NOTES AND REMARKS CHAPTER 1. The material of Chapter 1 is accessible in a number of sources. We refer the reader to various monographs and textbooks rather than to original sources. An early, but very readable, treatment is Schatten's monograph [154]. Chapter 9 in Diestel and Uhl [55] is also helpful. Among textbooks which treat the tensor product of linear topological spaces, are those of Treves [165], Day [41], Kothe [113], and Schaefer [153]. The article by Gilbert and Leih [61] contains a succinct account of tensor-product theory. The norms a p discussed in 1.43 to 1.46 were introduced independently by Saphar [152] and Chevet [36]. The paper of Saphar [151] contains a wealth of information about them. The proof of the triangle inequality in 1.46 is from [151]. The isomorphism theorem 1.52 was apparently obtained independently by Chevet [35] and Persson [140]. CHAPTER 2. The first half of Chapter 2 deals with the proximinality of G <2> Y in X 0 a Y, where X and Y are Banach spaces, G is a subspace of X, and a is a crossnorm. Results 2.1 to 2.3 and 2.5 to 2.7 are from Franchetti and Cheney [61]. The distance formula in C(S, Y) (2.4) was given, along with many other important results, by Buck [25]. The part of 2.4 which refers to £00 (S, Y) is from von Golitschek and Cheney [16]. The results 2.10 to 2.13, concerning proximinality in spaces LdS, Y), are contained implicitly or explicitly in Khalil [112]. The result 2.13 generalizes work in [121], where techniques like those in 2.8 gave a weaker result. An alternative approach to some of these proximinality questions is via the theory of Chebyshev centers. These ideas are explored in Franchetti and Cheney [62]. For recent results in the abstract theory of Chebyshev centers, the paper of Amir, Mach, and Saatkamp [1] is recommended. In the second half of Chapter 2, the theme is the proximinality of subspaces X <2> H + G <2> Y in X 0 a Y. Results 2.14 to 2.18 are from Respess and Cheney [146]. Corollary 2.19 was given first in [121]. Results 2.21 to 2.25 are also from [146]. Theorem 2.22 has been called the "Sitting Duck Theorem" because it seems likely that its hypotheses can be drastically weakened. The situation in LdS x T) is much more satisfactory, as can be seen by comparing 2.22 with 2.26. The latter is from Holland, Light, and Sulley [102]. The problem of characterizing best approximations in spaces G 0 Y + X 0 H has not received much attention. The general theorems of Singer [160] are of course applicable. Havinson [99] proved a characterization theorem for best approximations in C(S x T) by elements of the subspace C(S) +C(T). In [120], there is a characterization theorem similar to Havinson's for subspaces C(S) ® H + G ® C(T), where G and H are finite dimensional. Approximation in C(D), where D is a proper subset of S x T, differs in essential aspects from appproximation in 8 x T. As 2.27 shows, the proximinality of C(8) + C(T) in C(D) depends on the geometry of D. The reader should consult the papers of Ofman [137] and von Golitschek [69] for further information. Theorem 2.22 requires a proximity map of C(8) satisfying a Lipschitz condition. Unfortunately, this is a rather restrictive hypothesis, as the subspaces which are usually encountered in practical problems do not have such proximity maps. For example, Haar subspaces of dimension 2 or greater never have this property. Indeed, subspaces having the Haar property at a single point do not have Lipschitz proximity maps. See Respess and Cheney [145]. One can further prove that the proximity map onto a Haar subspace of dimension 2 or greater is not uniformly continuous on bounded sets. By a theorem of Freud, the proximity map of C(8) onto a Haar subspace is locally Lipschitz, but this property seems to be inadequate for the proof of theorem 2.22. In general, C(8) will contain subspaces of any dimension (finite or infinite) which possess Lipschitz proximity maps [145]. The proximinality of spaces C(8) 181 IIm(T) + IIn(8) 181 C(T), where 8 and Tare intervals and II n is the space of polynomials of degree at most n, is an open question, except when min(n, m) is O. Von Golitschek and Cheney in [76] prove that each function in C(8 x T) possesses a best approximation in e; (8) 181 IIdT) + IId8) 181 eoo (T) which is continuous on the interior of the rectangle 8 x T. CHAPTER 3. According to Deutsch [51], the alternating algorithm appeared first in mimeographed lecture notes of von Neumann in 1933. These notes were later published in [1351. References to subsequent work on this algorithm for Hilbert space projections will be found in Deutsch [49]. Applications of the algorithm to diverse problems, such as the multigrid method for partial differential equations and to computerized tomography, are mentioned in Deutsch [48]. Result 3.4 is from Franchetti and Light [63]. Item 3.5 is from [51]. The theorem in 3.6 derives from a paper by Baillon, Bruck and Reich [19]; item 3.7 is from [63]. The theorem in 3.8 is of course von Neumann's original result [135]. Results 3.9 and 3.10 are from [51]; the sequence 3.11 to 3.16, from [63]. Franchetti and Light in [63] have described a class of uniformly convex Banach spaces in which the alternating algorithm is effective for any pair of closed subspaces. This class properly includes the Hilbert spaces. CHAPTER 4. The concept of a central proximity map comes from Golomb's work [78]. Items 4.1, 4.2, and 4.3 are from [18], although the ideas are present in Diliberto and Straus [56]. Item 4.7 is also from [78]. CHAPTER 5. Lemmas 5.1 and 5.2 are implicit in Diliberto and Straus [56]. Items 135 5.3 and 5.4 are from Aumann [6]. Item 5.6 comes from [56], but the proof is due to Aumann [6]. The result 5.8 derives from [6], but the proof is from Light and Cheney [122]. It was proved by Dyn [58] that the Diliberto-Straus algorithm will fail if applied to the subspaces C(8) @ IIdT) and IIo(8) @ C{T). Her method of proof was used by von Golitschek and Cheney [75] to prove that the algorithm fails for C(8) @ Hand G@ C{T) whenever G and H are Haar subspaces such that dim G > 1 and dim H > O. There exist non-Haar subspaces G and H of arbitrary dimension, however, for which the algorithm does work. See Respess and Cheney [145]. The convergence of the Diliberto-Straus algorithm can be arbitrarily slow, as is shown by examples in von Golitschek and Cheney [77]. An application of the algorithm to the scaling (or "preconditioning") of matrices in the numerical solution of partial differential equations can be found in several papers of Bank [20,21]. Often, one or two steps of the algorithm suffice to produce a satisfactory preconditioning. The theoretical basis for this is discussed in [77]' where it is shown, for example, that the third and fourth iterates in the algorithm satisfy the inequalities IIz311 2dist (z,W) and llz411 1.5dist (z,W). CHAPTER 6. The results in this chapter are due to von Golitschek in a series of papers [69-74], especially [69]. For simplicity of exposition, we have limited the discussion to domains 8 X T, although the original work includes certain types of closed subsets in 8 x T. CHAPTER 7. The results 7.1 to 7.12 derive from Light and Holland [123]. Example 7.13 is from Light, McCabe, Phillips, and Cheney [124]. Lemma 7.15 comes from [124]. Items 7.16 to 7.19 are from [117] and [118]. CHAPTER 8. Result 8.2 is due to Jameson and Pinkus [109]. Theorem 8.8 is from Halton and Light [89]. Results 8.13 and 8.14 come from Franchetti and Cheney [60], and are based upon the work in [109]. We thank Dr. Shekhtman for communicating Example 8.15 to us. CHAPTER 9. Theorem 9.1 is due to Rudin [150]. The techniques used in 9.2 were kindly communicated to us by Y. Benyamini. Corollary 9.4 is from Halton and Light [90]. Theorem 9.5 comes from Halton and Light [88]' while results 9.7 to 9.9 are from [90]. It is possible to obtain Theorem 9.9 directly from 9.1 without using the discretization involved in 9.2 to 9.8. Details of this approach are contained in a preprint by W.A.Light entitled "Minimal Projections in Tensor Product Spaces". 136 CHAPTER 10. The material on the Bochner integral can be found in Diestel and Uhl [55]. CHAPTER 11. Lemma 11.1 on the Boolean sum of two projections is probably due to Gordon [81]; see also Gordon and Cheney [84]. Theorem 11.2 is from Respess and Ch­ eney [146]. In Theorem 11.3, the equivalence of (i) and (ii) occurs in Rudin [149, p.130]. The equivalence of (i) and (vi) is due to Reiter [144]. The proof of Auerbach's Lemma is given by A.F. Ruston in "Auerbach's lemma and tensor products of Banach spaces," Proc, Cambridge Phil.