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Developments in Schauder basis theory


Author: C. W. McArthur
Journal: Bull. Amer. Math. Soc. 78 (1972), 877-908
MSC (1970): Primary 46A35, 46A40, 46A45, 46B15; Secondary 46-02
DOI: https://doi.org/10.1090/S0002-9904-1972-13048-9
MathSciNet review: 0313766
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  • 1. R. D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515-519. MR 32 #8298. MR 190888
  • 2. M. G. Arsove, The Paley-Wiener theorem in metric linear spaces, Pacific J. Math. 10 (1960), 365-379. MR 23 # A2731. MR 125429
  • 3. M. G. Arsove and R. E. Edwards, Generalized bases in topological linear spaces, Studia Math. 19 (1960), 95-113. MR 22 #5871. MR 115068
  • 4. K. I. Babenko, On bases in Hilbert space, Dokl. Akad. Nauk SSSR 57 (1947), 427-430. (Russian) MR 9, 142. MR 21991
  • 5. G. F. Bachelis and H. P. Rosenthal, On unconditionally converging series and biorthogonal systems in a Banach space, Pacific J. Math. 37 (1971), 1-5. MR 312216
  • 6. S. Banach, Théorie des opérations linéaires, Monografie Mat., PWN, Warsaw, 1932; reprint, Chelsea, New York, 1955. MR 17, 175. MR 71726
  • 7. G. Bennett and J. B. Cooper, Weak bases in (F)- and (LF)-spaces, J. London Math. Soc. 44 (1969), 505-508. MR 39 #742. MR 239385
  • 8. Cz. Bessaga, On topological classification of complete linear metric spaces, Fund. Math. 56 (1964/65), 251-288. MR 31 #2580. MR 178322
  • 9. Cz. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. MR 22 #5872. MR 115069
  • 10. Cz. Bessaga and A. Pelczynski, A generalization of results of R. C. James concerning absolute bases in Banach spaces, Studia Math. 17 (1958), 165-174. MR 22 #5874. MR 115071
  • 11. Cz. Bessaga and A. Pelczynski, Properties of bases in spaces of type B, Prace Mat. 3 (1959), 123-142. (Polish) MR 23 # A3986. MR 126691
  • 12. Cz. Bessaga, A. Pełczyński and S. Rolewicz, On diametral approximative dimension and linear homogeneity of F-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9 (1961), 677-683. MR 24 #A2219. MR 132374
  • 13. Cz. Bessaga and A. Pełczyński, Some remarks on homeomorphisms of F-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 265-270. MR 25 #3344. MR 139917
  • 14. P. Billard, Sur les bases de Schauder dans les espaces de Banach H, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A36-A38. MR 275129
  • 15. Ja. M. Ceĭtlin, Unconditionality of a basis and partial order, Izv. Vysš. Učebn. Zaved. Matematika 1966, no. 2 (51), 98-104; English transl., Amer. Math. Soc. Transl. (2) 90 (1970), 17-25. MR 33 #6362; MR 41 #8191. MR 198203
  • 16. Z. Ciesielski, A construction of bases in C, Studia Math. 33 (1969), 243-247. MR 40 #1759. MR 248507
  • 17. T. A. Cook, Schauder decompositions and semi-reflexive spaces, Math. Ann 182 (1969), 232-235. MR 40 # 3255. MR 250014
  • 18. W. J. Davis, D. W. Dean and Bor-Luh Lin, Bibasic sequences and norming basic sequences (to appear). MR 313763
  • 19. W. J. Davis, D. W. Dean and I. Singer, Multipliers and unconditional convergence of biorthogonal expansions, Pacific J. Math. 37 (1971), 35-39. MR 313762
  • 20. W. J. Davis and W. B. Johnson, On the existence of fundamental and total bounded biorthogonal systems in Banach spaces, Studia Math. (to appear). MR 331021
  • 21. M. M. Day, Normed linear spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete Neue Folge. Heft 21. Reihe: Reelle Funktionen, Springer-Verlag, Berlin, 1958. MR 20 #1187. MR 94675
  • 22. M. M. Day, On the basis problem in normed spaces, Proc. Amer. Math. Soc. 13 (1962), 655-658. MR 25 # 1435. MR 137987
  • 23. D. W. Dean, Schauder decompositions in (m), Proc. Amer. Math. Soc. 18 (1967), 619-623. MR 36 # 664. MR 217575
  • 24. M. De Wilde, On the equivalence of weak and Schauder bases, Proc. Internat. Coll. on Nuclear Spaces and Ideals in Operator Algebras (Warsaw, 1969), Studia Math. 38 (1970), 457.
  • 25. J. Dieudonné, On biorthogonal systems, Michigan Math. J. 2 (1954), 7-20. MR 16, 47. MR 62946
  • 26. M. M. Dragilev, Standard form of basis for the spaces of analytic functions, Uspehi Mat. Nauk 15 (1960), no. 1 (92), 181-188. (Russian) MR 22 #6992. MR 116197
  • 27. M. M. Dragilev, On regular bases in nuclear spaces, Mat. Sb. 68 (110) (1965), 153-173; English transl., Amer. Math. Soc. Transl. (2) 93 (1970), 61-82. MR 33 #536; MR 42 #4. MR 192310
  • 28. E. Dubinsky and J. R. Retherford, Schauder bases in compatible topologies, Studia Math. 28 (1967), 221-226. MR 36 #640. MR 217551
  • 29. E. Dubinsky and J. R. Retherford, Schauder bases and Köthe sequence spaces, Trans. Amer. Math. Soc. 130 (1968), 265-280. MR 38 #510. MR 232184
  • 30. A. S. Dynin and B. Mitjagin, Criterion for nuclearity in terms of approximative dimension, Bull. Acad. Polon. Sci. Sér. Sci. Math/ Astronom. Phys. 8 (1960), 535-540. MR 24 #A2221. MR 132376
  • 31. P. Enflo, A Banach space with basis constant greater than 1 (to appear).
  • 32. K. Floret, Bases in sequentially retractive limit spaces, Proc. Internat. Coll. on Nuclear Spaces and Ideals in Operator Algebras (Warsaw, 1969), Studia Math. 38 (1970). MR 276712
  • 33. M. Fréchet, Les espaces abstraits, Gauthier-Villars, Paris, 1928.
  • 34. R. E. Fullerton, Geometric properties of a basis in a Banach space, Proc. Internat. Congress Math. (Amsterdam, 1954), vol. 2, North-Holland, Amsterdam; Noordhoff, Groningen, 1954, p. 109.
  • 35. R. E. Fullerton, Geometric structure of absolute basis systems in a linear topological space, Pacific J. Math. 12 (1962), 137-147. MR 26 #5401. MR 147888
  • 36. V. F. Gapoškin, Unconditional bases in Orlicz spaces, Uspehi Mat. Nauk 22 (1967), no. 2 (134), 113-114. (Russian) MR 34 #8164. MR 208354
  • 37. D. J. H. Garling, Symmetric bases of locally convex spaces, Studia Math. 30 (1968), 163-181. MR 37 #5670. MR 230105
  • 38. D. J. H. Garling and Y. Gordon, Relations between some constants associated with finite dimensional Banach spaces, Israel J. Math. 9 (1971), 346-361. MR 412775
  • 39. B. R. Gelbaum, Notes on Banach spaces and bases, An. Acad. Brasil. Ci. 30 (1958), 29-36. MR 20 #5419. MR 98974
  • 40. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955), 140 pp. MR 17, 763. MR 75539
  • 41. V. I. Gurariĭ, Inclination of subspaces and conditional bases in Banach spaces, Dokl. Akad. Nauk SSSR 145 (1962), 504-506 = Soviet Math. Dokl. 3 (1962), 1028-1030. MR 27 028-1030. MR 152863
  • 42. N. I. Gurariĭ, On sequences of coefficients of expansions with respect to bases in Hilbert and Banach spaces, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 216-223 = Math. USSR Izv. 5 (1971), 226-232. MR 43 #3778. MR 278046
  • 43. L. A. Gurevič, Conic tests for bases of absolute convergence, Problems of Math. Phys. and Theory of Functions, II, "Naukova Dumka", Kiev, 1964, pp. 12-21. (Russian) MR 33 # 7820. MR 199677
  • 44. J. Hofler, Continuous lattice ordering by Schauder basis cones, Proc. Amer. Math. Soc. 30 (1971), 527-532. MR 415264
  • 45. R. C. James, Bases and reflexitivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518-527. MR 12, 616. MR 39915
  • 46. R. C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 174-177. MR 13, 356. MR 44024
  • 47. R. C. James, Separable conjugate spaces, Pacific J. Math. 10 (1960), 563-571. MR 22 # 8307. MR 117528
  • 48. W. B. Johnson, Markuschevich bases and duality theory, Trans. Amer. Math. Soc. 149 (1970), 171-177. MR 41 #5927. MR 261312
  • 49. W. B. Johnson, On the existence of strongly series summable Markuschevich bases in Banach spaces, Trans. Amer. Math. Soc. 157 (1971), 481-486. MR 282201
  • 50. W. P. Johnson and H. P. Rosenthal, On w*-basic sequences and their applications to the study of Banach spaces, Studia Math. (to appear). MR 310598
  • 51. W. B. Johnson, H. P. Rosenthal and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506. MR 280983
  • 52. M. Ĭ. Kadeć, Bi-orthogonal systems and summation bases, Funkcional. Anal. i Primenen. (Trudy 5 Konf. po Funkcional' nomu Analizu i ego Primeneniju), Izdat. Akad. Nauk Azerbaĭdžan. SSR, Baku, 1961, pp. 106-108. (Russian) MR 26 #2858. MR 145327
  • 53. M. Ĭ. Kadeć, A proof of the topological equivalence of all separable infinite-dimensional Banach spaces, Funkcional. Anal. i Prilozen. 1 (1967), 61-70 = Functional Anal. Appl. 1 (1967), 53-62. MR 35 #700. MR 209804
  • 54. M. Ĭ. Kadeć, On complementably universal Banach spaces, Studia Math. 40 (1971), 85-89. MR 313764
  • 55. M. Ĭ. Kadeć and A. Pełczyński, Basic sequences, bi-orthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297-323. (Russian) MR 31 # 6112. MR 181886
  • 56. N. J. Kalton, Schauder bases and reflexivity, Colloq. on Nuclear Spaces and Ideals in Operator Algebras, Studia Math. 38 (1970), 255-266. MR 276717
  • 57. N. J. Kalton, A barreled space without a basis, Proc. Amer. Math. Soc. 26 (1970), 465-466. MR 42 # 8233. MR 273353
  • 58. S. Karlin, Bases in Banach spaces, Duke Math. J. 15 (1948), 971-985. MR 10, 548. MR 29103
  • 59. Nguen Van Khue, Tests for convergence basis, Izv. Vysš. Učebn. Zaved. Matematika 1968, no. 2 (69), 68-74. (Russian) MR 36 # 5667. MR 222617
  • 60. Y. Kōmura and T. Kōmura, Über die Einbellung der nuklearen Räume in (s), Math. Ann. 162 (1965/66), 284-288. MR 32 #6190. MR 188754
  • 61. Y. Kōmura and S. Koshi, Nuclear vector lattices, Math. Ann. 163 (1966), 105-110. MR 33 # 553. MR 192327
  • 62. G. Köthe and O. Toeplitz, Linear Räume mit unendlich vielen Koordinaten und Ringe unendlicher Matrizen, J. Reine Angew. Math. 171 (1934), 193-226.
  • 63. A. Lazar and J. R. Retherford, Nuclear spaces, Schauder bases and Choquet simplexes, Pacific J. Math. 37 (1971), 409-419. MR 308730
  • 64. M. Levin and C. W. McArthur, Order characterizations of unconditional and absolute Schauder bases, Rev. Roumaine Math. Pures Appl. (to appear). MR 326341
  • 65. J. Lindenstrauss, On James' paper "Separable conjugate spaces", Israel J. Math. 9 (1971), 279-284. MR 279567
  • 66. J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in L, Studia Math. 29 (1968), 275-326. MR 37 #6743. MR 231188
  • 67. J. Lindenstrauss and A. Pełczyński, Contributions to the theory of classical Banach spaces, J. Functional Analysis 8 (1971), 225-249. MR 291772
  • 68. J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Israel J. Math. 9 (1971), 264-270. MR 276734
  • 69. J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math. 10 (1971), 379-390. MR 313780
  • 70. J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis, J. Functional Analysis 3 (1969), 115-125. MR 38 #4963. MR 236668
  • 71. E. R. Lorch, Bicontinuous linear transformations in certain vector spaces, Bull. Amer. Math. Soc. 45 (1939), 564-569. MR 1, 58. MR 346
  • 72. A. Markuševič, Sur les basis (au sens large) dans les espaces linéaires, C. R. (Dokl. ) Acad. Sci. URSS 41 (1943), 227-229. MR 6, 69. MR 10778
  • 73. J. T. Marti, Introduction to the theory of bases, Springer Tracts in Natural Philosophy, vol. 18, Springer-Verlag, Berlin and New York, 1969. MR 438075
  • 74. J. T. Marti and D. R. Sherbert, A note on bases in ordered locally convex spaces, Comment. Math. Helv. 45 (1970), 299-302. MR 42 #5010. MR 270117
  • 75. C. W. McArthur, The weak basis theorem, Colloq. Math. 17 (1967), 71-76. MR 35 #7103. MR 216268
  • 76. C. W. McArthur, The projective equicontinuous topology, Proc. Conf. on Projections, Clemson, S. C., 1967. MR 216268
  • 77. C. W. McArthur, On a theorem of Orlicz and Pettis, Pacific J. Math. 22 (1967), 297-302. MR 35 #297-302. MR 213848
  • 78. C. W. McArthur, Convergence of monotone nets in ordered topological vector spaces, Studia Math. 34 (1970), 1-16. MR 41 #4197. MR 259559
  • 79. C. W. McArthur, In what spaces is every closed normal cone regular?, Proc. Edinburgh Math. Soc. 17 (1970), part 2, 121-125. MR 283545
  • 80. C. W. McArthur, I. Singer and M. Levin, On the cones associated with biorthogonal systems and bases in Banach spaces, Canad. J. Math. 21 (1969), 1206-1217. MR 41 #4207. MR 259569
  • 81. V. D. Mil'man, The geometric theory of Banach spaces, Uspehi Mat. Nauk 25 (1970), no. 3 (153), 132-138. (Russian) MR 280985
  • 82. V. D. Mil'man and Ju. B. Tumarkin, Properties of sequences in locally convex spaces, Dokl. Akad. Nauk SSSR 184 (1969), 278-281 = Soviet Math. Dokl. 10 (1969), 63-66. MR 39 #6045. MR 244731
  • 83. B. S. Mitjagin, Nuclearity and other properties of spaces of type S, Trudy Moskov. Mat. Obšč. 9 (1960), 317-328; English transl., Amer. Math. Soc. Transl. (2) 93 (1970), 45-59. MR 23 # A2041; MR 42 # 4. MR 124730
  • 84. B. S. Mitjagin, Approximative dimension and bases in nuclear spaces, Uspehi Mat. Nauk 16 (1961), no. 4 (100), 63-132 = Russian Math. Surveys 16 (1961), no. 4, 59-127. MR 27 # 2837. MR 152865
  • 85. B. S. Mitjagin, Fréchet spaces with a unique unconditional basis, Studia Math. 38 (1970), 23-34. MR 278033
  • 86. W. F. Newns, On the representation of analytic functions by infinite series, Philos. Trans. Roy. Soc. London Ser. A 245 (1953), 429-468. MR 14, 968. MR 54717
  • 87. V. N. Nikol'skiĭ, Some questions of best approximation in a function space, Uč. Zap. Kalininsk. Pedagog. Inst. 16 (1953), 119-160. (Russian) MR 17, 175. MR 71728
  • 88. A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228. MR 23 #A3441. MR 126145
  • 89. A. Pełczyński, Some problems on bases in Banach and Fréchet spaces, Israel J. Math. 2 (1964), 132-138. MR 30 #3356. MR 173141
  • 90. A. Pełczyński, Universal bases, Studia Math. 32 (1969), 247-268. MR 39 #3290. MR 241954
  • 91. A. Pełczyński, Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis, Studia Math. 40 (1971), 239-243. MR 308753
  • 92. A. Pełczyński and S. Kwapien, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43-68. MR 42 #5011. MR 270118
  • 93. A. Pełczyński and I. Singer, On non-equivalent bases and conditional bases in Banach spaces, Studia Math. 25 (1964/65), 5-25. MR 31 #3831. MR 179583
  • 94. A. Pełczyński and P. Wojtaszczyk, Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional subspaces, Studia Math. 40 (1971), 91-108. MR 313765
  • 95. J. R. Retherford, Bases, basic sequences and reflexivity of linear topological spaces, Math. Ann. 164 (1966), 280-285. MR 33 #6351. MR 198192
  • 96. W. H. Ruckle, The infinite sum of closed subspaces of an F-space, Duke Math. J. 31 (1964), 543-554. MR 29 #3862. MR 166589
  • 97. B. L. Sanders, Decompositions and reflexivity in Banach spaces, Proc. Amer. Math. Soc. 16 (1965), 204-208. MR 30 #2318. MR 172092
  • 98. S. Saxon, Basis cone base theory, Dissertation, Florida State University, Tallahassee, Fla., 1969.
  • 99. H. H. Schaefer, Halbgeordnete lokalkonvexe Vektorräume, Math. Ann. 135 (1958), 115-141. MR 21 #5134. MR 106401
  • 100. H. H. Schaefer, Topological vector spaces, Macmillan, New York, 1966. MR 33 # 1689. MR 193469
  • 101. J. Schauder, Zur Theorie Stetiger abbildungen in Funktionalraumen, Math. Z. 26 (1927), 47-65. MR 1544841
  • 102. S. Schonefeld, Schauder bases in spaces of differentiable functions, Bull. Amer. Math. Soc. 75 (1969), 586-590. MR 39 # 6067. MR 244753
  • 103. J. H. Shapiro, On convexity and compactness in F-spaces with bases, Indiana Univ. Math. J. 21 (1972), 1073-1090. MR 295037
  • 104. J. E. Shirey and R. E. Zink, On unconditional bases in certain Banach function spaces, Studia Math. 36 (1970), 169-175. MR 275142
  • 105. I. Singer, Sur les espaces de Banach à base absolue, canoniquement équivalents à un dual d'espace de Banach, C. R. Acad. Sci. Paris 251 (1960), 620-621. MR 22 # 5869. MR 115066
  • 106. I. Singer, Sur les espaces de Banach à base absolue, canoniquement équivalents à un dual d'espace de Banach. II, C. R. Acad. Sci. Paris 251 (1960), 2456-2458. MR 22 #9841. MR 119075
  • 107. I. Singer, Basic sequences and reflexivity of Banach spaces, Studia Math. 21 (1961/62), 351-369. MR 26 #4155. MR 146635
  • 108. I. Singer, Bases in Banach spaces. I, Acad. R. P. Romîne Stud. Cerc. Mat. 14 (1963), 533-585. (Romanian) MR 32 # 8121a. MR 190709
  • 109. I. Singer, Bases in Banach spaces. II, Acad. R. P. Romîne Stud. Cerc. Mat. 15 (1964), 157-208. (Romanian) MR 32 # 8121b. MR 190709
  • 110. I. Singer, Bases in Banach spaces. III, Acad. R. P. Romîne Stud. Cerc. Mat. 15 (1964), 675-725. (Romanian) MR 32 # 8121b. MR 190710
  • 111. I. Singer, On the basis problem in topological linear spaces, Rev. Roumaine Math. Pures Appl. 10 (1965), 453-457. MR 33 #3081. MR 194875
  • 112. I. Singer, Bases in Banach spaces. I, Die Grundlehren der math. Wissenschaften, Band 154, Springer-Verlag, New York, 1970. MR 298399
  • 113. S. Smith, Cone relationships of biorthogonal systems, Pacific J. Math. 35 (1970), 787-794. MR 276725
  • 114. W. J. Stiles, On properties of subspaces of l 0 < p < 1, Trans. Amer. Math. Soc. 149 (1970), 405-415. MR 261315
  • 115. Ju. B. Tumarkin, Locally convex spaces with basis, Dokl. Akad. Nauk SSSR 195 (1970), 1278-1281 = Soviet Math. Dokl. 11 (1970), 1672-1675. MR 42 #6575. MR 271694
  • 116. L. J. Weill, Unconditional and shrinking bases in locally convex spaces, Pacific J. Math. 29 (1969), 467-483. MR 39 #7389. MR 246083
  • 117. W. Wojtyński, On conditional bases in non-nuclear Fréchet spaces, Studia Math. 35 (1969), 467-483. MR 39 #7389. MR 271695
  • 118. P. Wojtaszczyk, On separable Banach spaces containing all separable reflexive Banach spaces, Studia Math. 37 (1971), 197-202. MR 308750
  • 119. M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. Math. 4 (1966), 265-272. MR 35 #705. MR 209809
  • 120. M. Zippin, A remark on bases and reflexivity in Banach spaces, Israel J. Math. 6 (1968), 74-79. MR 38 #4972. MR 236677
  • 121. M. Zippin, A remark on Pełczyński's paper "Universal bases", Proc. Amer. Math. Soc. 26 (1970), 294-300.

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DOI: https://doi.org/10.1090/S0002-9904-1972-13048-9

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