Normal structure and the sum-property
Thomas Landes · Pacific Journal of Mathematics · 1986
For Z = (R^, I I) with symmetric norm | | the Z-direct sum of the normed spaces X l9 ...,X N is its product space with norm \\(x ι ,..., x N )\\ = 1(11*21|,--,l)*/v!l)lA 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).It turns out that the class of spaces having the SP is the largest subclass of the class of spaces having NS which is closed under each finite Z-direct sum operation.The SP is characterized by the property that limit-affine (i.e., the functional Λ(x) = lim||x n -x\\ is defined and affine on conv({x n })) sequences {x n } with non-decreasing {Λ (x n )} are constant.In contrast to a previous conjecture it is shown that every infinite dimensional separable normed space can be renormed to have NS and not the SP.Moreover, in order that NS is inherited from X λ ,...,X N to its Z-direct sum, it is not only sufficient (as previously shown) but also necessary that each line segment (if there is any) in the unit sphere of Z lies in a hyperpiane {z\z t = a] for some i < N, a Φ 0. In fact, if Z does not satisfy this condition, and if infinite dimensional separable normed spaces X λJ ..., X N are given, then there are normed spaces Y ι with NS isomorphic to X,-whose Z-direct sum does not have NS.Finally, it is shown that a normed space with a symmetric (not necessarily countable) basis can be renormed to have NS if and only if it can be renormeά to be uniformly convex in eyery direction.In particular, c o (I) can be renormed to have NS if and only if / is countable.As a counter-example, a reflexive normed space with an unconditional basis is given which has the SP but cannot be renormed to be uniformly convex in every direction.All results hold also for weakly NS and the weak SP.