Normal structure and the sum-property
The first permanence result for normal structure is due to Belluce, Kirk and Steine: The \(\ell^ N_{\infty}\)-direct sum of N normally structured normed spaces again has normal structure. The author of this paper continues for discussion on it. For \(Z=(R^ N,| \cdot |)\) with symmetric norm \(| \cdot |\) the Z-direct sum of the normed spaces \(X_ 1,...,X_ N\) is their product space with norm \(\| (x_ i)^ N_{i=1}\| =| (\| x_ i\|)^ N_{i=1}|\). A normed space X is said to have the sum-property (SP) if each Z-direct sum of finitely many copies of X has normal structure (NS). The following results are mainly proved in this paper: (1) NS is preserved under the Z-direct-sum-operation iff each line segment (if there is any) in the unit sphere of Z lies in a hyperplane \(\{z|\) \(z_ i=\alpha\}\) for some \(i\leq N\), \(\alpha\neq 0.\) (2) Every infinite dimensional separable normed space can be renormed to have NS and not the SP. All results hold also for weakly NS and weak SP.
scientific article; zbMATH DE number 4024009