On fundamental biorthogonal systems in some conjugate Banach spaces. (Russian) Zbl 0546.46007

Godun, B. V.

Commentat. Math. Univ. Carol. 24, 431-436 (1983).

Let X be a Banach space. Given a fundamental biorthogonal system \(\{x_ i,X^*_ i\}_{i\in I}\subset X\times X^*\) define \(p(\{x_ i\})=\sup_{i\in I}\| x^*_ i\|\| x_ i\|,\quad q(\{x_ i\})=\sup_{i\in I}\sup_{\| x\|\leq 1}\| x-x^*_ i(x)x_ i\|.\) Furthermore, let \(p(X)=\inf p(\{x_ i\})\) and \(q(X)=\inf q(\{x_ i\}),\) where the infimum is taken over all fundamental biorthogonal systems of X. Earlier, A. Pełczyński [Stud. Math. 55, 295-304 (1976; Zbl 0336.46023)] has proved that \(p(X)=1\) and q(X)\(\leq 2\) for every separable Banach space X. Also, from the papers of V. F.Babenko, S. A. Pichugov [Ukr. Mat. Zh. 33, 491-492 (1981; Zbl 0492.47018)] and I. K. Daugavet [Usp. Mat. Nauk No.5(113) 157-158 (1963; Zbl 0138.386)] it is known that q(X)\(\geq 2\) if \(X=C[0,1]\) or \(X=L_ 1[0,1]\). The main result of this paper is as follows:
If X is a Banach space with an unconditional orthogonal basis and its dual \(X^*\) is nonseparable, then \(q(X^*)\geq 2\). Particularly, many properties of the space \(\ell_{\infty}\) are given.

Reviewer: T.Precupanu

MSC:

46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B20	Geometry and structure of normed linear spaces
46B10	Duality and reflexivity in normed linear and Banach spaces

Keywords:

fundamental biorthogonal system; unconditional orthogonal basis

Citations:

Zbl 0336.46023; Zbl 0492.47018; Zbl 0138.386