Remarks on independent sequences and dimension in topological linear spaces
Most of the paper has its origin in the theorem that a linearly independent sequence in a Banach space has an \(\omega\)-independent subsequence, which is due essentially to P. Erdős and E. G. Straus (1953). This theorem, clearly, extends to topological linear spaces that admit a continuous norm. In the class of locally convex \(F\)-spaces a converse holds, as shown by V. M. Kadets (1993). We generalize this equivalence to the class of metrizable locally convex spaces. In this connection, we establish some properties of the space \(\varphi\) of scalar sequences with finite support equipped with the product topology. In particular, we prove some results on the containment of \(\varphi\) in topological linear spaces, generalizing the corresponding results on the containment of \(\omega\) \((=\mathbb{R}^{\mathbb{N}})\) in \(F\)-spaces due to C. Bessaga, A. Pełczyński and S. Rolewicz (1957, 1959). Moreover, we strengthen the Erdős-Straus theorem to the effect that a linearly independent sequence in a dual Banach space has a subsequence which is \(\omega\)-independent with respect to the weak* topology. (The author has recently noticed that if the predual space is separable, then this last result reduces to the Erdős-Straus theorem via a standard argument; see, e.g. \textit{M. A. Rieffel} [Trans. Am. Math. Soc. 131, 466-487 (1968; Zbl 0169.46803), Corollary 5.4]). We also present two different proofs of the theorem that the algebraic dimension of a closed convex subset of an \(F\)-space is either finite or at least \(2^{\aleph_0}\).
scientific article; zbMATH DE number 1186311