Abstract
In the last years the theory of bases in Banach spaces has rapidly developped*). We shall make here some remarks on a few recent results and raise some new problems in this field. The paper has partially an expository character, since we want to make it accessible also to those who are not assumed to know basis theory.
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References
E. J. Akutowicz, Construction of a Schauder basis in some spaces of holomorphic functions in the unit disc. Colloq. Math. 15 (1966), 287–296.
K. I. Babenko, On conjugate functions. Dokl. Akad. Nauk. SSSR 57 (1948), 157–160 (Russian).
S. Banach, Théorie des opérations linéaires. Monografie Matematyczne, Warszawa 1932.
S. Banach und S. Mazur, Zur Theorie der linearen Dimension. Studia Math. 4 (1933), 100–112.
N. K. Bari, Biorthogonal systems and bases in Hilbert space. Moskov. Gos. Univ. Uceneye Zapiski 148, Matematika 4 (1951), 69–107 (Russian).
C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces. Studia Math. 17 (1958)151–164.
H. F. Bohnenblust, Subspaces of l p,n spaces. Amer. J. Math. 63 (1941)64–72.
A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type C(K). Canad. J. Math. 5 (1953)129–173.
A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16 (1955)140 pp.
I. M. Gelfand, Remark on the paper of N. K. Bari “Biorthogonal systems and bases in Hilbert space”. Moscov. Gos. Univ. Uceneye Zapiski 148, Matematika 4, (1951), 224–225 (Russian)
V. I. Gurarii, On some geometric characteristics of subspaces and of bases in Banach spaces Coll. Math. 13 (1964)59–63 (Russian).
I. Kaplansky, Functional Analysis. In: Surveys in Applied Math. IV, 3–34. Wiley, New York; Chapman and Hall, London 1957.
G. Köthe, Probleme der linearen Algebra in topologischen Vektorräumen. Proc. Internat. Sympos. on Linear Spaces (July 5–12, 1960), 290–298. Jerusalem Academic Press; Pergamon, Oxford 1961.
J. Lindenstrauss and A. Pelczynski, Absolutely summing operators in 𝓛-spaces and their applications. Studia Math. 39 (1968)275–326.
J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis. (To appear.)
A. Pelczynski, Projections in certain Banach spaces. Studia Math. 19 (1960)209–228.
A. Pelczynski, Some open questions in functional analysis. A lecture given to Louisiana State University (dittoed notes), 1966.
A. Pelczynski, Universal bases. Studia Math. (To appear.)
A. Pelczynski and I. Singer, On non-equivalent bases and conditional bases in Banach spaces. Studia Math. 25 (1964)5–25.
J. Schauder, Zur Theorie des stetigen Abbildungen in Funktionalräumen. Math. Z. 26 (1927) 47–65.
I. Singer, On Banach spaces with a symmetric basis. Rev. Math. Pures Appl. 6 (1961)159–166 (Russian).
I. Singer, Some characterizations of symmetric bases in Banach spaces. Bull. Acad. Polon. Sci.Sér. Sci. Math. Astronom. Phys. 10 (1962)185–192.
I. Singer, Bases in Banach spaces. I, Studii si cercet. mat. 14 (1963)533–585
I. Singer, Bases in Banach spaces. II, Studii si cercet. mat., 15 (1964)157–208
I. Singer, Bases in Banach spaces. III, Studii si cercet. mat., 15 (1964)675–725 (Romanian).
I. Singer, On the basis problem in topological linear spaces. Rev. Roumaine Math. Pures Appl. 10 (1965)453–457.
I. Singer, On a theorem of N. K. Bari and I. M. Gelfand. Arch. Math. (To appear.)
I. Singer, Bases in Banach spaces (monograph). To appear.
M. Zippin, On a certain basis in c o . Israel J. Math. 4 (1966)199–204.
M. Zippin, On perfectly homogeneous bases in Banach spaces. Israel J. Math. 4 (1966)265–272.
M. Zippin, A remark on bases and reflexivity in Banach spaces. Israel J. Math. 6 (1968), 74–79.
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Singer, I. (1969). Some Remarks and Problems on Bases in Banach Spaces. In: Butzer, P.L., Szőkefalvi-Nagy, B. (eds) Abstract Spaces and Approximation / Abstrakte Räume und Approximation. ISNM International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 10. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5869-4_13
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DOI: https://doi.org/10.1007/978-3-0348-5869-4_13
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