ON THE PROJECTION CONSTANTS OF SOME TOPOLOGICAL SPACES AND SOME APPLICATIONS

It is well known that any Banach space Y can be isometrically embedded into l∞( ) for some index set ( is usually taken to be UY ∗ where Y ∗ denotes the dual space of Y and UY ∗ denotes the set {f : f ∈ Y ∗, ‖f ‖ ≤ 1}) and that if Y is complemented in l∞( ), then it is complemented in every Banach space containing it as a closed subspace, that is, Y is injective. We also know that for any such embedding the supremum in (1.2) is attained, that is, λ(Y ) =


Introduction
If Y is a closed subspace of a Banach space X, then the relative projection constant of Y in X is defined by λ(Y, X) := inf P : P is a linear projection from X onto Y . (1.1) And the absolute projection constant of Y is defined by λ(Y ) := sup λ(Y, X) : X contains Y as a closed subspace . (1. 2) It is well known that any Banach space Y can be isometrically embedded into l ∞ ( ) for some index set ( is usually taken to be U Y * where Y * denotes the dual space of Y and U Y * denotes the set {f : f ∈ Y * , f ≤ 1}) and that if Y is complemented in l ∞ ( ), then it is complemented in every Banach space containing it as a closed subspace, that is, Y is injective.We also know that for any such embedding the supremum in (1.2) is attained, that is, λ(Y ) = λ(Y, l ∞ ( )) (see [1,4]).For each finite-dimensional space Y n with dim Y n = n, Kadets and Snobar [6] proved that λ(Y n ) ≤ √ n.König [7] showed that for each prime number n the space l ∞ n 2 contains an n-dimensional subspace Y n with projection constant König and Lewis [9] verified the strict inequality λ(Y n ) < √ n in case n ≥ 2. Lewis [14] showed that König and Tomczak-Jaegermann [11] also showed that there is a sequence In fact, it is shown in [9] that for each Banach space The improvement of these results was given in [12], where an upper estimate for λ(Y n ) was found in the form , in the complex field. ( The precise values of l 1 n , l 2 n , and l p n , 1 < p < ∞, p = 2, have been calculated by Grünbaum [4], Rutovitz [15], Gordon [3], and Garling and Gordon [2].In the case of 1 < p < 2, the improvement of these results was given by König, Schütt, and Tomczak-Jaegermann in [10], they showed that , in the complex field. (1.7) Some other results are mentioned in [2,3,13,15].For finite codimensional subspaces, Garling and Gordon [2] showed that if Y is a finite codimensional subspace of the Banach space X with codimension n, then for every > 0 there exists a projection P from X onto Y with norm (1.8)

Notations and basic definitions
The sets X, Y , Z, and E denote Banach spaces, X * denotes the conjugate space of X and U X denotes the unit ball of the space X.Elements of X, Y , X * , and Y * will be denoted by x, u, . . ., y, v, . . ., f, h, . . ., and g, k, . . ., respectively.The Entisarat El-Shobaky et al. 301 injective tensor product X ⊗ ∨ Y between the normed spaces X and Y is defined as the completion of the smallest cross norm on the space X ⊗ Y and the norm on the space X ⊗ Y is defined by where the supremum is taken over all functionals f ∈ U X * and g ∈ U Y * .The projective tensor product X ⊗ ∧ Y between the normed spaces X and Y is defined as the completion of the largest cross norm on the space X ⊗ Y and the norm on X ⊗ Y is defined by where the infimum is taken over all equivalent representations m j =1 u j ⊗ v j ∈ X ⊗ Y of n i=1 x i ⊗ y i (see [5]).If X is a Banach space on which every linear bounded operator from X into any Banach space Y is nuclear (this is the case in all finite-dimensional Banach spaces X), then for any Banach space Y the space X⊗ ∨ Y is isomorphically isometric to X ⊗ ∧ Y (see [16]).
The set We start with the following two lemmas.
Lemma 2.1.For Banach spaces X and Y there is a norm one projection from Proof.Since the space l ∞ ( ) has the 1-extension property, it is sufficient to show that l ∞ ( ) can be isometrically embedded in the space l in the space l ∞ ( ), (note that the norm in this Banach space is given by ) by the following formulas: Since both the injective and the projective tensor products are cross norms, The mapping J defined by the formula J (F) = F is the required isometric embedding.
Lemma 2.2.Let X and Y be two Banach spaces.Then Proof.It is also sufficient to show that the space X⊗ ∨ Y can be isometrically embedded in l ∞ ( ).In fact, every element F = n i=1 x i ⊗ y i ∈ X⊗ ∨ Y defines a scalar-valued bounded function F ∈ l ∞ ( ) by the formula F((f, g)) = i=1 f (x i )g(y i ).Using definition (2.1) for the injective tensor product, we have F ∨ = F l ∞ ( ) .The mapping i defined by the formula i(F) = F is the required isometric embedding.
We have the following theorem.
Theorem 2.3.(1) If Y 1 and Y 2 are complemented subspaces of Banach spaces X 1 and X 2 , respectively, then the injective (resp., projective) tensor product (2.5) Proof.Let P 1 and P 2 be any projections from X 1 onto Y 1 and from X 2 onto Y 2 , respectively.Then the operator P from the space X 1 ⊗ ∨ X 2 onto the space Y 1 ⊗ ∨ Y 2 (resp., from the space X 1 ⊗ ∧ X 2 onto the space Y 1 ⊗ ∧ Y 2 ) defined by is a projection and its norm P is not exceeding P 1 P 2 .In fact, let n i=1 x i ⊗ y i be any element of the space X 1 ⊗ (∨ or ∧) X 2 .Then, in the case of projective tensor product we have for all equivalent representations m j =1 u j ⊗ v j of n i=1 x i ⊗ y i .So And in the case of injective tensor product we have (2.9) Thus in both cases, P ≤ P 1 P 2 .Taking the infimum of each side with respect to all such P 1 and P 2 , we get inequality (2.4).To prove inequality (2.5), we apply inequality (2.4) and get in particular We claim that the sign ≥ is an equal sign.In fact, if P is any projection from l ∞ (U X * ) ⊗ ∨ l ∞ (U Y * ) onto X⊗ ∨ Y and J is the embedding given in Lemma 2.1, then Ṕ = P J is a projection from l ∞ ( ) onto X⊗ ∨ Y with Ṕ ≤ P .This is the sufficient condition for the two infimum Using inequality (2.10), we get (2.5).
Corollary 2.5.For any finite sequence {X i } n i=1 of Banach spaces with complemented subspaces {Y i } n i=1 , the relative projection constant of the injective (resp., projective) tensor product n i=1 Y i of the spaces Y i in the space n i=1 X i satisfies (2.12) (2.13)

a finite sequence of finite-dimensional Banach spaces. Then the relation between the absolute projection constant of the projective (or injective) tensor product
Proof.In fact, the proof is a combination of Corollary 2.5 and the results of [3,Theorem 4].

Applications
In this section, using Theorem 2.3, we obtain new results.
(1) For finite-dimensional Banach spaces X and Y with dimensions n and m, respectively, we have Entisarat El-Shobaky et al. 305 in the real field and in the complex field.Compare this result with the result in (1.6).
(2) For any positive integer m (not necessarily prime) with a prime factorization m = n i=1 q i where the numbers q i are distinct prime numbers, the space where Comparing this result with (1.3), we mention that the m 2 -dimension of the space n i=1 l ∞ is not a square of a prime number, so it gives a new subspace Y with a new projection constant.

The projection constants of operators
Now we start with our basic definitions of the projection constants of operators.
Definition 4.1.(1) A linear bounded operator A from a Banach space X into a Banach space Y is said to be left complemented with respect to a Banach space Z (Z contains Y as a closed subspace) if and only if there exists a linear bounded operator B from Z into X such that the composition AB is a projection from Z onto Y .In this case Z is said to be a left complementation of A.
If P Z (A) denotes the convex set of all operators B from Z into X such that the composition AB is a projection, then (2) the left relative projection constant of the operator A with respect to the space Z is defined as λ l (A, Z) := inf AB : B ∈ P Z (A) .Remark 4.2.We notice the following.
(1) From the definition of λ l (A, Z), the infimum in (4.1) is taken only with respect to the projections that are factored (through X) into two operators one of them is A and the other is an operator from Z into X, so 1 ≤ λ(Y, Z) ≤ λ l (A, Z) for every left complementation Z of A.
(2) If A is a projection from X onto Y , then A is left complemented with respect to Y .In fact AJ is a projection for any embedding J from Y into X.
(3) If I Y is the identity operator on Y and X contains Y as a complemented subspace, then I Y P = P for every projection P from X onto Y and hence I Y is left complemented with respect to X.Moreover, λ l (I Y , X) = λ(Y, X), that is, the relative projection constant of the identity operator on the space Y with respect to the space X is the relative projection constant of the space Y in the space X.
(4) If Z is a left complementation of the linear bounded operator A : X → Y , then Y is complemented in Z and the operator A is onto.
(5) If Z is a separable or reflexive Banach space and X is a Banach space, then for any index set the space Z is not a right complementation of any linear bounded operator from l ∞ ( ) into X.In particular, if X is a Banach space, then for any index set , the space l ∞ ( ) is not a left complementation of any linear bounded operator from X into the space c 0 .
The following lemma is parallel to that lemma mentioned in [8] for Banach spaces and we omit the proof since the proof is nearly similar.Lemma 4.3.Let be an index set such that Y is isometrically embedded into l ∞ ( ) and let A be a linear bounded operator from X onto Y such that l ∞ ( ) is one of its left complementation.Then for a given B ∈ P l ∞ ( ) (A), (1) For all Banach spaces E, Z, E ⊆ Z and every linear bounded operator T from E into Y there is an operator T from Z into Y extending the operator T with T ≤ AB T , that is, the space Y has AB -extension property, and in particular, if Z ⊇ X, the operator A has a linear extension Â from Z into Y with Â ≤ AB A .That is, the extension constant c(A) of the operator A defined by (c(A) := sup X⊂Z inf{ Â : Â is an extension of A and Â : Z → Y }) satisfies c(A) ≤ AB A .
(2) For every Banach space Z ⊇ Y , there exists a projection P from Z onto Y such that P ≤ AB .
The following theorem is also parallel to that given in (1.3) for Banach spaces.for every Banach space Z containing Y as a closed subspace, that is, λ l (A) attains its supremum at l ∞ ( ).Therefore, λ l (A) = λ l A, l ∞ ( ) , c(A) ≤ A λ l (A).(4.4)

( 2 . 7 )
Entisarat El-Shobaky et al. 303 n i=1 Y i and the direct sum n i=1 Y i (with the supremum norm) is as follows:

(4. 1 )( 3 )
And the left absolute projection constant of A is defined as λ l (A) := sup λ l (A, Z) : Z is a left complementation of the operator A .(4.2) We define the same analogy from the right.
Entisarat El-Shobaky et al. 307 Theorem 4.4.Let Y be isometrically embedded in l ∞ ( ) and let A be a linear bounded operator from X onto Y such that l ∞ ( ) is a left complementation of A. Then A is left complemented with respect to any other Banach space Z containing Y as a closed subspace.Moreover, λ l (A, Z) ≤ λ l A, l ∞ ( ) (4.3)