We consider the relation
between the following two statements for E and F a pair of normed spaces.
(SI) For each absolutely divergent series Σnχn in E there is a continuous linear
mapping T from E into F such that ΣnTxn diverges absolutely.
(LI) The finite dimensional subspaces of E are uniformly isomorphic to
subspaces of F under isomorphisms which extend to all of E without increase of
norm.
Our main result is that (SI) implies (LI) when F is isometric to F × F with a
certain type of norm. We also observe that if a normed space E is not isomorphic to
a subspace of an Lρ(μ) space, then for each r with 1 ≦ r < ∞ there is a series Σnxn
in E such that Σn∥Txn∥r < ∞ for each continuous linear mapping T from E into lp
but Σn∥xn∥r = ∞.
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