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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 902694, 9 pages
http://dx.doi.org/10.1155/2013/902694
Research Article

Intersection Properties of Balls in Banach Spaces

Stat-Math Unit, Indian Statistical Institute, R. V. College Post, Bangalore 560059, India

Received 29 May 2013; Revised 25 July 2013; Accepted 20 August 2013

Academic Editor: Miguel Martin

Copyright © 2013 C. R. Jayanarayanan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a weaker notion of central subspace called almost central subspace, and we study Banach spaces that belong to the class (GC), introduced by Veselý (1997). In particular, we prove that if is an almost central subspace of a Banach space such that is in the class (GC), then is a central subspace of . We also prove that if  is a semi -ideal in a Banach space such that is an almost central subspace of , then is an -ideal in . Certain stability results for quotient spaces, injective tensor product spaces, and polyhedral direct sums of Banach spaces are also derived.

1. Introduction

In [1], Veselý studied a new class of Banach spaces, namely, the class (GC), which were defined in terms of the existence of weighted Chebyshev centers (see below for definition and details). In the same paper, he characterized such spaces using intersection properties of balls. In [2], Bandyopadhyay and Rao considered some general results about the class (GC) by introducing a new class of subspaces called “central subspaces” of Banach spaces. Using this concept, they characterized the class (GC) and produced several examples of Banach spaces which belong to the class (GC) (see [2] for details). In this paper, we introduce and study a weaker notion of central subspace called almost central subspace (see Section 2 for definition and details). Using this concept, we obtain some new results about the class (GC) and also about some of the other types of intersection properties of balls studied in the literature.

For a Banach space , we denote by the closed ball in of radius around and by the closed unit ball of . In this paper, we restrict ourselves to real scalars and all subspaces we consider are assumed to be closed. Under the canonical embedding, we will consider as a subspace of . Also, if a Banach space is isometric to a subspace of the Banach space , then, without loss of generality, we will consider as a subspace of . Our notations are otherwise standard. Any unexplained terminology can be found in [3].

We now recall the definition of the class (GC) from [1].

Definition 1. Let be a Banach space. Let and . Minimizers of the function defined by are called weighted Chebyshev centers with the weight . Classical Chebyshev centers are the weighted Chebyshev centers with the weight .

Theorem 2 (see [1, Theorem 2.7]). For a Banach space and , the following assertions are equivalent.(i)If and , then . (ii) admits weighted Chebyshev centers for all weights , where for all .

Definition 3 (see [1, Definition 2.8]). One shall denote by (GC) the class of all Banach spaces such that for every positive integer and every , one of the equivalent conditions (i) or (ii) of Theorem 2 is satisfied.

Next we recall the definition of a central subspace which generalizes the notion of (GC).

Definition 4 (see [2, Definition 2.1]). Let be a Banach space. One says that a subspace is a central subspace of if every finite family of closed balls with centers in that intersects in also intersects in .

Clearly if and only if is a central subspace of . It follows from [2, Proposition 2.2(a)] that is a central subspace of a Banach space if and only if, for any finite set and , there exists a such that for .

An infinite version of central subspace called almost constrained subspace was investigated in [4, 5].

Definition 5. A subspace of a Banach space is said to be an almost constrained (AC) subspace of if any family of closed balls centered at points of that intersects in also intersects in .

We recall that a subspace of a Banach space is called -complemented in if there exists a projection of norm one on with range . One can easily observe that -complemented subspaces are AC-subspaces, and hence they are also central subspaces. The notion of an “ideal,” which is weaker than being a -complemented subspace, was introduced by Godefroy et al. in [6].

Definition 6. A subspace of a Banach space is said to be an ideal in if is the kernel of a norm one projection on .

Clearly -complemented subspaces are ideals. Also, every Banach space is an ideal in its bidual. For, if is a Banach space, then the projection defined by is a projection of norm one with kernel . It is well known that is an ideal in but it is not the range of a projection of norm one in .

An important concept in the -structure theory which is closely related to ball intersection properties is the well known concept called -ideal (see [7] for details).

Definition 7. A projection on a Banach space is called an -projection (-projection) if () for all . A subspace of is called an -summand (-summand) if it is the range of an -projection (-projection). A subspace of is called an -ideal if is an -summand in . For two Banach spaces and , one denotes by and the direct sum of and , equipped with the -norm and supremum norm, respectively.

For , we recall that a subspace of a Banach space is said to have the (strong) -ball property if, given closed balls in such that and for all , then for every . It is well known that -ideals are precisely the subspaces having the -ball property for all (see [7, Chapter I, Theorem 2.2] for details). There is a weaker notion of -ideal called semi -ideal, which is precisely the subspace having the -ball property (see [7, Page 43] for details).

In Section 2, we define an almost central subspace of a Banach space by a relative intersection property of balls. We will use this to give some sufficient conditions for subspaces to be central. We also consider some general results about the class (GC). In particular, we prove that an almost central subspace of a Banach space is in the class (GC) if and only if it is a central subspace of . We also derive several sufficient conditions for a semi -ideal to be an -ideal in terms of these intersection properties of balls.

In Section 3, we prove the stability of some of the ball intersection properties in quotient spaces, direct sums, vector-valued continuous function spaces, and injective tensor product spaces (see Chapter VIII of [3] for the theory of injective tensor product spaces). In quotient spaces, we prove that for Banach spaces , and with , if is almost central or ideal in , then is almost central or ideal in , respectively, and we also prove the converse when is an -predual (that is , for some positive measure ) and is an -ideal in . In the case of injective tensor product spaces, we show that if is an -predual space, then, for any almost central subspace of a Banach space , the injective tensor product is an almost central subspace of . We also prove that properties of being a central subspace and an AC-subspace are stable under a recently introduced concept called polyhedral direct sums of Banach spaces (see [8, Definition 2.1]).

2. Almost Central Subspaces

We begin this section with the definition of an “almost central subspace” of a Banach space which is the generalization of the concept central subspace, defined in [2].

Definition 8. A subspace of a Banach space is called an almost central subspace if, for every finite set , , and , there exists a such that for .

Remark 9. Clearly central subspaces of Banach spaces are almost central. As in the case of central subspace, it is easy to observe that is an almost central subspace of a Banach space if and only if, for each family of closed balls in having nonempty intersection in , the family of closed balls in has nonempty intersection in for all . On the other hand, by a weak*-compactness argument, it is easy to see that weak*-closed almost central subspace of a dual space is a central subspace. Moreover, if is an almost central subspace of a Banach space and is an almost central subspace of a Banach space , then is an almost central subspace of .

Lemma 10. Let be a Banach space and let be an ideal in . Then is an almost central subspace of .

Proof. Let , , and . Choose an such that for all . Define . Since is an ideal in , by [9, Theorem 1], there exists an operator such that Now define . Then and for , Hence is an almost central subspace of .

Since every Banach space is an ideal in its bidual, we have the following result.

Corollary 11. Every Banach space is almost central in its bidual.

Since every -ideal is an ideal, by Lemma 10, -ideals are almost central. We now give an example to show that a semi -ideal may not be an almost central subspace.

Example 12. Let denote the three-dimensional space , endowed with the norm for . Now consider the subspace of defined as . Then, by [7, Chapter I, Remark 2.3(a)], is a semi -ideal in . But is not a central subspace of . For, let , and let . Then and . Clearly for all . Suppose there is an such that for all . Then But (3) shows that both of and cannot be positive. But the symmetric inequalities (4) and (5) rule out other possibilities. Thus is not a central subspace of . Then, by a compactness argument, we can see that is not an almost central subspace of .

In [1, Example 5.6], Veselý gave an example of a three-dimensional Banach space such that is not a central subspace of its bidual. Since every Banach space is an ideal in its bidual, the same example shows that an ideal (in particular, an almost central subspace) need not be a central subspace. We now give a sufficient condition for an almost central subspace to be a central subspace.

Theorem 13. Let be an almost central subspace of a Banach space such that . Then is a central subspace of .

Proof. Let and . Since , by [2, Proposition 2.9], it is enough to show that for all .
Now let . Choose such that for all . Since is an almost central subspace of , there exists a such that for all . Hence for all , and the result follows.

Our next result gives a sufficient condition for an almost central subspace to be an AC-subspace.

Proposition 14. Let be an almost central subspace of a Banach space such that is isometric to the range of a projection of norm one in some dual space. Then is an AC-subspace of .

Proof. Let be a Banach space and be a projection of norm one such that is isometric to . Let be the corresponding onto isometry. Now let be any family of closed balls in and be such that for all . Consider the family . Since is an almost central subspace of , any finite collection of balls from this family has nonempty intersection in . Hence any finite collection of balls from the family has nonempty intersection in . Now, by weak* compactness, there exists a such that for all and for all . Hence for all . Now define . Then, for all , we have Hence is an AC-subspace of .

We now give a class of Banach spaces where almost central subspaces are central. We recall that a Banach space whose dual is isometric to for some positive measure is called an -predual.

Proposition 15. Let be an -predual and let be an almost central subspace of . Then is an -predual. In particular, is a central subspace of .

Proof. Let be any family of balls in such that any two of them intersect in . Since is an -predual, by [10, Theorem 6.1], there exists an such that for all . Also, since is an almost central subspace of , for all . Then, by [10, Lemma 4.2 and Theorem 6.1], it follows that is an -predual. Now let be a family of balls in that has nonempty intersection in . It is well known that two balls intersect if and only if the distance between the centers is less than or equal to the sum of the radii. Thus is a pairwise intersecting family in . Since is an -predual, by [10, Theorem 6.1], it follows that intersects in . Hence is a central subspace of .

Remark 16. Following the same line of argument as in the proof of [2, Theorem 3.3], we can observe that a Banach space is an -predual if and only if is an almost central subspace of every Banach space that contains it.

Since every ideal is almost central, our next result generalizes Proposition 14 of [11].

Proposition 17. Let be an almost central subspace of a Banach space . Then is a central subspace of if and only if .

Proof. If is a central subspace of , then, by [2, Proposition 2.2(d)], .
Conversely suppose . Since is an almost central subspace of and is an almost central subspace of , by Remark 9, is an almost central subspace of . Hence, by Theorem 13, it follows that is a central subspace of .

By a similar transitivity argument used in the proof of the Proposition 17, we can easily observe the following corollaries.

Corollary 18. Let be a subspace of such that is an almost central subspace of . Then is an almost central subspace of . In addition, if , then is a central subspace of .

It is well known that a semi -ideal need not be an -ideal (see [7, Chapter I, Remarks 2.3(a)] for example). Our next theorem gives a sufficient condition for a semi -ideal to be an -ideal.

Theorem 19. Let be a semi -ideal in a Banach space such that is an almost central subspace of . Then is an -ideal in .

Proof. Since is a semi -ideal, by [12, Theorem 6.14], is a semi -ideal in . Also, since is a weak*-closed almost central subspace of , is an AC-subspace of . Hence, for any , by [4, Proposition 2.2], is -complemented in and hence is an ideal in . Now, for any , since is a semi -ideal in , by [11, Proposition 23], it follows that is an -ideal in . Hence, by [7, Chapter I, Theorem 2.2], is an -ideal in . Since is a weak*-closed -ideal in , by [7, Chapter II, Corollary 3.6], is an -summand in . Hence, by [7, Chapter I, Theorem 1.9], there exists an -summand in such that . Then, by the duality between - and -projections, we get and hence is an -ideal in .

We now give a sufficient condition for a semi -ideal to be an -summand.

Theorem 20. Let be an AC-subspace of a Banach space . Then is a semi -ideal in if and only if is an -summand in .

Proof. Suppose is a semi -ideal in and is an AC-subspace of . Since is an AC-subspace of , by [4, Proposition 2.2], is -complemented in for all . Also, since is a semi -ideal in , is a semi -ideal in for all . Thus, by [11, Proposition 23], is an -ideal in for all . Then, by [7, Chapter I, Corollary 1.3], is an -summand in for all . Hence, by [7, Chapter II, Proposition 3.2], is an -summand in .

Our next theorem gives another sufficient condition for a semi -ideal to be an -ideal. In fact, this result improves Proposition 23 of [11].

Theorem 21. Let be a subspace of a Banach space such that is an ideal in for all . Then is a semi -ideal in if and only if is an -ideal in .

Proof. Suppose is a semi -ideal in and is an ideal in for all . Then, by [11, Proposition 23], is an -ideal in for all . Hence, by [7, Chapter I, Theorem 2.2], it follows that is an -ideal in .

We now recall the following theorem of Bandyopadhyay and Dutta that characterizes an AC-subspace of finite codimension in the space of all continuous real-valued functions on a compact Hausdorff space , endowed with the supremum norm.

Theorem 22 (see [5, Theorem 1.1]). Let be a compact Hausdorff space and be a subspace of codimension of . Then the following are equivalent.(i) is an AC-subspace of . (ii) is -complemented in . (iii)There exist measures on and distinct isolated points of such that(a), (b).

In our next proposition, we observe a simple proof for the implication of Theorem 22 when is an extremely disconnected space.

We recall that a compact Hausdorff space is extremely disconnected if the closure of each open set in is again open in (see [13, Section 7] for details).

For any infinite discrete set , denotes the space of all bounded real-valued functions on , endowed with the supremum norm, and denotes its subspace consisting of all functions such that the set is finite for all . Also, for any infinite discrete set , denotes the space of all functions such that with the norm .

The following lemma is the uncountable version of the main theorem of [14] for the space . As the proof is similar to that of the theorem of [14], we omit the proof here.

Lemma 23. Let be any infinite discrete set and be a subspace of codimension in . Then is -complemented in if and only if there exist distinct elements in and linearly independent functionals in such that (a) with , ,(b), (c).

Our next result shows that the space cannot have a finite codimensional -complemented subspace containing .

Corollary 24. Let be a Banach space such that for some infinite discrete space . If is a finite codimensional subspace of , then cannot be -complemented in .

Proof. Suppose is -complemented in . Then, by Lemma 23, there exist distinct elements in and linearly independent functionals in such that(a) with , ,(b), (c). Since , for all . Hence for all . Since , we get for all . Then, by (c), and hence for all . This contradiction proves that cannot be -complemented in .

Let be a compact Hausdorff space and be a closed subset of . Also, let be the class of Borel subsets of . Now, for , we define as

Lemma 25. Let be a compact Hausdorff space and let be a closed subset of such that there exists a continuous map which is identity on and let   . If is -complemented in , then is -complemented in .

Proof. Let be a projection of norm one with range .
Now define by Since we get and hence is well defined. Clearly is a linear map.
Now let . Since is identity on , we have Thus, and . Therefore and hence is a projection on with range . Now, since , . Hence is the required projection.

Proposition 26. Let be an extremely disconnected space. If there exist measures on and distinct isolated points of such that for , then is -complemented in .

Proof. Let be a dense subset of . Since each ’s are isolated points of , for all . Now consider with the discrete topology and its Stone-Čech compactification . Then, by [13, Section 7, Lemma 3 and Theorem 3], is homeomorphically embedded into , and also there exists a continuous map such that is identity on . Now consider measures on such that for any Borel set disjoint from and for any Borel set . Since , . Since is isometric to , by Lemma 23, is -complemented in . Then, by Lemma 25, is -complemented in .

In an -predual space, we do not know whether every AC-subspace of finite codimension is the range of a norm one projection and/or is the intersection of AC-subspaces of codimension one.

3. Stability Results

Coming to quotient spaces, one can easily observe that if is -complemented in a Banach space , then, for any subspace of , is 1-complemented in . Motivated by this, we consider the following problem. Let be a subspace of a Banach space having some property () in . If is a subspace of , then when can we say that has the property () in ? We study this problem when the property () under consideration is almost constrained, almost central, central, and ideal.

For a subspace of a Banach space and , we denote by the equivalence class in containing .

Our next result solves the above problem for AC-subspaces.

Proposition 27. Let be an AC-subspace of and let be a subspace of . Then is an AC-subspace of .

Proof. Let be a family of balls in and also let be such that . Then, for each and , there exists a such that We now consider the family of closed balls in . Clearly . Since is an AC-subspace of , there exists a such that . Then, for , we have Therefore for all , and hence is an AC-subspace of .

We now prove the stability of ideals in quotient spaces.

Proposition 28. Let be an ideal in and let be a subspace of . Then is an ideal in .

Proof. Since is an ideal in , by [9, Theorem 1], is -complemented in . Then is -complemented in . But is isometric to , and this isometry takes onto . Hence is -complemented in . Then, again by [9, Theorem 1], is an ideal in .

Our next result proves the stability of almost central subspaces in quotient spaces.

Proposition 29. Let be an almost central subspace of and let be a subspace of . Then is an almost central subspace of .

Proof. Let , and . Then, for , there exists an element such that Since is an almost central subspace of , there exists an element such that Now, for , we have Hence is an almost central subspace of .

Now, for Banach spaces , , with , our next set of results give some sufficient conditions for to be a central subspace of .

Combining Proposition 29 and Theorem 13, we get the following.

Corollary 30. Let be an almost central subspace of and let be a subspace of . If , then is a central subspace of .

As a consequence of the above corollary, we have the following result.

Corollary 31. Let be a subspace of and let be a subspace of such that . If is an almost central subspace of , then is a central subspace of .

Proof. By Corollary 18, is an almost central subspace of . Hence is an almost central subspace of . Since , by Corollary 30, is a central subspace of .

We recall that a subspace of a Banach space is said to be a factor reflexive subspace if the quotient space is reflexive. Since any reflexive spaces are in the class , the following corollary is easy to see.

Corollary 32. Let be a subspace of a Banach space such that is an almost central subspace of . Then, for any factor reflexive subspace of , is a central subspace of .

We now prove the converse of Proposition 29 under some additional assumptions.

Proposition 33. Let be an -predual, let be an -ideal in , and let be a subspace of such that . If is almost central in , then is a central subspace of .

Proof. Let , , and . Then, by assumption, there exists a such that Let be such that for all . Now consider the finite collection of balls in . Since this is a pairwise intersecting family of balls in and is an -predual, . Also, since is an -predual, by [12, Proposition 6.5], it follows that has the strong -ball property in . Hence there exists an element such that for all . Therefore is an almost central subspace of and hence, by Proposition 15, is a central subspace of .

The following corollary is the converse of Proposition 28 under some additional assumptions.

Corollary 34. Let be an -predual, let be an -ideal in , and let be a subspace of such that . If is an ideal in , then is an ideal in .

Proof. Since is an ideal in , by Lemma 10, is almost central in . Thus, by Proposition 33, is a central subspace of . Then, by Proposition 15, is an -predual. Hence, by [15, Proposition 1], is an ideal in .

We recall that, for any collection of Banach spaces, -sum of () and -sum of () are defined as and , respectively, and equip both spaces with supremum norm. For a finite family of Banach spaces , -sum of () is denoted by .

Remark 35. It is easy to observe that, for any family of Banach spaces, if is an almost central subspace of a Banach space , then is an almost central subspace of .

We now prove the stability of almost central subspaces in vector-valued continuous function spaces. For a compact Hausdorff space and a Banach space , we denote by the space of all -valued continuous functions defined on , endowed with the supremum norm.

Let be a compact Hausdorff space and let be a Banach space. Then, for and , an element is defined as for .

Proposition 36. Let be an almost central subspace of Banach space and let be a compact Hausdorff space. Then is an almost central subspace of .

Proof. Let , , and . Then, by the proof of [10, Page 43, Corollary 2], for the finite family , there exists a partition of unity and a closed subspace of spanned by the elements of the form with such that for and is isometric to . Similarly for , there exists a partition of unity and a closed subspace of spanned by the elements of the form with such that , and is isometric to . Now let be such that and let be such that for .
Case 1. Since is isometric to , is an -summand in (up to an isometry). Since -summands are central, by Remark 35 and Remark 9, is an almost central subspace of . Then there exists a such that for . Hence we have
Case 2. In this case, we can isometrically embed into . Since is isometric to , by Remark 35, is an almost central subspace of (up to an isometry). Then there exists an element such that for . Hence we have Therefore in all cases there exists a such that for . Hence is an almost central subspace of .

For a central subspace of a Banach space and for a compact Hausdorff space , it is not known whether is a central subspace of . But if and is almost central in , then, by Proposition 36 and Theorem 13, is a central subspace of . Now for a Banach space , Theorem 3.6 of [8] gives a sufficient condition for to be in the class . Precisely, if is a polyhedral Banach space such that and is finite for each with , then (by , we denote the set of all extreme points of and a Banach space is called polyhedral if the unit ball of each of its finite dimensional subspace is a polytope). In particular, by [8, Fact 1.3(e)], if is a finite dimensional polyhedral space, then . This information together with Proposition 36 give the following corollary.

Corollary 37. Let be an almost central subspace of a Banach space and let be a compact Hausdorff space. If is a polyhedral Banach space such that and is finite for each with , then is a central subspace of . In particular, if is a finite dimensional polyhedral central subspace of , then is a central subspace of .

We now discuss the stability problem in injective tensor product spaces.

Proposition 38. Let be a compact Hausdorff space and let be an almost central subspace of . Then, for any Banach space , the injective tensor product is almost central in .

Proof. Since is an almost central subspace of , by Proposition 15, is an -predual. Then, by [15, Proposition 1], is an ideal in . Hence, by [15, Lemma 2], is an ideal in . Since (up to an isometry), by Lemma 10, is almost central in .

Theorem 39. Let be a compact Hausdorff space and let be an almost central subspace of . If is an almost central subspace of a Banach space , then the injective tensor product is an almost central subspace of . In particular, is an almost central subspace of .

Proof. By Proposition 38, is almost central in . Then, by Proposition 36 and Remark 9, is an almost central subspace of . Since , is an almost central subspace of .

Corollary 40. Let be an -predual. Then, for any almost central subspace of a Banach space , the injective tensor product is an almost central subspace of .

Proof. Since is an -predual, by [15, Proposition 1], is isometric to an ideal in for some compact Hausdorff space . Then, by Lemma 10, is an almost central subspace of (up to an isometry). Therefore, by Theorem 39, is an almost central subspace of .

We now answer a question raised in [2] and also improve Theorem 6 of [11].

Proposition 41. Let be an index set and let be Banach spaces. Then is a central subspace of .

Proof. Let and . Let . Since , there exists a finite set such that whenever .
Define as Now for ,  if , then and  if , then .Hence for all .

Corollary 42. The class is stable under -direct sum of Banach spaces.

Proof. If for all , then, by [2, Theorem 4.7] and Proposition 41, is a central subspace of . Hence the result follows.

In [8], Veselý defined a new direct sum called polyhedral direct sum. We now prove the stability of some ball intersection properties under polyhedral direct sums.

Definition 43 (see [8]). A norm on is called polyhedral if it is of the form , where . In this case, we say that the family generates . Now Lemma 1.5 of [8] shows that if is a minimal family generating , then for all and .
We say that a Banach space is the polyhedral direct sum of Banach spaces if and the norm on is of the form , where is a polyhedral nondecreasing norm on . In this case, we write .

Our next theorem proves that the property of being a central subspace is stable under polyhedral direct sums.

For and , we denote by the th canonical unit vector of .

Theorem 44. Let be a polyhedral direct sum of Banach spaces    and let be a subspace of   . Let be the corresponding polyhedral norm and suppose for all . Then the polyhedral sum of    is a central subspace of if and only if is a central subspace of for all .

Proof. Suppose is a central subspace of . Fix an . Let and . Define and () as Then there exists a such that for . Therefore, for , we have Since for all , we get for . Hence is a central subspace of .
Conversely suppose is central in for all . Let and   . Then, for , there exists a such that for . Define as   . Now, by the monotonicity of , Hence is a central subspace of .

An argument similar to the one used to prove Theorem 44 gives the following.

Theorem 45. Let be a polyhedral direct sum of Banach spaces    and let be a subspace of   . Let be the corresponding polyhedral norm and suppose for all . Then the polyhedral sum of    is an AC-subspace of if and only if is an AC-subspace of for all .

Acknowledgments

The author would like to thank Professor T. S. S. R. K. Rao for many helpful discussions and valuable suggestions. The author also thanks the referees for their extensive comments that lead to an improved version of the paper.

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