Köthe spaces with preequivalent unconditional bases
Let \(X\) be a metrizable Köthe space and let \(X'\) be the conjugate space of \(X\). The author proves that (1) in \(X\), all of its unconditional bases are pre-equivalent if and only if \(X\) is isomorphic to one of the spaces \(\ell_1\), \(\ell_2\), \(c_0\) and \(\omega\) (Theorem 1), and (2) in \(X'= \{X',\sigma^*(X', X))\), all of its unconditional Schauder bases are pre-equivalent if and only if \(X'\) is isomorphic to one of the spaces \(\varphi\), \((\ell_1,\sigma^*(\ell_1,c_0))\), \((\ell_2,\sigma^*(\ell_2,\ell_2))\), \((\ell_\infty, \sigma^*(\ell_\infty, \ell_1))\), where \(\sigma^*\) stands for normal topology (Corollary 2). Note that two bases in a topological vector space are said to be pre-equivalent if they can be reduced to the coincidence by a linear automorphism of the space and the norming of elements.
scientific article; zbMATH DE number 1752712