Non-barrelled dense beta phi subspaces

[摘要] A sequence space S with its strong beta(S, phi) topology is called a beta phi space [W. H. Ruckle, Pacific J. Math. 42 (1972), 235-249], where phi is the space of eventually zero sequences. The familiar Banach sequence spaces are beta phi spaces, and all barrelled subspaces thereof are necessarily beta phi subspaces; when does the converse hold? Our answer differs from Ruckle's symmetric case [Canad. J, Math. 37 (1985), 235-249] and settles his recent questions with discovery of non-barrelled dense beta phi subspaces of l(1). Their images under the canonical isometry from l(1) onto b upsilon(o) prove to be (non-barrelled dense) beta phi subspaces of b upsilon(o). In contrast, every beta phi subspace of c(o) or l(p) (1 < p < infinity) is barrelled. Surprisingly, all dense subspaces of l* are beta phi subspaces, some non-barrelled. These and other of our results are vital in joint papers with Ruckle that extend and unify classical sectional convergence/multiplier theorems, where we prove: Let R be a dense beta phi subspace [of either l* or c(o)] (of b upsilon(o)). An AD space S that is FK or beta phi has [unconditional AK] (AK) if and only if RS subset of S or RS subset of (S) over bar, respectively. (C) 1995 Academic Press, Inc.

[发布日期] 1995-12-01