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**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 x_{n} independent in the following sense: if C^{k}_{n} are scalars such that (1) \sup_{k} |C^{k}_{n}| < \phi(n), (2) **lim**_{k ––> oo} **sum**_{n = 1}^{oo} C^{k}_{n}x_{n} = 0, then **lim**_{k ––> oo} C^{k}_{n} = 0 for n = 1,2,.... This implies a fortiori that these vectors are linearly independent in the following sense: if **sum**_{n = 1}^{oo} c_{n} x_{n} = 0, then c_{n} = 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 x_{ni} by x_{n}; p. 691, line 20 replace Q by O.]

**Reviewer: ** A.Alexiewicz

**Classif.: ** * 46B99 Normed linear spaces and Banach spaces

**Index Words: ** functional analysis

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