Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis

• Autorzy: Paolo Terenzi
• Języki publikacji: EN

Abstrakty

EN

Every separable, infinite-dimensional Banach space X has a biorthogonal sequence ${z_n, z*_n}$, with $span{z*_n}$ norming on X and ${∥z_n∥ + ∥z*_n∥}$ bounded, so that, for every x in X and x* in X*, there exists a permutation {π(n)} of {n} so that
$x ∈ \overline{conv} {finite subseries of ∑_{n=1}^{∞} z*_n(x)z_n} and x*_n(x) = ∑_{n=1}^∞ z*_{π(n)}(x)x*(z_{π(n)})$.

Kategorie tematyczne

• 46B15: Summability and bases

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994
• Tom: 111
• Numer: 3
• Strony: 207-222

Daty

wydano

1994

otrzymano

1990-12-01

poprawiono

1991-12-06

poprawiono

1992-12-23

poprawiono

1993-06-30

Twórcy

autor

Paolo Terenzi

• Dipartimento di Matematica del Politecnico, Piazza Leonardo da Vinci, 32 20133 Milano, Italy

Bibliografia

• [1] S. Banach, Théorie des opérations linéaires, Chelsea, New York, 1932.
• [2] 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.
• [3] W. J. Davis and I. Singer, Boundedly complete M-bases and complemented subspaces in Banach spaces, Trans. Amer. Math. Soc. 175 (1973), 187-194.
• [4] A. Dvoretzky, Some results on convex bodies and Banach spaces, in: Proc. Internat. Sympos. on Linear Spaces, Jerusalem Academic Press, 1961, 123-160.
• [5] P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309-317.
• [6] V. P. Fonf, Operator bases and generalized summation bases, Dokl. Akad. Nauk Ukrain. SSR Ser. A 1986 (11), 16-18 (in Russian).
• [7] V. I. Gurarii and M. I. Kadec, On permutations of biorthogonal decompositions, Istituto Lombardo, 1991.
• [8] E. Indurain and P. Terenzi, A characterization of basis sequences in Banach spaces, Rend. Accad. dei XL 18 (1986), 207-212.
• [9] M. I. Kadec, Nonlinear operator bases in a Banach space, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 2 (1966), 128-130 (in Russian).
• [10] M. I. Kadec and A. Pełczyński, Basic sequences, biorthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297-323 (in Russian).
• [11] G. W. Mackey, Note on a theorem of Murray, Bull. Amer. Math. Soc. 52 (1946), 322-325.
• [12] P. Mankiewicz and N. J. Nielsen, A superreflexive Banach space with a finite dimensional decomposition so that no large subspace has a basis, Odense University Preprints, 1989.
• [13] A. Markushevich, Sur les bases (au sens large) dans les espaces linéaires, Dokl. Akad. Nauk SSSR 41 (1943), 227-229.
• [14] A. M. Olevskiĭ, Fourier series of continuous functions with respect to bounded orthonormal systems, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 387-432.
• [15] 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), 149-159.
• [16] A. Pełczyński, All separable Banach space admit for every ε > 0 fundamental total and bounded by 1 + ε biorthogonal sequences, ibid. 55 (1976), 295-304.
• [17] A. Plans and A. Reyes, On the geometry of sequences in Banach spaces, Arch. Math. (Basel) 40 (1983), 452-458.
• [18] W. H. Ruckle, Representation and series summability of complete biorthogonal sequences, Pacific J. Math. 34 (1970), 511-528.
• [19] W. H. Ruckle, On the classification of biorthogonal sequences, Canad. J. Math. 26 (1974), 721-733.
• [20] I. Singer, Bases in Banach Spaces II, Springer, 1981.
• [21] S. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math. 159 (1987), 81-98.
• [22] P. Terenzi, Representation of the space spanned by a sequence in a Banach space, Arch. Math. (Basel) 43 (1984), 448-459.
• [23] P. Terenzi, On the theory of fundamental norming bounded biorthogonal systems in Banach spaces, Trans. Amer. Math. Soc. 299 (1987), 497-511.
• [24] P. Terenzi, On the properties of the strong M-bases in Banach spaces, Sem. Mat. Garcia de Galdeano, Zaragoza, 1987.