AMUCActa Mathematica Universitatis Comenianae
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([??]) Assume that X is AMUC, and let [bar.[delta]]([??]) be its AMUC modulus.
By the AMUC property, there exists a subspace U [subset] X of finite codimension corresponding to the element in X of finite support x = [([x.sub.n]).sup.N.sub.n=1] and the given value of [epsilon] > 0.
Then by AMUC, for either [theta] = 1 or [theta] = -1 we have [mathematical expression not reproducible].
([??]) Assume that [([X.sub.n], [[parallel]*[parallel].sub.n]).sup.[infinity].sub.n=1] are uniformly AMUC and E is uniformly monotone.
For n [member of] B, [[parallel][y.sub.n][parallel].sub.n] [greater than or equal to] [[epsilon]/3][[parallel][x.sub.n][parallel].sub.n], so by uniform AMUC, [[parallel][x.sub.n] + [[theta].sub.n][y.sub.n][parallel].sub.n]
First suppose that [L.sub.p] (X) is AMUC. To show X is UC, fix [??] [member of] (0,1) and let x,y [member of] [S.sub.X] such that [parallel]x - y[parallel] = [??].
Thus [mathematical expression not reproducible] by the AMUC property for [L.sub.p] (X).