On nonseparable Banach spaces

Author:
Spiros A. Argyros

Journal:
Trans. Amer. Math. Soc. **270** (1982), 193-216

MSC:
Primary 46B20; Secondary 03E35, 03E50

DOI:
https://doi.org/10.1090/S0002-9947-1982-0642338-2

MathSciNet review:
642338

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Abstract: Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions of the paper, in conjunction with already known results, give complete answers to problems of the theory of Banach spaces. An interesting point here is that some questions of Banach spaces theory are independent of Z.F.C. So, for example, the answer to a conjecture of Pełczynski that states that the isomorphic embeddability of into implies, for any infinite cardinal , the isomorphic embedding of into , gets the following form:

if , has been proved from Pełczynski;

if , the proof is given in this paper;

if , in , an example discovered by Haydon gives a negative answer;

if , in , is also proved in this paper.

**[1]**S. Argyros,*On dimension of injective Banach spaces*, Proc. Amer. Math. Soc.**78**(1980), 267-268. MR**550510 (81c:46013)****[2]**-,*Weak compactness in**and injective Banach spaces*, Israel J. Math.**37**(1980), 21-33. MR**599299 (82a:46016)****[3]**S. Argyros and S. Negrepontis,*Universal embeddings of**into**and*, Colloq. Math. Soc. Janós Bolyai Topology, vol. 23, Budapest, 1978.**[4]**W. W. Comfort and S. Negrepontis,*The theory of ultrafilters*, Band 211, Springer-Verlag, Berlin, 1974. MR**0396267 (53:135)****[5]**L. Dor,*On projection*, Ann. of Math. (2)**102**(1978), 463-474. MR**0420244 (54:8258)****[6]**J. Hagler,*On the structure**and**for**dyadic*, Trans. Amer. Math. Soc.**214**(1975), 415-427. MR**0388062 (52:8899)****[7]**J. Hagler and C. Stegall,*Banach spaces whose duals contain complemented subspaces isomorphic to*, J. Funct. Anal.**13**(1973), 233-251. MR**0350381 (50:2874)****[8]**A. Hajnal,*Proof of a conjecture of S. Ruziewicz*, Fund. Math.**50**(1961), 123-128. MR**0131986 (24:A1833)****[9]**R. Haydon,*On Banach spaces which contain**and types of measures on compact spaces*, Israel J. Math.**28**(1977), 313-324, MR**0511799 (58:23514)****[10]**-,*On dual**spaces and injective bidual Banach spaces*, Israel J. Math.**7**(1978), 142-152.**[11]**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**0280983 (43:6702)****[12]**A. Pełczynski,*Projections in certain Banach spaces*, Studia Math.**19**(1960), 209-228. MR**0126145 (23:A3441)****[13]**-,*On Banach spaces containing*, Studia Math.**30**(1968), 231-246.**[14]**H. P. Rosenthal,*On injective Banach spaces and the spaces**for finite measures*, Acta Math.**127**(1970), 205-248. MR**0257721 (41:2370)****[15]**-,*On relatively disjoint families of measures with some applications to Banach space theory*, Studia Math.**37**(1970), 13-36. MR**0270122 (42:5015)****[16]**M. E. Rudin,*Martin's axiom*, Handbook of Mathematical Logic, (J. Barwise, ed.), North-Holland, Amsterdam, 1977, pp. 481-502. MR**0457132 (56:15351)****[17]**T. W. Starbird,*Subspaces of**containing*, Dissertation, Univ. of California, Berkeley, 1976.**[18]**C. Stegall,*Banach spaces whose duals contain**with applications to the study of dual**spaces*, Trans. Amer. Math. Soc.**17**(1973), 463-477. MR**0315404 (47:3953)**

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DOI:
https://doi.org/10.1090/S0002-9947-1982-0642338-2

Article copyright:
© Copyright 1982
American Mathematical Society