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

FRAMES AND RIESZ BASES FOR BANACH SPACES 181(c) F (xn)(X ∗ ) = B s .(d) (x n ) has a biorthogonal sequence and F (xn)(X ∗ ) is closed in B s .(e) For every n x n ∉ span{x i } i≠n and R (xn)(Y s ) is closed in X.Use the operators in Corollary 2.6 to show Corollary 3.9.Corollary 3.9. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s and let (x ∗ n) ∈ Bs w∗ (X ∗ ). Then the following are equivalent.(a) (x ∗ n) is a Y s -Riesz basic sequence for X ∗ .(b) F (x ∗ n )(X) = B s .(c) (x ∗ n) has a biorthogonal sequence in X and F (x ∗ n )(X) is closed in B s .Corollaries 3.10 and 3.11 extend [11, Proposition 2.7].Corollary 3.10. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s and let (x n ) be a B s -frame for X ∗ . Then the following areequivalent.(a) (x n ) is a Y s -Riesz basis for X.(b) For (β n ) ∈ Y s , if ∑ n β nx n = 0, then β n = 0 for all n.(c) F (xn)(X ∗ ) = B s .(d) (x n ) has a biorthogonal sequence.(e) For every n x n ∉ span{x i } i≠n .Corollary 3.11. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s and let (x ∗ n) be a B s -frame for X. Then the following areequivalent.(a) (x ∗ n) is a Y s -Riesz basis for X ∗ .(b) F (x ∗ n )(X) = B s .(c) (x ∗ n) has a biorthogonal sequence in X.Now we obtain some applications for Riesz bases.Theorem 3.12. Suppose that (e n ) is a Schauder basis for B s and (f n ) is aSchauder basis for Y s . If (x n ) is a B s -Riesz basis for X, then there exists aY s -Riesz basis (x ∗ n) for X ∗ , which is a biorthogonal sequence for (x n ), so thatx = ∑ nfor every x ∈ X and x ∗ ∈ X ∗ .x ∗ n(x)x n ,x ∗ = ∑ nx ∗ (x n )x ∗ nProof. We have shown that if (x n ) is a B s -Riesz basis for X, then (x n ) is aY s -frame for X ∗ and F (xn)(X ∗ ) = Y s . Consider the sequence (F −1(x f n) n) in X ∗ .Then F −1(x f n) n(x k ) = ϕ k (F (xn)F −1(x f n) n) = ϕ k (f n ), where ϕ k is the k-th coordinatefunctional for Y s . Therefore (F −1(x f n) n) is a biorthogonal sequence for (x n ) and forevery x ∗ ∈ X ∗x ∗ = F −1(x F n) (x n)x ∗ = F −1(x n) [(x∗ (x n ))]= F −1(x n)( ∑ )x ∗ (x n )f n = ∑nnx ∗ (x n )F −1(x n) f n.