Book Review
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MathSciNet review:
1567364
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Book Information:
Author:
Ivan Singer
Title:
Bases in Banach spaces.
Additional book information:
Springer-Verlag, Berlin and New York, 1981, viii + 880 pp., $78.00.
M. G. Arsove and R. E. Edwards, Generalized bases in topological linear spaces, Studia Math. 19 (1960), 95–113. MR 115068, DOI 10.4064/sm-19-1-95-113
2. S. Banach, Théorie des opérations linéaires, Monogrofje Matematyczne, Warszawa, 1932.
C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
C. Bessaga and A. Pełczyński, A generalization of results of R. C. James concerning absolute bases in Banach spaces, Studia Math. 17 (1958), 165–174. MR 115071, DOI 10.4064/sm-17-2-165-174
A. M. Davie, The approximation problem for Banach spaces, Bull. London Math. Soc. 5 (1973), 261–266. MR 338735, DOI 10.1112/blms/5.3.261
William J. Davis, David W. Dean, and Bor Luh Lin, Bibasic sequences and norming basic sequences, Trans. Amer. Math. Soc. 176 (1973), 89–102. MR 313763, DOI 10.1090/S0002-9947-1973-0313763-9
W. J. Davis and W. B. Johnson, On the existence of fundamental and total bounded biorthogonal systems in Banach spaces, Studia Math. 45 (1973), 173–179. MR 331021, DOI 10.4064/sm-45-2-173-179
David W. Dean, Schauder decompositions in $(m)$, Proc. Amer. Math. Soc. 18 (1967), 619–623. MR 217575, DOI 10.1090/S0002-9939-1967-0217575-9
R. E. Edwards, Integral bases in inductive limit spaces, Pacific J. Math. 10 (1960), 797–812. MR 115065
Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
11. T. Figiel, Further counterexamples to the approximation problem, Dittoed notes.
T. Figiel and W. B. Johnson, The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc. 41 (1973), 197–200. MR 341032, DOI 10.1090/S0002-9939-1973-0341032-5
Orrin Frink Jr., Series expansions in linear vector space, Amer. J. Math. 63 (1941), 87–100. MR 3452, DOI 10.2307/2371279
Bernard R. Gelbaum, Expansions in Banach spaces, Duke Math. J. 17 (1950), 187–196. MR 35396
M. M. Grinblyum, On the representation of a space of type $B$ in the form of a direct sum of subspaces, Doklady Akad. Nauk SSSR (N.S.) 70 (1950), 749–752 (Russian). MR 0033977
16. D. F. Hale, Integral bases in Banach spaces, Dissertation, Ohio State Univ., 1969.
William B. Johnson, Finite-dimensional Schauder decompositions in $\pi _{\lambda }$ and dual $\pi _{\lambda }$ spaces, Illinois J. Math. 14 (1970), 642–647. MR 265917
William B. Johnson, Markuschevich bases and duality theory, Trans. Amer. Math. Soc. 149 (1970), 171–177. MR 261312, DOI 10.1090/S0002-9947-1970-0261312-3
W. B. Johnson and H. P. Rosenthal, On $\omega ^{\ast }$-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77–92. MR 310598, DOI 10.4064/sm-43-1-77-92
M. Ĭ. Kadec′, Bi-orthogonal systems and summation bases, Funkcional′nyĭ Analiz i ego Primenenie (Trudy 5 Konf. po Funkcional′nomu Analizu i ego Primeneniju), Izdat. Akad. Nauk Azerbaĭdžan. SSR, Baku, 1961, pp. 106–108 (Russian). MR 0145327
M. Ĭ. Kadec′ and A. Pelčin′ski, Basic sequences, bi-orthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297–323 (Russian). MR 181886
V. Ya. Kozlov, On a generalization of the concept of basis, Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 643–646 (Russian). MR 0036446
Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580
Joram Lindenstrauss, On James’s paper “Separable conjugate spaces”, Israel J. Math. 9 (1971), 279–284. MR 279567, DOI 10.1007/BF02771677
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
26. I. Maddaus, On completely continuous linear transformations, Bull. Amer. Math. Soc. 44 (1938), 279-282.
A. Markouchevitch, Sur les bases (au sens large) dans les espaces linéaires, C. R. (Doklady) Acad. Sci. URSS (N.S.) 41 (1943), 227–229 (French). MR 0010778
R. I. Ovsepian and A. Pełczyński, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in $L^{2}$, Studia Math. 54 (1975), no. 2, 149–159. MR 394137, DOI 10.4064/sm-54-2-149-159
A. Pelčinskiĭ and T. Figel′, On Enflo’s method of constructing Banach spaces without the approximation property, Uspehi Mat. Nauk 28 (1973), no. 6 (174), 95–108 (Russian). Translated from the English by V. L. Levin. MR 0450941
A. Pełczyński and I. Singer, On non-equivalent bases and conditional bases in Banach spaces, Studia Math. 25 (1964/65), 5–25. MR 179583, DOI 10.4064/sm-25-1-5-25
J. R. Retherford, Some remarks on Schauder bases of subspaces, Rev. Roumaine Math. Pures Appl. 11 (1966), 787–792. MR 205034
William H. Ruckle, The infinite sum of closed subspaces of an $F$-space, Duke Math. J. 31 (1964), 543–554. MR 166589
William H. Ruckle, Representation and series summability of complete biorthogonal sequences, Pacific J. Math. 34 (1970), 511–528. MR 267317
William H. Ruckle, On the classification of biorthogonal sequences, Canadian J. Math. 26 (1974), 721–733. MR 346494, DOI 10.4153/CJM-1974-067-0
B. L. Sanders, On the existence of [Schauder] decompositions in Banach spaces, Proc. Amer. Math. Soc. 16 (1965), 987–990. MR 180835, DOI 10.1090/S0002-9939-1965-0180835-2
Ivan Singer, On biorthogonal systems and total sequences of functionals, Math. Ann. 193 (1971), 183–188. MR 350387, DOI 10.1007/BF02052389
37. B. S. Tsirel'son, Not every Banach space contains an imbedding of l, Functional Anal. Appl. 8 (1974), 138-141.
- 1.
- M. G. Arsove and R. E. Edwards, Generalized bases in topological linear spaces, StudiaMath. 19 (1960), 95-113. MR 0115068
- 2.
- S. Banach, Théorie des opérations linéaires, Monogrofje Matematyczne, Warszawa, 1932.
- 3.
- C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. MR 115069
- 4.
- C. Bessaga and A. Pełczyński, A generalization of results of R. C. James concerning absolute bases in Banach spaces, Studia Math. 17 (1958), 165-174. MR 115071
- 5.
- A. M. Davie, The approximation problem for Banach spaces, Bull. London Math. Soc. 5 (1973), 261-266. MR 338735
- 6.
- W. J. Davis, D. W. Dean and Bor-Luh Lin, Bibasic sequences and norming basic sequences, Trans. Amer. Math. Soc. 176 (1973), 89-102. MR 313763
- 7.
- W. J. Davis and W. B. Johnson, On the existence of fundamental and total bounded biorthogonal systems in Banach spaces, Studia Math. 45 (1973), 173-179. MR 331021
- 8.
- D. W. Dean, Schauder decompositions in (m), Proc. Amer. Math. Soc. 18 (1967), 619-623. MR 217575
- 9.
- R. E. Edwards, Integral bases in inductive limit spaces, Pacific J. Math. 10 (1960), 797-812. MR 115065
- 10.
- P. Enflo, A counterexample to the approximation problem, Acta Math. 13(1973), 309-317. MR 402468
- 11.
- T. Figiel, Further counterexamples to the approximation problem, Dittoed notes.
- 12.
- T. Figiel and W. B. Johnson, The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc. 41 (1973), 197-200. MR 341032
- 13.
- O. Frink, Jr., Series expansions in linear spaces, Amer. J. Math. 63 (1941), 87-100. MR 3452
- 14.
- B. R. Gelbaum, Expansions in Banach spaces, Duke Math. J. 17 (1950), 187-196. MR 35396
- 15.
- M. M. Grunblium, On the representation of a space of type B as a direct sum of subspaces, Dokl. Akad. Nauk SSSR 70 (1950), 749-752. (Russian) MR 33977
- 16.
- D. F. Hale, Integral bases in Banach spaces, Dissertation, Ohio State Univ., 1969.
- 17.
- W. B. Johnson, Finite-dimensional Schauder decompositions in IIλ and dual IIλ spaces, Illinois J. Math. 14 (1970), 642-647. MR 265917
- 18.
- W. B. Johnson, Markuschevich bases and duality theory, Trans. Amer. Math. Soc. 149 (1970), 171-177. MR 261312
- 19.
- W. B. Johnson and H. P. Rosenthal, On w*-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77-92. MR 310598
- 20.
- M. I. Kadec, Bi-orthogonal systems and summation bases, Funkcional'nyĭ Analiz i ego Primenenie, Izdat. Akad. Nauk Azerbaĭdžan SSR Baku 1961, pp. 106-108. (Russian) MR 145327
- 21.
- M. I. Kadec and A. Pełczyński, Basic sequences, biorthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297-323. (Russian) MR 181886
- 22.
- V. Ya. Kozlov, On a generalization of the notion of basis, Dokl. Akad. Nauk SSSR 73 (1950), 643-646. (Russian) MR 36446
- 23.
- J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 1-112. MR 179580
- 24.
- J. Lindenstrauss, On James's paper "Separable conjugate spaces ", Israel J. Math. 9 (1971), 279-284. MR 279567
- 25.
- J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I: Sequence spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR 500056
- 26.
- I. Maddaus, On completely continuous linear transformations, Bull. Amer. Math. Soc. 44 (1938), 279-282.
- 27.
- A. Markouchevitch, Sur les bases (au sens large) dans les espaces linéaires, Dokl. Akad. Nauk SSSR 41 (1943), 227-229. MR 10778
- 28.
- R. I. Ovsepian and A. Pełczyński, The existence in every Banach space of a fundamental total and bounded biorthogonal sequence and related constructions of uniformly bounded orthonormal systems in L, Studia Math. 54 (1975), 149-159. MR 394137
- 29.
- A. Pełczyński and T. Figiel, On Enflo's method of construction of Banach spaces without the approximation property, Uspehi Mat. Nauk 28 (1973) no. 6(174), 95-108. (Russian) MR 450941
- 30.
- A. Pełczyński and I. Singer, On non-equivalent bases and conditional bases in Banach spaces, Studia Math. 25 (1964), 5-25. MR 179583
- 31.
- J. R. Retherford, Some remarks on Schauder bases of subspaces, Rev. Roumaine Math. Pures Appl. 11 (1966), 787-792. MR 205034
- 32.
- W. H. Ruckle, The infinite sum of closed subspaces of an F-space, Duke Math. J. 31 (1964), 543-554. MR 166589
- 33.
- W. H. Ruckle, Representation and series summability of complete biorthogonal sequences, Pacific J. Math. 34 (1970), 511-528. MR 267317
- 34.
- W. H. Ruckle, On the classification of biorthogonal sequences, Canad. J. Math. 26 (1974), 721-733. MR 346494
- 35.
- B. L. Sanders, On the existence of Schauder decompositions in Banach spaces, Proc. Amer. Math. Soc. 16 (1965), 987-990. MR 180835
- 36.
- I. Singer, On biorthogonal systems and total sequences of functionals, Math. Ann. 193 (1971), 183-188. MR 350387
- 37.
- B. S. Tsirel'son, Not every Banach space contains an imbedding of l, Functional Anal. Appl. 8 (1974), 138-141.
Review Information:
Reviewer:
Charles W. McArthur
Journal:
Bull. Amer. Math. Soc.
6 (1982), 478-486
DOI:
https://doi.org/10.1090/S0273-0979-1982-15023-6
