Non-Barreled Dense β-φ Subspaces

A sequence space S with its strong β(S, φ) topology is called a βφ space [W. H. Ruckle, Pacific J. Math.42 (1972), 235-249], where φ is the space of eventually zero sequences. The familiar Banach sequence spaces are βφ spaces, and all barrelled subspaces thereof are necessarily βφ 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 βφ subspaces of l1. Their images under the canonical isometry from l1 onto bv0 prove to be (non-barrelled dense) βφ subspaces of bv0. In contrast, every βφ subspace of c0 or lp (1 < p < ∞) is barrelled. Surprisingly, all dense subspaces of l∞ are βφ 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 βφ subspace 〈of eitherl∞or c0〉 (ofbv0). An AD space S that is FK or βφ has 〈unconditional AK〉 (AK) if and only if RS ⊂ S or RS ⊂ S̃, respectively.