Vol. 58, No. 2, 1975

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Absolutely divergent series and isomorphism of subspaces

William Henry Ruckle

Vol. 58 (1975), No. 2, 605–615

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 ΣnTxnr < for each continuous linear mapping T from E into lp but Σnxnr = .

Mathematical Subject Classification 2000
Primary: 46B15
Received: 22 February 1974
Published: 1 June 1975
William Henry Ruckle