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Zbl 0534.46015
Landes, Thomas
Permanence properties of normal structure. (English)
[J] Pac. J. Math. 110, 125-143 (1984). ISSN 0030-8730

A list of equivalent conditions for a subset C of a normed space to have NS (normal structure) is given, among them, the following new characterization: \par C has NS iff each limit-constant sequence with convex hull in C is constant. \par Here, $(x\sb n)$ is limit-constant (limit-affine) if the functional $\Lambda x=\lim\sb{n\to \infty}\Vert x\sb n-x\Vert$ is defined and constant (affine) on the convex hull of $(x\sb n)$. It is shown that NS is preserved under the Z-DSO (direct-sum-operation) - i.e., $\prod\sp{N}\sb{i=1}X\sb i$ with norm $\Vert x\Vert =\vert(\Vert x\sb 1\Vert,...,\Vert x\sb N\Vert)\vert$, the Z-DS (direct sum) of the $X\sb i$, has NS if all $X\sb i$ have NS - provided $Z=({\bbfR}\sp N,\vert \vert)$ is symmetric and: \par (SC) If $z\sb i,\hat z\sb i\ge 0$ for all i and $\vert z\vert =\vert \hat z\vert =frac{1}{2}\vert z+\hat z\vert =1$ then $z\sb i=\hat z\sb i>0$ for some i. \par All strictly convex Z and $Z=\ell\sp N\sb p$, $1<p\le \infty$, satisfy (SC), but not $Z=\ell\sp N\sb 1$. NS is also preserved under any Z-DSO with unifomly convex symmetric Z with basis $(e\sb i)\sb{i\in I}$, I an arbitrary index set. \par It is shown that a normed space has INS isonormal structure - i.e., it is isomorphic to a normed space with NS - iff it can be mapped by a bounded 1-1 linear operator into some normed space with NS. Every separable space has INS, in particular $c\sb o({\bbfN})$. The problem is raised for which I $c\sb 0(I)$ has INS. The normed space X is said to have the SP (sum- property) if every non-decreasingly $((\Lambda x\sb n)$ is non- decreasing) limit-affine sequence $(x\sb n)$ in X is constant. It is shown that the SP is preserved under any Z-DSO with finite dimensional symmetric Z. The SP clearly implies NS. There is no known condition sufficient (but not necessary) for NS which does not imply the SP (shown in the appendix). So, it is conjectured that NS implies the SP.

MSC 2000:: *46B20 Geometry and structure of normed spaces

Keywords: normal structure; limit-constant; limit-affine; direct-sum-operation; isonormal structure; sum-property

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