A set-theoretic approach to complete minimal 
systems in Banach spaces of bounded functions

L. Halbeisen

Using independent families from combinatorial set theory, it is shown that for every infinite cardinal $\kappa$, $\ell_\infty(\kappa)^*$ contains a subspace which is isomorphic to a Hilbert space of dimension $\mathfrak{2}^\kappa$. This provides a new proof for the first step in the construction of complete minimal systems in Banach spaces of bounded functions.

1. Plichko A. A Banach space without a fundamental biorthogonal system, Doklady Akademii Nauk SSSR, 254 (1980), № 4 798–801 (in Russian); translation in Soviet Mathematical Doklady, 22 (1980), № 2, 450–453.

2. Godun B. V., Kadets M. I. Banach spaces without complete minimal systems, Funktsional'nyi Analiz i Ego Prilozheniya 14 (1980), № 4, 67–68 (in Russian); translation in Functional Analysis and its Applications, 14 (1980), № 4, 301–302.

3. Davis W. J., Johnson W. B. On the existence of fundamental and total bounded biorthogonal systems in Banach spaces Studia Mathematica, 45 (1973), 173–179.

4. Godun B. V. Complete biorthogonal systems in Banach spaces, Funktsional'nyi Analiz i Ego Prilozheniya, 17 (1983), № 1, 1–7 (in Russian); translation in Functional Analysis and its Applications 17 (1983), № 1, 1–5.

5. Rosenthal H.~P. On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from $L^p(\mu)$ to $L^r(\mu)$, Journal of Functional Analysis, 4 (1969), 176--214.

6. Kunen K. Set Theory, an Introduction to Independence Proofs, North Holland, Amsterdam (1983).

7. Hausdorff F. Über zwei Sätze von G. Fichtenholz und L. Kantorovitch, Studia Mathematica, 6 (1936), 18–19.

	A set-theoretic approach to complete minimal systems in Banach spaces of bounded functions
Author	L. Halbeisen halbeis@qub.ac.uk Department of Pure Mathematics, Queen's University Belfast, Belfast BT7 1NN, Northern Ireland
Abstract	Using independent families from combinatorial set theory, it is shown that for every infinite cardinal $\kappa$, $\ell_\infty(\kappa)^*$ contains a subspace which is isomorphic to a Hilbert space of dimension $\mathfrak{2}^\kappa$. This provides a new proof for the first step in the construction of complete minimal systems in Banach spaces of bounded functions.
Keywords	infinite cardinal, Hilbert space, bounded function
DOI	doi:10.30970/ms.20.2.162-166
Reference	1. Plichko A. A Banach space without a fundamental biorthogonal system, Doklady Akademii Nauk SSSR, 254 (1980), № 4 798–801 (in Russian); translation in Soviet Mathematical Doklady, 22 (1980), № 2, 450–453. 2. Godun B. V., Kadets M. I. Banach spaces without complete minimal systems, Funktsional'nyi Analiz i Ego Prilozheniya 14 (1980), № 4, 67–68 (in Russian); translation in Functional Analysis and its Applications, 14 (1980), № 4, 301–302. 3. Davis W. J., Johnson W. B. On the existence of fundamental and total bounded biorthogonal systems in Banach spaces Studia Mathematica, 45 (1973), 173–179. 4. Godun B. V. Complete biorthogonal systems in Banach spaces, Funktsional'nyi Analiz i Ego Prilozheniya, 17 (1983), № 1, 1–7 (in Russian); translation in Functional Analysis and its Applications 17 (1983), № 1, 1–5. 5. Rosenthal H.~P. On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from $L^p(\mu)$ to $L^r(\mu)$, Journal of Functional Analysis, 4 (1969), 176--214. 6. Kunen K. Set Theory, an Introduction to Independence Proofs, North Holland, Amsterdam (1983). 7. Hausdorff F. Über zwei Sätze von G. Fichtenholz und L. Kantorovitch, Studia Mathematica, 6 (1936), 18–19.
Pages	162-166
Volume	20
Issue	2
Year	2003
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

A set-theoretic approach to complete minimal systems in Banach spaces of bounded functions