Frames and Riesz bases for Banach spaces, and Banach spaces of ...

186 K. CHO, J.M. KIM, H.J. LEE(a) There exists an AD for X with respect to a Banach sequence space in whichthe sequence of the canonical unit vectors is a shrinking Schauder basis for it.(b) There exists a strongly shrinking AD for X.(c) There exists a shrinking AD for X.(d) X ∗ is separable and has the BAP.(e) X ∗ is separable and has the AP.Proof. (b)=⇒(c) is clear and (a)=⇒(b) follows from [3, Proposition 1.9]. (c)=⇒(d)is [3, Corollary 1.5] and (d)⇐⇒(e) is well known; cf. [5, Theorem 3.6].(d)=⇒(a) From [5, Theorem 4.9] there exists a Banach space Z with a shrinkingbasis (z n ) such that X embeds complementably into Z. Put{B s = (α n )| ∑ }α n z n converges in Znwith ‖(α n )‖ Bs = ‖ ∑ n α nz n ‖ Z . Then we see that B s is a Banach sequence spacein which the sequence of the canonical unit vectors is a shrinking Schauder basisfor it. Since X embeds complementably into Z, we can find an AD for X withrespect to B s .□5. Banach spaces consisting of Bessel or Riesz sequencesRecall the vector spacesB w s (X) = {(x n ) in X : (x ∗ (x n )) ∈ B s for every x ∗ ∈ X ∗ },Bsw∗(X ∗ ) = {(x ∗ n) in X ∗ : (x ∗ n(x)) ∈ B s for every x ∈ X},B s R(X) = {(x n ) in X : ∑ α n x n converges for every (α n ) ∈ B s }.nThen by boundedness of the analysis and synthesis operators, for every (x n ) ∈Bs w (X), (x ∗ n) ∈ Bsw∗ (X ∗ ), (x n ) ∈ B s R(X), respectively,and‖(x n )‖ B w s (X) =are all finite.We now havesupx ∗ ∈B X ∗‖(x ∗ (x n ))‖ Bs , ‖(x ∗ n)‖ B ws∗ (X ∗ ) = sup ‖(x ∗ n(x))‖ Bs ,x∈B X∥ ∥∥ ∑ ∥ ∥∥‖(x n )‖ BsR(X) = sup α n x n(α n)∈B BsProposition 5.1. (Bs w (X), ‖ · ‖ B w s (X)) and (Bsw∗ (X ∗ ), ‖ · ‖ B w ∗s (X )) are Banach∗spaces.Proof. The proofs of the two cases are the same and so we only prove the firstcase. It is easy to check that ‖ · ‖ B w s (X) is a norm on Bs w (X). Let ((x (k)n )) k be aCauchy sequence in Bs w (X) and let m ∈ N be fixed. Let ε > 0 be given. Thenthere exists an N ∈ N so that k, l ≥ N implies‖(x (k)n ) − (x (l)n )‖ B w s (X) ≤ε‖e ∗ m‖ ,n