Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  053.08002
Autor:  Erdös, Pál; Straus, E.G.
Title:  On linear independence of sequences in a Banach space. (In English)
Source:  Pac. J. Math. 3, 689-694 (1953).
Review:  This paper gives an answer to a problem raised by A.Dvoretzky: given a sequence of (algebraically) linearly independent unit vectors of a Banach space, does there exists a subsequence linearly independent in some stronger sense? The authors prove that, given any positively valued function \phi(n), every sequence of linearly independent unit vectors of a Banach space contains a subsequence xn independent in the following sense: if Ckn are scalars such that

(1) \supk |Ckn| < \phi(n),    (2) limk ––> oo sumn = 1oo Cknxn = 0,

then limk ––> oo Ckn = 0 for n = 1,2,.... This implies a fortiori that these vectors are linearly independent in the following sense: if sumn = 1oo cn xn = 0, then cn = 0 for n = 1,2,.... The authors prove also that if the condition (1) is dropped in the definition of linear independence, their theorem is no longer true.
[The following misprints are to be noted: p. 689, line 13 the inequality is to be read |C(k)n| < \phi(n); p. 690, line 16 replace xni by xn; p. 691, line 20 replace Q by O.]
Reviewer:  A.Alexiewicz
Classif.:  * 46B99 Normed linear spaces and Banach spaces
Index Words:  functional analysis

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag