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For Z = (RN,|⋅|) with
symmetric norm |⋅| the Z-direct sum of the normed spaces X1,…,XN is its product
space with norm ∥(x1,…,xN)∥ = |(∥x1∥,…,∥xN∥)|. 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). 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∥xn − x∥ is defined
and affine on conv({xn})) sequences {xn} with non-decreasing {Λ(xn)} 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 X1,…,XN 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|zi = α} for
some i ≤ N, α≠0. In fact, if Z does not satisfy this condition, and if infinite
dimensional separable normed spaces X1,…,XN are given, then there are normed
spaces Y i with NS isomorphic to Xi, 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 renormed to be
uniformly convex in eyery direction. In particular, c0(I) can be renormed to have NS
if and only if I 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.
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