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References
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A Somewhat Annotated Bibliography
R. D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1961) 515-519. The finishing touches are applied to the Anderson-Kadec theorem: all separable infinite dimensional Fréchet spaces are homeomorphic.
R. D. Anderson and R. H. Bing, A complete elementary proof that Hilbert space is homeomorphic to the countably infinite product of lines, Bull. Amer. Math. Soc. 72 (1969), 771–792. The title says it all.
C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. Contains one of the three 1958 proofs that each infinite dimensional Banach space has a subspace with a Schauder basis. Also gives some sharp forms of the Orlicz-Pettis Theorem in spaces withoutc 0-subspaces.
C. Bessaga and A. Pelczynski, A topological proof that every separable Banach space is homeomorphic to a countable product of lines, Bull. Acad. Polon. Sci. 17 (1969), 487–493.
M. M. Day,Normed Linear Spaces, Ergebnisse d. Math. 21 (1958), Springer-Verlag. A classic for its brevity, crispness and completeness. Contains a solid hint at one of the three 1958 proofs of the existence of a basic sequence in any Banach space.
M. M. Day, On the basis problem in normed spaces, Proc. Amer. Math. Soc. 13 (1962), 655–658. Fills in details for those unable to follow the hint provided inNormed Linear Spaces!
E. Dubinsky, Every separable Frechet space contains a nonstable dense subspace, Studia Math. 40 (1971), 77–79. An infinite dimensional separable Fréchet spaceE can have a subspaceF of co-dimension one withE andF non-isomorphic.
P. Enfio, A counterexample to the approximation problen in Banach spaces, Acta Math. 130 (1973), 309–317. A counterexample of the title resides insidec 0 and so provides an example of a separable Banach space without a Schauder basis. May well be the Emancipation Proclamation of Banach Space Theory. The solution of these problems freed a number of mathematicians to work on other aspects of Banach space theory.
B. R. Gelbaum, Banach spaces and bases, An Acad. Brasil Ci. 30 (1958), 29–36. The third of the 1958 proofs that each Banach space has a subspace with a Schauder basis.
M. I. Kadec, A proof of the topological equivalence of all separable infinite dimensional Banach spaces, Funkcional Anal, i Prilozen 1 (1967), 55–62. Kadec gives a coordinate system to each separable Banach space resulting in the result alluded to in the title.
M. I. Kadec and A. Pelczynski, Basic sequences, biorthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297–323. Mazur’s method of producing basic sequences in presented here, along with much, much more.
J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces, Volumes I, II, II, IV,..., Ergebnisse d. Math., Springer-Verlag. It’s a big step from the last two chapters ofOpérationes Linéaires to modern Banach space theory but anyone interested in so proceeding must go through these multivolumed treasures. BEWARE: This is not for the faint-hearted.
W. Orlicz, Beiträge zur Théorie der Orthogonalentwicklungen II, Studia Math. 1 (1929), 241–255.
W. Orlicz, Über unbedingte Konvergenz in Funktionenräumen I, Studia Math. 4 (1933), 33–37; II, Studia Math 4 (1933), 41-47
B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277–304. The Orlicz-Pettis Theorem.
A. Pelczynski and C. Bessaga, Some aspects of the present theory of Banach spaces,Oeuvres (Stefan Banach) Volume II, Académie Polonaise des Sciences, Institut Mathématique, Warszawa, 1979 An updating of the “Remarks” ofOpérations Linéaires’.
S. Rolewicz, An example of a normed space non-isomorphic to its product by the real line, Studia Math. 40 (1971), 71–75.
H. Torunczyk, Characterizing Hilbert space topology, preprint. A consequence of the results in this paper is the stunning fact that every Fréchet space is homeomorphic to a Hilbert space.
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Smullyan, R.M., Abikoff, W., Kra, I. et al. About Books. The Mathematical Intelligencer 4, 39–54 (1982). https://doi.org/10.1007/BF03022996
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DOI: https://doi.org/10.1007/BF03022996