Basic sequences and subspaces in Lorentz sequence spaces without local convexity

Nicolae Popa
1981 Transactions of the American Mathematical Society  
After some preliminary results ( § 1), we give in §2 another proof of the result of N. J. Kalton [5] concerning the unicity of the unconditional bases of lp, 0<p< 1. 1. ||x|| = 0 if and only if x = 0. 2. ||ax|| = \a\p||;c|| for x G X and a G R. 3. H* + y\\ < \\x\\ + \\y\\ for x,y G X. Then the subsets U" = {x G X: \\x\\ < n~x), for n G N, constitute a fundamental system of neighbourhoods of zero for a metric linear topology of X. If X is complete with respect to this topology we say that X is ap-Banach space.
doi:10.1090/s0002-9947-1981-0594418-7 fatcat:s66ojsngkvfdbp3wufov22f7mq