Absolutely divergent series and isomorphism of subspaces

William Henry Ruckle

Vol. 58 (1975), No. 2, 605–615

DOI: 10.2140/pjm.1975.58.605

Abstract

We consider the relation between the following two statements for E and F a pair of normed spaces.

(SI) For each absolutely divergent series Σ_nχ_n in E there is a continuous linear mapping T from E into F such that Σ_nTx_n 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 L_ρ(μ) space, then for each r with 1 ≦ r < ∞ there is a series Σ_nx_n in E such that Σ_n∥Tx_n∥^r < ∞ for each continuous linear mapping T from E into l_p but Σ_n∥x_n∥^r = ∞.

Mathematical Subject Classification 2000

Primary: 46B15

Milestones

Received: 22 February 1974

Published: 1 June 1975