × [ b ] → Y ( Z T:Z xx[b]rarr Y(ZT: Z \times[b] \rightarrow Y(ZT:Z×[b]→Y(Z a subspace of X X XXX and b ∈ X ∖ { 0 } b ∈ X ∖ { 0 } b in X\\{0}b \in X \backslash\{0\}b∈X∖{0} has a norm-preserving extension T ~ : X × [ b ] → Y T ~ : X × [ b ] → Y widetilde(T):X xx[b]rarr Y\widetilde{T}: X \times[b] \rightarrow YT~:X×[b]→Y. Using these terms, Proposition 4.1 can be restated as follows:
COROLLARY 4.2 A seminormed space ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) has the restricted extension property for bilinear operators if it has the extension property for linear operators.
Observing that the proof of sufficiency part of Nachbin's theorem [14, p.31] remains valid when all spaces are supposed to be seminormed we get:
COROLLARY 4.3 ([1, Th. 2.1]). If a seminormed space ( Y , q Y , q Y,qY, qY,q ) has the binary intersection property, then it has the restricted extension property for bilinear operators.
Remark. We do not know whether the necessity part of Nachbin's theorem remains valid for bilinear operators too: Must a seminormed space ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) verifying the restricted extension property for bilinear operators have the binary intersection property?
The extension result for operators defined on condimension one subspaces, proved by J. Lindenstrauss [12, Lemma 5.2], can be transposed to bilinear operators too.
Proposition 4.4 Let ( X , ‖ ⋅ ‖ ) ( X , ‖ ⋅ ‖ ) (X,||*||)(X,\|\cdot\|)(X,‖⋅‖) be a 2 -normed space, Z Z ZZZ a codimension one subspace of X X XXX and b ∈ X ∖ { 0 } b ∈ X ∖ { 0 } b in X\\{0}b \in X \backslash\{0\}b∈X∖{0}. A bounded bilinear operator T T TTT from Z × [ b ] Z × [ b ] Z xx[b]Z \times[b]Z×[b] to a seminormed space ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) admits a norm-preserving extension T ~ : X × [ b ] → Y T ~ : X × [ b ] → Y widetilde(T):X xx[b]rarr Y\widetilde{T}: X \times[b] \rightarrow YT~:X×[b]→Y if and only if there exists u ∈ X ∖ Z u ∈ X ∖ Z u in X\\Zu \in X \backslash Zu∈X∖Z such that
(4.1) ⋂ { B q ( T ( z , b ) , v ( T ) ‖ u − z , b ‖ : z ∈ Z ) } ≠ ∅ (4.1) ⋂ B q ( T ( z , b ) , v ( T ) ‖ u − z , b ‖ : z ∈ Z ) ≠ ∅ {:(4.1)nnn{B_(q)(T(z,b),v(T)||u-z,b||:z in Z)}!=O/:}\begin{equation*} \bigcap\left\{B_{q}(T(z, b), v(T)\|u-z, b\|: z \in Z)\right\} \neq \varnothing \tag{4.1} \end{equation*}(4.1)⋂{Bq(T(z,b),v(T)‖u−z,b‖:z∈Z)}≠∅
Proof. If u ∈ X ∖ Z u ∈ X ∖ Z u in X\\Zu \in X \backslash Zu∈X∖Z then X = Z + R u X = Z + R u X=Z+RuX=Z+\mathrm{R} uX=Z+Ru and any bilinear extension T ~ : X × [ b ] → Y T ~ : X × [ b ] → Y widetilde(T):X xx[b]rarr Y\widetilde{T}: X \times[b] \rightarrow YT~:X×[b]→Y of T T TTT is completely determined through the formula
(4.2) T ~ ( z + α u , β b ) = T ( z , β b ) + α β y 0 , (4.2) T ~ ( z + α u , β b ) = T ( z , β b ) + α β y 0 , {:(4.2) widetilde(T)(z+alpha u","beta b)=T(z","beta b)+alpha betay_(0)",":}\begin{equation*} \widetilde{T}(z+\alpha u, \beta b)=T(z, \beta b)+\alpha \beta y_{0}, \tag{4.2} \end{equation*}(4.2)T~(z+αu,βb)=T(z,βb)+αβy0,
by its value y 0 y 0 y_(0)y_{0}y0 at ( u , b ) ( u , b ) (u,b)(u, b)(u,b). Consequently, T ~ T ~ widetilde(T)\widetilde{T}T~ is a norm-preserving extension of T T TTT if and only if
(4.3) q ( T ~ ( z + α u , β b ) ≤ v ( T ) ⋅ ‖ z + α u , β b ‖ ) , (4.3) q ( T ~ ( z + α u , β b ) ≤ v ( T ) ⋅ ‖ z + α u , β b ‖ ) , {:(4.3)q( widetilde(T)(z+alpha u","beta b) <= v(T)*||z+alpha u","beta b||)",":}\begin{equation*} q(\widetilde{T}(z+\alpha u, \beta b) \leq v(T) \cdot\|z+\alpha u, \beta b\|), \tag{4.3} \end{equation*}(4.3)q(T~(z+αu,βb)≤v(T)⋅‖z+αu,βb‖),
for all z ∈ Z z ∈ Z z in Zz \in Zz∈Z and all α , β ∈ R α , β ∈ R alpha,beta inR\alpha, \beta \in \mathbf{R}α,β∈R. Supposing α ⋅ β ≠ 0 α ⋅ β ≠ 0 alpha*beta!=0\alpha \cdot \beta \neq 0α⋅β≠0 and deleting by | α ⋅ β | > 0 | α ⋅ β | > 0 |alpha*beta| > 0|\alpha \cdot \beta|>0|α⋅β|>0, one obtains successively:
q ( T ~ ( z + α u , β b ) ) ≤ v ( T ) ⋅ ‖ z + α u , β b ‖ ⇔ q ( T ~ ( α − 1 z + u , b ) ) ≤ v ( T ) ⋅ ‖ α − 1 z + u , b ‖ ⇔ q ( y 0 − T ( z ′ , b ) ) ≤ v ( T ) ⋅ ‖ u − z ′ , b ‖ q ( T ~ ( z + α u , β b ) ) ≤ v ( T ) ⋅ ‖ z + α u , β b ‖ ⇔ q T ~ α − 1 z + u , b ≤ v ( T ) ⋅ α − 1 z + u , b ⇔ q y 0 − T z ′ , b ≤ v ( T ) ⋅ u − z ′ , b {:[q( widetilde(T)(z+alpha u","beta b)) <= v(T)*||z+alpha u","beta b||<=>],[q(( widetilde(T))(alpha^(-1)z+u,b)) <= v(T)*||alpha^(-1)z+u,b||<=>],[q(y_(0)-T(z^('),b)) <= v(T)*||u-z^('),b||]:}\begin{gathered} q(\widetilde{T}(z+\alpha u, \beta b)) \leq v(T) \cdot\|z+\alpha u, \beta b\| \Leftrightarrow \\ q\left(\widetilde{T}\left(\alpha^{-1} z+u, b\right)\right) \leq v(T) \cdot\left\|\alpha^{-1} z+u, b\right\| \Leftrightarrow \\ q\left(y_{0}-T\left(z^{\prime}, b\right)\right) \leq v(T) \cdot\left\|u-z^{\prime}, b\right\| \end{gathered}q(T~(z+αu,βb))≤v(T)⋅‖z+αu,βb‖⇔q(T~(α−1z+u,b))≤v(T)⋅‖α−1z+u,b‖⇔q(y0−T(z′,b))≤v(T)⋅‖u−z′,b‖
for all z ′ = − α − 1 z ∈ Z z ′ = − α − 1 z ∈ Z z^(')=-alpha^(-1)z in Zz^{\prime}=-\alpha^{-1} z \in Zz′=−α−1z∈Z. This last relation is equivalent to:
y 0 ∈ ⋂ { B q ( T ( z , b ) , v ( T ) ⋅ ‖ u − z , b ‖ ) : z ∈ Z } y 0 ∈ ⋂ B q ( T ( z , b ) , v ( T ) ⋅ ‖ u − z , b ‖ ) : z ∈ Z y_(0)in nnn{B_(q)(T(z,b),v(T)*||u-z,b||):z in Z}y_{0} \in \bigcap\left\{B_{q}(T(z, b), v(T) \cdot\|u-z, b\|): z \in Z\right\}y0∈⋂{Bq(T(z,b),v(T)⋅‖u−z,b‖):z∈Z}
Since, for α = 0 q ( T ~ ( z , β b ) ) = q ( T ( z , β b ) ) ≤ ν ( T ) ⋅ ‖ z , β b ‖ α = 0 q ( T ~ ( z , β b ) ) = q ( T ( z , β b ) ) ≤ ν ( T ) ⋅ ‖ z , β b ‖ alpha=0quad q( widetilde(T)(z,beta b))=q(T(z,beta b)) <= nu(T)*||z,beta b||\alpha=0 \quad q(\widetilde{T}(z, \beta b))=q(T(z, \beta b)) \leq \nu(T) \cdot\|z, \beta b\|α=0q(T~(z,βb))=q(T(z,βb))≤ν(T)⋅‖z,βb‖ and for β = 0 T ~ ( z + α u , 0 ) = 0 β = 0 T ~ ( z + α u , 0 ) = 0 beta=0 widetilde(T)(z+alpha u,0)=0\beta=0 \widetilde{T}(z+\alpha u, 0)=0β=0T~(z+αu,0)=0, the proposition is proved ◻ ◻ ◻\square◻.
Remark. 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ From the proof it is clear that if relation (4.3) holds for an element u 0 ∈ X ∖ Z u 0 ∈ X ∖ Z u_(0)in X\\Zu_{0} \in X \backslash Zu0∈X∖Z, then it holds for any other element u ∈ X ∖ Z u ∈ X ∖ Z u in X\\Zu \in X \backslash Zu∈X∖Z.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ Proposition 4.4 appears in [1, Proposition 3.3] in a slightly different form.

5. COMPACT BILINEAR OPERATORS

The aim of this section is to show how some extension results for compact operators on normed spaces, proved by J. Lindestrauss [12], can be transposed to bilinear operators on 2-nomed spaces. The basic tool used in doing this will be again Proposition 3.2.
Roughly speaking, a compact bilinear operator is a bilinear operator mapping bounded sets into relatively compact ones. We shall consider three boundedness notions in 2-normed spaces and three corresponding compactness notions for bilinear operators.
Let ( X , ‖ ⋅ ‖ ) ( X , ‖ ⋅ ‖ ) (X,||*||)(X,\|\cdot\|)(X,‖⋅‖) be a 2-normed space and b ∈ X ∖ { 0 } b ∈ X ∖ { 0 } b in X\\{0}b \in X \backslash\{0\}b∈X∖{0}. A subset V V VVV of X X XXX is called p b p b p_(b)p_{b}pb-bounded if sup p b ( V ) < ∞ sup p b ( V ) < ∞ s u pp_(b)(V) < oo\sup p_{b}(V)<\inftysuppb(V)<∞. The set V V VVV is called bounded if it is p b p b p_(b)p_{b}pb-bounded for all b ∈ X ∖ { 0 } b ∈ X ∖ { 0 } b in X\\{0}b \in X \backslash\{0\}b∈X∖{0} (and obviously for all b ∈ X b ∈ X b in Xb \in Xb∈X ). This is nothing else than the boundedness of V V VVV with respect to the natural locally convex topology of X X XXX induced by the 2-norm |., |. Finally, we call a subset W W WWW of X × X X × X X xx XX \times XX×X 2-norm bounded provided sup { ‖ x , y ‖ : ( x , y ) ∈ W } < ∞ sup { ‖ x , y ‖ : ( x , y ) ∈ W } < ∞ s u p{||x,y||:(x,y)in W} < oo\sup \{\|x, y\|:(x, y) \in W\}<\inftysup{‖x,y‖:(x,y)∈W}<∞. The corresponding boundedness notions for sequences in X X XXX or in X × X X × X X xx XX \times XX×X are defined in an obvious way.
Let X 1 , X 2 X 1 , X 2 X_(1),X_(2)\mathrm{X}_{1}, X_{2}X1,X2 be linear subspaces of a 2-normed space ( X , ‖ ⋅ X , ‖ ⋅ X,||*X,\|\cdotX,‖⋅, ‖ ) a n d l e t ( Y , q ‖ ) a n d l e t ( Y , q ||)andlet(Y,q\| ) and let ( Y, q‖)andlet(Y,q ) be a seminormed space. A bilinear operator T : X 1 × X 2 → Y T : X 1 × X 2 → Y T:X_(1)xxX_(2)rarr YT: X_{1} \times X_{2} \rightarrow YT:X1×X2→Y, is called separately compact (s-compact for short) if { T ( x n , y n ) } T x n , y n {T(x_(n),y_(n))}\left\{T\left(x_{n}, y_{n}\right)\right\}{T(xn,yn)} contains a convergent subsequence for every bounded sequence { x n } x n {x_(n)}\left\{x_{n}\right\}{xn} in X 1 X 1 X_(1)X_{1}X1 and every bounded sequence { n } n {_(n)}\left\{_{n}\right\}{n} in X 2 X 2 X_(2)X_{2}X2. The operator T T TTT is called compact if { T ( x n , y n ) } T x n , y n {T(x_(n),y_(n))}\left\{T\left(x_{n}, \mathrm{y}_{n}\right)\right\}{T(xn,yn)}, contains a convergent subsequence for every 2-norm bounded sequence { ( x n , y n ) } x n , y n {(x_(n),y_(n))}\left\{\left(x_{n}, y_{n}\right)\right\}{(xn,yn)} in X 1 × X 2 X 1 × X 2 X_(1)xxX_(2)X_{1} \times X_{2}X1×X2.
We have:
Proposition 5.1 Every compact operator is bounded.
Proof. Let T : X 1 × X 2 → Y T : X 1 × X 2 → Y T:X_(1)xxX_(2)rarr YT: X_{1} \times X_{2} \rightarrow YT:X1×X2→Y be a compact bilinear operator. Supposing T T TTT not bounded then, by (3.2), we can choose a sequence { ( x n , y n ) } x n , y n {(x_(n),y_(n))}\left\{\left(x_{n}, y_{n}\right)\right\}{(xn,yn)} in X 1 × X 2 X 1 × X 2 X_(1)xxX_(2)X_{1} \times X_{2}X1×X2 such that ‖ x n , y n ‖ ≤ 1 x n , y n ≤ 1 ||x_(n),y_(n)|| <= 1\left\|x_{n}, y_{n}\right\| \leq 1‖xn,yn‖≤1 and q ( T ( x n , y n ) ) > n q T x n , y n > n q(T(x_(n),y_(n))) > nq\left(T\left(x_{n}, y_{n}\right)\right)>nq(T(xn,yn))>n, for all n ∈ N n ∈ N n inNn \in \mathrm{~N}n∈ N. It follows that the sequence { T ( x n , y n ) } T x n , y n {T(x_(n),y_(n))}\left\{T\left(x_{n}, y_{n}\right)\right\}{T(xn,yn)} has no convergent subsequences
Consider now the case of a bilinear operator T : Z × [ b ] → Y T : Z × [ b ] → Y T:Z xx[b]rarr YT: Z \times[b] \rightarrow YT:Z×[b]→Y, where Z Z ZZZ is a subspace of the 2 -normed space ( X , ‖ ⋅ X , ‖ ⋅ X,||*X,\|\cdotX,‖⋅, ‖ ) a n d b ∈ X ∖ { 0 } ‖ ) a n d b ∈ X ∖ { 0 } ||)andb in X\\{0}\| ) and b \in X \backslash\{0\}‖)andb∈X∖{0}. We call the operator T p b T p b Tp_(b)T p_{b}Tpb-compact if { T ( z n , α n b ) } T z n , α n b {T(z_(n),alpha_(n)b)}\left\{T\left(z_{n}, \alpha_{n} b\right)\right\}{T(zn,αnb)} contains a convergent subsequence for every p b p b p_(b)p_{b}pb-bounded sequence { z n , } z n , {z_(n),}\left\{z_{n},\right\}{zn,} in Z Z ZZZ and every bounded sequence { α n } α n {alpha_(n)}\left\{\alpha_{n}\right\}{αn} in R . In this case these three notions of compactness are related as follows:
Proposition 5.2 Let T : Z × [ b ] → Y T : Z × [ b ] → Y T:Z xx[b]rarr YT: Z \times[b] \rightarrow YT:Z×[b]→Y be a bilinear oper ator:
Then
T compact ⇒ T p b − compact ⇒ T s − compact. T  compact  ⇒ T p b −  compact  ⇒ T s −  compact.  T" compact "=>Tp_(b)-" compact "=>Ts-" compact. "T \text { compact } \Rightarrow T p_{b}-\text { compact } \Rightarrow T s-\text { compact. }T compact ⇒Tpb− compact ⇒Ts− compact. 
Proof. T compact ⇒ T p b − ⇒ T p b − =>Tp_(b)-\Rightarrow T p_{b}-⇒Tpb− compact
If { z n } z n {z_(n)}\left\{z_{n}\right\}{zn} is a p b p b p_(b)p_{b}pb-bounded sequence in Z Z ZZZ and { α n } α n {alpha_(n)}\left\{\alpha_{n}\right\}{αn} is a bounded sequence of real numbers, then the equality ‖ z n , α n b ‖ = | α n | ⋅ ‖ z n , b ‖ z n , α n b = α n ⋅ z n , b ||z_(n),alpha_(n)b||=|alpha_(n)|*||z_(n),b||\left\|z_{n}, \alpha_{n} b\right\|=\left|\alpha_{n}\right| \cdot\left\|z_{n}, b\right\|‖zn,αnb‖=|αn|⋅‖zn,b‖ implies sup n ‖ z n , α n b ‖ < ∞ n z n , α n b < ∞ _(n)||z_(n),alpha_(n)b|| < oo{ }_{n}\left\|z_{n}, \alpha_{n} b\right\|<\inftyn‖zn,αnb‖<∞. The operator T T TTT being compact it follows that { T ( z n , α n b ) } T z n , α n b {T(z_(n),alpha_(n)b)}\left\{T\left(z_{n}, \alpha_{n} b\right)\right\}{T(zn,αnb)} contains a convergent subsequence.
To prove the second implication we need a lemma:
LEMMA 5.3 Let b ∈ X ∖ { 0 } b ∈ X ∖ { 0 } b in X\\{0}b \in X \backslash\{0\}b∈X∖{0} and { α n } ⊂ R α n ⊂ R {alpha_(n)}subR\left\{\alpha_{n}\right\} \subset \mathbf{R}{αn}⊂R. The sequence { α n b } α n b {alpha_(n)b}\left\{\alpha_{n} b\right\}{αnb} is bounded in X X XXX if and only if the sequence { α n } α n {alpha_(n)}\left\{\alpha_{n}\right\}{αn} is bounded in R R R\mathbf{R}R.
Proof. By definition, a 2-normed space has dimension at least 2, so that there exist a ∈ X a ∈ X a in Xa \in Xa∈X with ‖ a , b ‖ > 0 ‖ a , b ‖ > 0 ||a,b|| > 0\|a, b\|>0‖a,b‖>0 (Axiom BN 1). The boundedness of { ‖ α n b , a ‖ } α n b , a {||alpha_(n)b,a||}\left\{\left\|\alpha_{n} b, a\right\|\right\}{‖αnb,a‖}, and the equality ‖ α n b , a ‖ = | α n | ⋅ ‖ a , b ‖ α n b , a = α n ⋅ ‖ a , b ‖ ||alpha_(n)b,a||=|alpha_(n)|*||a,b||\left\|\alpha_{n} b, a\right\|=\left|\alpha_{n}\right| \cdot\|a, b\|‖αnb,a‖=|αn|⋅‖a,b‖ imply the boundedness of the sequence { α n } α n {alpha_(n)}\left\{\alpha_{n}\right\}{αn}.
Conversely, if { α n } α n {alpha_(n)}\left\{\alpha_{n}\right\}{αn} is a bounded sequence of real numbers then the equality ‖ α n b , c ‖ = | α n | ⋅ ‖ b , c ‖ α n b , c = α n ⋅ ‖ b , c ‖ ||alpha_(n)b,c||=|alpha_(n)|*||b,c||\left\|\alpha_{n} b, c\right\|=\left|\alpha_{n}\right| \cdot\|b, c\|‖αnb,c‖=|αn|⋅‖b,c‖ implies that the sequence { ‖ α n b , c ‖ } α n b , c {||alpha_(n)b,c||}\left\{\left\|\alpha_{n} b, c\right\|\right\}{‖αnb,c‖} is bounded for every c ∈ X c ∈ X c in Xc \in Xc∈X. Lemma is proved.
Prove now that:
T p b − compact ⇒ T s − compact T p b −  compact  ⇒ T s −  compact  Tp_(b)-" compact "=>Ts-" compact "T p_{b}-\text { compact } \Rightarrow T s-\text { compact }Tpb− compact ⇒Ts− compact 
Let { z n } ⊆ Z z n ⊆ Z {z_(n)}sube Z\left\{z_{n}\right\} \subseteq Z{zn}⊆Z and { α n b } α n b {alpha_(n)b}\left\{\alpha_{n} b\right\}{αnb} be bounded sequences. It follows that the sequence { z n } z n {z_(n)}\left\{z_{n}\right\}{zn} is p b p b p_(b)p_{b}pb-bounded and, by Lemma 5.3, the sequence { α n } α n {alpha_(n)}\left\{\alpha_{n}\right\}{αn} is bounded too. Since T T TTT is p b p b p_(b)p_{b}pb-compact, the sequence { T ( z n , α n b ) } T z n , α n b {T(z_(n),alpha_(n)b)}\left\{T\left(z_{n}, \alpha_{n} b\right)\right\}{T(zn,αnb)} will contain a convergent subsequence, proving that T T TTT is a p b p b p_(b)p_{b}pb-compact bilinear operator ◻ ◻ ◻\square◻.
Concerning the compactness properties of a bilinear operator T : Z × [ b ] → Y T : Z × [ b ] → Y T:Z xx[b]rarr YT: Z \times[b] \rightarrow YT:Z×[b]→Y and of the associated operator A : Z → Y A : Z → Y A:Z rarr YA: Z \rightarrow YA:Z→Y (in the sense of Proposition 3.2) one can prove:
Proposition 5.4 A bilinear operator T : Z × [ b ] → Y T : Z × [ b ] → Y T:Z xx[b]rarr YT: Z \times[b] \rightarrow YT:Z×[b]→Y is p b p b p_(b)p_{b}pb-compact if and only if the associated linear operator A : ( Z , p b ) → ( Y , q ) A : Z , p b → ( Y , q ) A:(Z,p_(b))rarr(Y,q)A:\left(Z, p_{b}\right) \rightarrow(Y, q)A:(Z,pb)→(Y,q) is compact.
Proof. Suppose that the bilinear operator T : Z × [ b ] → Y T : Z × [ b ] → Y T:Z xx[b]rarr YT: Z \times[b] \rightarrow YT:Z×[b]→Y is p b p b p_(b)p_{b}pb-compact and let { z n } z n {z_(n)}\left\{z_{n}\right\}{zn} be a bounded sequence in the seminormed space ( Z , p b Z , p b Z,p_(b)Z, p_{b}Z,pb ). It follows that { z n } z n {z_(n)}\left\{z_{n}\right\}{zn} is a p b p b p_(b)p_{b}pb-bounded sequence in Z Z ZZZ and, consequently, { T ( z n , b ) } T z n , b {T(z_(n),b)}\left\{T\left(z_{n}, b\right)\right\}{T(zn,b)} will contain a convergent subsequence { T ( z n k , b ) } T z n k , b {T(z_(n_(k)),b)}\left\{T\left(z_{n_{k}}, b\right)\right\}{T(znk,b)}. Since { T ( z n k , b ) } = A z n k T z n k , b = A z n k {T(z_(n_(k)),b)}=Az_(n_(k))\left\{T\left(z_{n_{k}}, b\right)\right\}=A z_{n_{k}}{T(znk,b)}=Aznk it follows that { A z n k } A z n k {Az_(n_(k))}\left\{A z_{n_{k}}\right\}{Aznk} is a convergent subsequence of { A z n k } A z n k {Az_(n_(k))}\left\{A z_{n_{k}}\right\}{Aznk}, showing that the operator A A AAA is compact.
Conversely, let A : ( Z , p b ) → ( Y , q ) A : Z , p b → ( Y , q ) A:(Z,p_(b))rarr(Y,q)A:\left(Z, p_{b}\right) \rightarrow(Y, q)A:(Z,pb)→(Y,q) be a compact linear operator. If { z n } z n {z_(n)}\left\{z_{n}\right\}{zn} is a p b p b p_(b)p_{b}pb-bounded sequence in Z Z ZZZ and { α n } α n {alpha_(n)}\left\{\alpha_{n}\right\}{αn} is a bounded sequence in R , then { A z n } A z n {Az_(n)}\left\{A z_{n}\right\}{Azn} contains a convergent subsequence { A z n k } A z n k {Az_(n_(k))}\left\{A z_{n_{k}}\right\}{Aznk}. Taking a convergent subsequence { α n k } α n k {alpha_(n_(k))}\left\{\alpha_{n_{k}}\right\}{αnk} of { α n k } α n k {alpha_(n_(k))}\left\{\alpha_{n_{k}}\right\}{αnk} it follows that T ( z n k , α n k j b ) = α n k j ⋅ A z n k j , j ∈ N T z n k , α n k j b = α n k j ⋅ A z n k j , j ∈ N T(z_(n_(k)),alpha_(n_(kj))b)=alpha_(n_(kj))*Az_(n_(kj)),j inNT\left(z_{n_{k}}, \alpha_{n_{k j}} b\right)=\alpha_{n_{k j}} \cdot A z_{n_{k j}}, j \in \mathrm{~N}T(znk,αnkjb)=αnkj⋅Aznkj,j∈ N, is a convergent subsequence of { T ( z n , α n b ) } T z n , α n b {T(z_(n),alpha_(n)b)}\left\{T\left(z_{n}, \alpha_{n} b\right)\right\}{T(zn,αnb)}. Therefore the operator T T TTT is p b p b p_(b)p_{b}pb-compact ◻ ◻ ◻\square◻.
The following result was proved by J. Lindestrauss [12, Th. 5.4], in the case. of linear operators on normed spaces and by I. Beg and M. Iqbal [1, Th. 3.5] in the case of bilinear operators on 2 -normed spaces. A normed space ( Y , q Y , q Y,qY, qY,q ) is said to have the finite 2 -intersection property (F.2.I.P.) if any finite collection of mutually intersecting closed balls in Y Y YYY has nonvoid intersection.
Proposition 5.5 Let ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) be a Banach space with the (F.2.I.P.), ( X , ‖ ⋅ ‖ ) ( X , ‖ ⋅ ‖ ) (X,||*||)(X,\|\cdot\|)(X,‖⋅‖) a 2-normed space, Z Z ZZZ a codimension one subspace of X , b ∈ X ∖ { 0 } X , b ∈ X ∖ { 0 } X,b in X\\{0}X, b \in X \backslash\{0\}X,b∈X∖{0} and T : Z × [ b ] → Y T : Z × [ b ] → Y T:Z xx[b]rarr YT: Z \times [b] \rightarrow YT:Z×[b]→Y a p b p b p_(b)p_{b}pb-compact bilinear operator. Then, for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 there exists an extension T ~ : X × [ b ] → Y T ~ : X × [ b ] → Y widetilde(T):X xx[b]rarr Y\widetilde{T}: X \times[b] \rightarrow YT~:X×[b]→Y of T T TTT verifying v ( T ~ ) ≤ ( 1 + ε ) ⋅ v ( T ) v ( T ~ ) ≤ ( 1 + ε ) ⋅ v ( T ) v( widetilde(T)) <= (1+epsi)*v(T)v(\widetilde{T}) \leq(1+\varepsilon) \cdot v(T)v(T~)≤(1+ε)⋅v(T).
Proof. Let A : ( Z , p b ) → ( Y , q ) A : Z , p b → ( Y , q ) A:(Z,p_(b))rarr(Y,q)A:\left(Z, p_{b}\right) \rightarrow(Y, q)A:(Z,pb)→(Y,q) be the linear operator associated to T T TTT according to Proposition 3.2. By Proposition 5.4, the operator A A AAA is compact and, by J . Lindenstrauss [12, Th. 5.4], there exists an extension A ~ : ( X , p b ) → ( Y , q ) A ~ : X , p b → ( Y , q ) tilde(A):(X,p_(b))rarr(Y,q)\tilde{A}:\left(X, p_{b}\right) \rightarrow(Y, q)A~:(X,pb)→(Y,q) of A A AAA, verifying ‖ A ~ ‖ ≤ ( 1 + ε ) ⋅ ‖ A ‖ ‖ A ~ ‖ ≤ ( 1 + ε ) ⋅ ‖ A ‖ || widetilde(A)|| <= (1+epsi)*||A||\|\widetilde{A}\| \leq(1+\varepsilon) \cdot\|A\|‖A~‖≤(1+ε)⋅‖A‖. Appealing again to Proposition 3.2 it follows that the desired extension of T T TTT is given by T ~ ( x , α b ) = α ⋅ A ~ x , x ∈ X , α ∈ R ◻ T ~ ( x , α b ) = α ⋅ A ~ x , x ∈ X , α ∈ R ◻ widetilde(T)(x,alpha b)=alpha* widetilde(A)x,x in X,alpha inR◻\widetilde{T}(x, \alpha b)=\alpha \cdot \widetilde{A} x, x \in X, \alpha \in \mathbf{R} \squareT~(x,αb)=α⋅A~x,x∈X,α∈R◻.
Remark. 1 ∘ I 1 ∘ I 1^(@)I1^{\circ} \mathrm{I}1∘I. Beg and M. Iqbal [1, Th. 3.5] proved Proposition 5.5 for compact bilinear operators following the ideas of the proof given by J. Lindenstrauss [12] for compact linear operators. By Proposition 5.2, a compact bilinear operator T : Z × × [ b ] → Y T : Z × × [ b ] → Y T:Z xx xx[b]rarr YT: Z \times \times[b] \rightarrow YT:Z××[b]→Y is p b p b p_(b)p_{b}pb-compact, so that the result of I . Beg and M . Iqbal follows from Proposition 5.5.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ We have used a seminormed version of Lindenstrauss' result which can be proved in the same way as in the case of normed spaces (The space Y Y YYY could be also supposed to be only seminormed too).

Brul 6. UNIQUE EXTENSION AND UNIQUE BEST APPROXIMATION

The aim of this section is to prove some duality results relating the extension properties for bilinear operators and best approximation in spaces of bilinear operators. In the case of linear functionals on normed spaces the problem was first studied by R. R. Phelps [16]. For other related results see I. Singer's book [17].
Recall that, for a 2 -normed space ( X , ‖ ⋅ ‖ X , ‖ ⋅ ‖ X,||*||X,\|\cdot\|X,‖⋅‖ ), a normed space ( Y , q Y , q Y,qY, qY,q ) and two subspaces X 1 , X 2 X 1 , X 2 X_(1),X_(2)X_{1}, X_{2}X1,X2 of X X XXX, we denote by L 2 ( W , Y ) L 2 ( W , Y ) L_(2)(W,Y)L_{2}(W, Y)L2(W,Y) the normed space of all bounded bilinear operators from W = X 1 × X 2 W = X 1 × X 2 W=X_(1)xxX_(2)W=X_{1} \times X_{2}W=X1×X2 to Y Y YYY. If X ~ 1 ⊃ X 1 X ~ 1 ⊃ X 1 widetilde(X)_(1)supX_(1)\widetilde{X}_{1} \supset X_{1}X~1⊃X1 and X ~ 2 ⊃ X 2 X ~ 2 ⊃ X 2 widetilde(X)_(2)supX_(2)\widetilde{X}_{2} \supset X_{2}X~2⊃X2 are other two subspaces of X X XXX then the normed space L 2 ( W ~ , Y ) L 2 ( W ~ , Y ) L_(2)( tilde(W),Y)L_{2}(\tilde{W}, Y)L2(W~,Y) and W ~ W ~ tilde(W)\tilde{W}W~ are defined similarly. The norms in L 2 ( W , Y ) L 2 ( W , Y ) L_(2)(W,Y)L_{2}(W, Y)L2(W,Y) and L 2 ( W ~ , Y ) L 2 ( W ~ , Y ) L_(2)( widetilde(W),Y)L_{2}(\widetilde{W}, Y)L2(W~,Y) will be denoted by the same symbol v v vvv (see (3.2) and Proposition 3.1). For T ∈ L 2 ( W , Y ) T ∈ L 2 ( W , Y ) T inL_(2)(W,Y)T \in L_{2}(W, Y)T∈L2(W,Y) denote by E ( T ) E ( T ) E(T)\mathscr{E}(T)E(T) the set of all norm preserving extensions of T T TTT to W ~ W ~ widetilde(W)\widetilde{W}W~, i.e.
(6.1) E ( T ) = { T ~ ∈ L 2 ( W ~ , Y ) : T ~ | W = T and ν ( T ~ ) = ν ( T ) } (6.1) E ( T ) = T ~ ∈ L 2 ( W ~ , Y ) : T ~ W = T  and  ν ( T ~ ) = ν ( T ) {:(6.1)E(T)={( tilde(T))inL_(2)(( tilde(W)),Y):( tilde(T))|_(W)=T" and "nu(( widetilde(T)))=nu(T)}:}\begin{equation*} \mathscr{E}(T)=\left\{\tilde{T} \in L_{2}(\tilde{W}, Y):\left.\tilde{T}\right|_{W}=T \text { and } \nu(\widetilde{T})=\nu(T)\right\} \tag{6.1} \end{equation*}(6.1)E(T)={T~∈L2(W~,Y):T~|W=T and ν(T~)=ν(T)}
The annihilator of W W WWW is L 2 ( W ~ , Y ) L 2 ( W ~ , Y ) L_(2)( widetilde(W),Y)L_{2}(\widetilde{W}, Y)L2(W~,Y) is defined by
(6.2) W ⊥ = { S ∈ L 2 ( W ~ , Y ) : S ( W ) = { 0 } } . (6.2) W ⊥ = S ∈ L 2 ( W ~ , Y ) : S ( W ) = { 0 } . {:(6.2)W^(_|_)={S inL_(2)(( widetilde(W)),Y):S(W)={0}}.:}\begin{equation*} W^{\perp}=\left\{S \in L_{2}(\widetilde{W}, Y): S(W)=\{0\}\right\} . \tag{6.2} \end{equation*}(6.2)W⊥={S∈L2(W~,Y):S(W)={0}}.
As usual, for a nonvoid subset V V VVV of a nomed space E E EEE and x ∈ E x ∈ E x in Ex \in Ex∈E, denote by d ( x , V ) = inf { ‖ x − v ‖ : v ∈ V } d ( x , V ) = inf { ‖ x − v ‖ : v ∈ V } d(x,V)=i n f{||x-v||:v in V}d(x, V)=\inf \{\|x-v\|: v \in V\}d(x,V)=inf{‖x−v‖:v∈V} the distance from x x xxx to V V VVV. An element v 0 ∈ V v 0 ∈ V v_(0)in Vv_{0} \in Vv0∈V satisfying ‖ x − v 0 ‖ = d ( x , V ) x − v 0 = d ( x , V ) ||x-v_(0)||=d(x,V)\left\|x-v_{0}\right\|=d(x, V)‖x−v0‖=d(x,V) is called a nearest point to x x xxx in V V VVV (or a best approximation element). The set of nearest points to x x xxx in V V VVV is denoted by P V ( x ) P V ( x ) P_(V)(x)P_{V}(x)PV(x) and the set-valued operator P V : E → 2 V P V : E → 2 V P_(V):E rarr2^(V)P_{V}: E \rightarrow 2^{V}PV:E→2V is called the metric projection operator of E E EEE onto V V VVV. The set V V VVV is called proximinal if P V ( x ) ≠ ∅ P V ( x ) ≠ ∅ P_(V)(x)!=O/P_{V}(x) \neq \varnothingPV(x)≠∅ and Chebyshevian if P V ( x ) P V ( x ) P_(V)(x)P_{V}(x)PV(x) is a singleton, for all x ∈ E x ∈ E x in Ex \in Ex∈E.
We say that W W WWW has the extension property with respect to W ~ W ~ widetilde(W)\widetilde{W}W~ if every bounded bilinear operator T : W → Y T : W → Y T:W rarr YT: W \rightarrow YT:W→Y has a norm preserving extension T ~ ∈ L 2 ( W ~ , Y ) T ~ ∈ L 2 ( W ~ , Y ) widetilde(T)inL_(2)( widetilde(W),Y)\widetilde{T} \in L_{2}(\widetilde{W}, Y)T~∈L2(W~,Y). The following proposition shows that the extension properties of W W WWW and the best approximation properties of its amnihilator are closely related.
Proposition 6.1 If the subspace W W WWW has the extension property with respect. to W ~ W ~ widetilde(W)\widetilde{W}W~, then its annihilator W ⊥ W ⊥ W^(_|_)W^{\perp}W⊥ is a proximinal subspace of L 2 ( W ~ , Y ) L 2 ( W ~ , Y ) L_(2)( widetilde(W),Y)L_{2}(\widetilde{W}, Y)L2(W~,Y) and the following formulae hold
(6.3) d ( S , W ⊥ ) = v ( S | W ′ ) (6.3) d S , W ⊥ = v S W ′ {:(6.3)d(S,W^(_|_))=v(S|_(W^('))):}\begin{equation*} d\left(S, W^{\perp}\right)=v\left(\left.S\right|_{W^{\prime}}\right) \tag{6.3} \end{equation*}(6.3)d(S,W⊥)=v(S|W′)
and
(6.4) P W ⊥ ( S ) = S − E ( S | W ) , (6.4) P W ⊥ ( S ) = S − E S W , {:(6.4)P_(W^(_|_))(S)=S-E(S|_(W))",":}\begin{equation*} P_{W^{\perp}}(S)=S-\mathscr{E}\left(\left.S\right|_{W}\right), \tag{6.4} \end{equation*}(6.4)PW⊥(S)=S−E(S|W),
for any operator S ∈ L 2 ( W ~ , Y ) S ∈ L 2 ( W ~ , Y ) S inL_(2)( widetilde(W),Y)S \in L_{2}(\widetilde{W}, Y)S∈L2(W~,Y).
Proof. If S ∈ L 2 ( W ~ , Y ) S ∈ L 2 ( W ~ , Y ) S inL_(2)( widetilde(W),Y)S \in L_{2}(\widetilde{W}, Y)S∈L2(W~,Y) then ( S − T ) | W = S | W ( S − T ) W = S W (S-T)|_(W)=S|_(W)\left.(S-T)\right|_{W}=\left.S\right|_{W}(S−T)|W=S|W and, by the definition of the norm ν ν nu\nuν (formula (3.2)), we have
(), [8]
v ( S | W ) = v ( ( S − T ) | W ) ≤ v ( S − T ) , v S W = v ( S − T ) W ≤ v ( S − T ) , v(S|_(W))=v((S-T)|_(W)) <= v(S-T),v\left(\left.S\right|_{W}\right)=v\left(\left.(S-T)\right|_{W}\right) \leq v(S-T),v(S|W)=v((S−T)|W)≤v(S−T),
for every T ∈ W ⊥ T ∈ W ⊥ T inW^(_|_)T \in W^{\perp}T∈W⊥, implying v ( S | W ) ≤ d ( S , W ⊥ ) v S W ≤ d S , W ⊥ v(S|_(W)) <= d(S,W^(_|_))v\left(\left.S\right|_{W}\right) \leq d\left(S, W^{\perp}\right)v(S|W)≤d(S,W⊥). If S ∈ L 2 ( W ~ , Y ) S ∈ L 2 ( W ~ , Y ) S inL_(2)( widetilde(W),Y)S \in L_{2}(\widetilde{W}, Y)S∈L2(W~,Y), is a norm-preserving extension of S | W S W S|_(W)\left.S\right|_{W}S|W then T 0 = S − S 0 ∈ W ⊥ T 0 = S − S 0 ∈ W ⊥ T_(0)=S-S_(0)inW^(_|_)T_{0}=S-S_{0} \in W^{\perp}T0=S−S0∈W⊥ and since S 0 = S − T 0 S 0 = S − T 0 S_(0)=S-T_(0)S_{0}=S-T_{0}S0=S−T0 we can write
v ( S | W ) = v ( S 0 ) = v ( S − T 0 ) ≥ d ( S , W ⊥ ) , v S W = v S 0 = v S − T 0 ≥ d S , W ⊥ , v(S|_(W))=v(S_(0))=v(S-T_(0)) >= d(S,W^(_|_)),v\left(\left.S\right|_{W}\right)=v\left(S_{0}\right)=v\left(S-T_{0}\right) \geq d\left(S, W^{\perp}\right),v(S|W)=v(S0)=v(S−T0)≥d(S,W⊥),
showing that formula (6.3) holds and that S − S 0 S − S 0 S-S_(0)S-S_{0}S−S0 is a nearest point to S S SSS in W ⊥ W ⊥ W^(_|_)W^{\perp}W⊥ for any S 0 ∈ E ( S | W ) S 0 ∈ E S W S_(0)inE(S|_(W))S_{0} \in \mathscr{E}\left(\left.S\right|_{W}\right)S0∈E(S|W), i.e.
(6,5) S − E ( S W ) ⊆ P W ⊥ ( S ) (6,5) S − E S W ⊆ P W ⊥ ( S ) {:(6,5)S-E(S_(W))subeP_(W^(_|_))(S):}\begin{equation*} S-\mathscr{E}\left(S_{W}\right) \subseteq P_{W^{\perp}}(S) \tag{6,5} \end{equation*}(6,5)S−E(SW)⊆PW⊥(S)
Suppose now that T 0 T 0 T_(0)T_{0}T0 is a nearest point to S S SSS in W ⊥ W ⊥ W^(_|_)W^{\perp}W⊥ and let S 0 = S − T 0 S 0 = S − T 0 S_(0)=S-T_(0)S_{0}=S-T_{0}S0=S−T0. It follows S 0 | W = S | W S 0 W = S W S_(0)|_(W)=S|_(W)\left.S_{0}\right|_{W}=\left.S\right|_{W}S0|W=S|W and by (6.3)
v ( S | W ) = d ( S , W ⊥ ) = v ( S − T 0 ) = v ( S 0 ) v S W = d S , W ⊥ = v S − T 0 = v S 0 v(S|_(W))=d(S,W^(_|_))=v(S-T_(0))=v(S_(0))v\left(\left.S\right|_{W}\right)=d\left(S, W^{\perp}\right)=v\left(S-T_{0}\right)=v\left(S_{0}\right)v(S|W)=d(S,W⊥)=v(S−T0)=v(S0)
showing that S 0 S 0 S_(0)S_{0}S0 is a norm-preserving extension of S ∣ W S ∣ W S_(∣W)S_{\mid W}S∣W, i.e.
S − P W ⊥ ( S ) ⊆ E ( S | W ) S − P W ⊥ ( S ) ⊆ E S W S-P_(W^(_|_))(S)subeE(S|_(W))S-P_{W^{\perp}}(S) \subseteq \mathscr{E}\left(\left.S\right|_{W}\right)S−PW⊥(S)⊆E(S|W)
or equivalently
( 6.6 ) ( 6.6 ) (6.6)(6.6)(6.6) P W ⊥ ( S ) ⊆ S − E ( S | W ) P W ⊥ ( S ) ⊆ S − E S W P_(W^(_|_))(S)sube S-E(S|_(W))P_{W^{\perp}}(S) \subseteq S-\mathscr{E}\left(\left.S\right|_{W}\right)PW⊥(S)⊆S−E(S|W)
S-P_(W^(_|_))(S)subeE(S|_(W)) or equivalently (6.6) P_(W^(_|_))(S)sube S-E(S|_(W))| | $S-P_{W^{\perp}}(S) \subseteq \mathscr{E}\left(\left.S\right\|_{W}\right)$ | | :--- | :--- | | or equivalently | | | $(6.6)$ | $P_{W^{\perp}}(S) \subseteq S-\mathscr{E}\left(\left.S\right\|_{W}\right)$ |
which together with relation (6.5) give (6.4)
Let ( X , ‖ ⋅ ‖ ) ( X , ‖ ⋅ ‖ ) (X,||*||)(X,\|\cdot\|)(X,‖⋅‖) be a 2 -normed space, Z Z ZZZ a subspace of X X XXX and b ∈ X ∖ { 0 } b ∈ X ∖ { 0 } bin X\\{0}\mathrm{b} \in X \backslash\{0\}b∈X∖{0}. For a normed space ( Y , q ) ( Y , q ) (Y,q)(Y, q)(Y,q) let
Z b ⊥ = { T ∈ L 2 ( X × [ b ] , Y ) : T ( Z × [ b ] ) = { 0 } } Z b ⊥ = T ∈ L 2 ( X × [ b ] , Y ) : T ( Z × [ b ] ) = { 0 } Z_(b)^(_|_)={T inL_(2)(X xx[b],Y):T(Z xx[b])={0}}Z_{b}^{\perp}=\left\{T \in L_{2}(X \times[b], Y): T(Z \times[b])=\{0\}\right\}Zb⊥={T∈L2(X×[b],Y):T(Z×[b])={0}}
denote the amililator of Z × [ b ] Z × [ b ] Z xx[b]Z \times[b]Z×[b] in L 2 ( X × [ b ] , Y ) L 2 ( X × [ b ] , Y ) L_(2)(X xx[b],Y)L_{2}(X \times[b], Y)L2(X×[b],Y). In this case Proposition 6.1 and Corollary 4.3 give:
COROLLARY 6.2 Let ( Y , q Y , q Y,qY, qY,q ) be a normed space with the binary intersection property. Then Z b ⊥ Z b ⊥ Z_(b)^(_|_)Z_{b}^{\perp}Zb⊥ is a proximinal subspace of L 2 ( X × [ b ] , Y ) L 2 ( X × [ b ] , Y ) L_(2)(X xx[b],Y)L_{2}(X \times[b], Y)L2(X×[b],Y) and the following formulae
(6.7) d ( S , Z b ⊥ ) = v ( S | Z × [ b ] ) (6.7) d S , Z b ⊥ = v S Z × [ b ] {:(6.7)d(S,Z_(b)^(_|_))=v(S|_(Z xx[b])):}\begin{equation*} d\left(S, Z_{b}^{\perp}\right)=v\left(\left.S\right|_{Z \times[b]}\right) \tag{6.7} \end{equation*}(6.7)d(S,Zb⊥)=v(S|Z×[b])
and
(6.8) P Z b ⊥ ( S ) = S − E ( S | Z × [ b ] ) (6.8) P Z b ⊥ ( S ) = S − E S Z × [ b ] {:(6.8)P_(Z_(b)^(_|_))(S)=S-E(S|_(Z xx[b])):}\begin{equation*} P_{Z_{b}^{\perp}}(S)=S-\mathscr{E}\left(\left.S\right|_{Z \times[b]}\right) \tag{6.8} \end{equation*}(6.8)PZb⊥(S)=S−E(S|Z×[b])
hold for every S ∈ L 2 ( X × [ b ] , Y ) S ∈ L 2 ( X × [ b ] , Y ) S inL_(2)(X xx[b],Y)S \in L_{2}(X \times[b], Y)S∈L2(X×[b],Y).
Furthermore the amililator W ⊥ W ⊥ W^(_|_)W^{\perp}W⊥ is a Chebyshevian subspace of L 2 ( X × [ b ] , Y ) L 2 ( X × [ b ] , Y ) L_(2)(X xx[b],Y)L_{2}(X \times[b], Y)L2(X×[b],Y) if and only if every T ∈ L 2 ( Z × [ b ] , Y ) T ∈ L 2 ( Z × [ b ] , Y ) T inL_(2)(Z xx[b],Y)T \in L_{2}(Z \times[b], Y)T∈L2(Z×[b],Y) has a unique norm preserving extension to X × [ b ] X × [ b ] X xx[b]X \times[b]X×[b].

REFERENCES

  1. I. Beg, M. Iqbal, Extension of linear 2-operators, Mathematica Montesnigri 2 (1993), 1-10.
  2. S. Cobzaş, C. Mustăta, Extension of bilinear functionals and best approximation in 2-normed spaces, (to appear).
  3. N. Dunford, J. T. Schwartz, Linear Operators Part. I: General Theory, Interscience Publishers, New York 1958.
  4. I. Franić, An extension theorem for bounded linear 2 -functionals and applications, Math. Japonica 40 (1994), 79-85.
  5. S. Gähler, 2-Metrische Räume und ihre Topologische Struktur; Math. Nachr. 26 (1963/64), 115-148.
  6. S. Gähler, Linear 2-Normierte Räume, Math. Nachr. 28 (1965), 1-45.
  7. S. Gähler, Über 2-Banach-Räume, Math. Nachr. 42 (1969), 335-347.
  8. T. L. Hayden, The extension of bilinear functionals, Pacific I M
  9. R. B. Holmes 22 (1967), 99-108.
    R. B. Homposium on Infinite Dimensional Topology (R. D. Anderson Editor), Annals of Math. Studies, Princeton Unjv. Press, Princeton 1972, 137-158.
  10. R. B. Holmes, Geometric Functional Analysis and its Application, Springer-Verlag, Berlin 1975.
  11. K. Iseki, Mathematics on 2-normed spaces, Bull. Korean. Math. Soc., 13 (1976), 127-136.
  12. J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48, (1964)
  13. S. Mabizela, On bounded 2-inear functionals, Math. Japonica, 35 (1990), 51-55
  14. L. Nachbin, A theorem of Hahm-Banach tope for linear transformation, Trans. Amer. Math. Soc.,
  15. L. Nachbin, Some problems in extending and lifting continuous linear transformations, in Proc. International Symposium on Linear Spaces, Jenusalem, 1960.
  16. R. R. Phelps, Uniqueness of Hahn-Banach extension and unique best approximation, Trans Amer. Math. Soc., 95 (1960), 238-255.
  17. I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Publishing House of the Romanian Academy and Springer Verlag, Bucharest and Berlin,1970. 18. A. G. White Jr., 2-Banach spaces, Math. Nachr. 42 (1969), 43-60.
  18. A. G. White and Yeol Je Cho, Linear mappings on linear 2-normed spaces, Bull. Korean Math. Soc. 21 (1984),1-6.

Received 14.01.1996

S. Cobzas

Faculty of Mathematics
Ro-3400 Cluj-Napoca
Romania
C. Mustăta
"T. Popoviciu" Institute
of Numerical Analysis
O.P. 1 C.P. 68
Ro-3400 Cluj-Napoca
Romania
1996

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