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

Studia Mathematica (1994)

• Volume: 111, Issue: 3, page 207-222
• ISSN: 0039-3223

top

## Abstract

top
Every separable, infinite-dimensional Banach space X has a biorthogonal sequence ${z}_{n},z{*}_{n}$, with $spanz{*}_{n}$ norming on X and $\parallel {z}_{n}\parallel +\parallel z{*}_{n}\parallel$ bounded, so that, for every x in X and x* in X*, there exists a permutation π(n) of n so that $x\in \overline{conv}finitesubseriesof{\sum }_{n=1}^{\infty }z{*}_{n}\left(x\right){z}_{n}andx{*}_{n}\left(x\right)={\sum }_{n=1}^{\infty }z{*}_{\pi \left(n\right)}\left(x\right)x*\left({z}_{\pi \left(n\right)}\right)$.

## How to cite

top

Terenzi, Paolo. "Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis." Studia Mathematica 111.3 (1994): 207-222. <http://eudml.org/doc/216129>.

@article{Terenzi1994,
abstract = {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 $x ∈ \overline\{conv\} \{finite subseries of ∑_\{n=1\}^\{∞\} z*_n(x)z_n\} and x*_n(x) = ∑_\{n=1\}^∞ z*_\{π(n)\}(x)x*(z_\{π(n)\})$.},
author = {Terenzi, Paolo},
journal = {Studia Mathematica},
keywords = {biorthogonal sequence},
language = {eng},
number = {3},
pages = {207-222},
title = {Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis},
url = {http://eudml.org/doc/216129},
volume = {111},
year = {1994},
}

TY - JOUR
AU - Terenzi, Paolo
TI - Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis
JO - Studia Mathematica
PY - 1994
VL - 111
IS - 3
SP - 207
EP - 222
AB - 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 $x ∈ \overline{conv} {finite subseries of ∑_{n=1}^{∞} z*_n(x)z_n} and x*_n(x) = ∑_{n=1}^∞ z*_{π(n)}(x)x*(z_{π(n)})$.
LA - eng
KW - biorthogonal sequence
UR - http://eudml.org/doc/216129
ER -

## References

top
1. [1] S. Banach, Théorie des opérations linéaires, Chelsea, New York, 1932. Zbl0005.20901
2. [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. Zbl0256.46026
3. [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. Zbl0256.46027
4. [4] A. Dvoretzky, Some results on convex bodies and Banach spaces, in: Proc. Internat. Sympos. on Linear Spaces, Jerusalem Academic Press, 1961, 123-160. Zbl0119.31803
5. [5] P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309-317. Zbl0267.46012
6. [6] V. P. Fonf, Operator bases and generalized summation bases, Dokl. Akad. Nauk Ukrain. SSR Ser. A 1986 (11), 16-18 (in Russian). Zbl0629.46012
7. [7] V. I. Gurarii and M. I. Kadec, On permutations of biorthogonal decompositions, Istituto Lombardo, 1991.
8. [8] E. Indurain and P. Terenzi, A characterization of basis sequences in Banach spaces, Rend. Accad. dei XL 18 (1986), 207-212. Zbl0618.46014
9. [9] M. I. Kadec, Nonlinear operator bases in a Banach space, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 2 (1966), 128-130 (in Russian).
10. [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). Zbl0135.34504
11. [11] G. W. Mackey, Note on a theorem of Murray, Bull. Amer. Math. Soc. 52 (1946), 322-325. Zbl0063.03692
12. [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. Zbl0721.46007
13. [13] A. Markushevich, Sur les bases (au sens large) dans les espaces linéaires, Dokl. Akad. Nauk SSSR 41 (1943), 227-229. Zbl0061.24701
14. [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. [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. Zbl0317.46019
16. [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. Zbl0336.46023
17. [17] A. Plans and A. Reyes, On the geometry of sequences in Banach spaces, Arch. Math. (Basel) 40 (1983), 452-458. Zbl0517.46003
18. [18] W. H. Ruckle, Representation and series summability of complete biorthogonal sequences, Pacific J. Math. 34 (1970), 511-528. Zbl0202.39404
19. [19] W. H. Ruckle, On the classification of biorthogonal sequences, Canad. J. Math. 26 (1974), 721-733. Zbl0282.46013
20. [20] I. Singer, Bases in Banach Spaces II, Springer, 1981. Zbl0467.46020
21. [21] S. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math. 159 (1987), 81-98. Zbl0637.46013
22. [22] P. Terenzi, Representation of the space spanned by a sequence in a Banach space, Arch. Math. (Basel) 43 (1984), 448-459. Zbl0581.46010
23. [23] P. Terenzi, On the theory of fundamental norming bounded biorthogonal systems in Banach spaces, Trans. Amer. Math. Soc. 299 (1987), 497-511. Zbl0621.46013
24. [24] P. Terenzi, On the properties of the strong M-bases in Banach spaces, Sem. Mat. Garcia de Galdeano, Zaragoza, 1987.

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.