THE SECOND DUAL SPACES OF THE SETS OF Λ-STRONGLY CONVERGENT AND BOUNDED SEQUENCES

We give the second β-, γ-, and f -duals of the sets w 0 (Λ), w p ∞(Λ) (0<p <∞), c 0 (Λ), c p(Λ), and c ∞(Λ) (0 < p ≤ 1) and the second continuous dual spaces of w 0 (Λ), c 0 (Λ), and c p(Λ) for 0< p ≤ 1. Furthermore, we determine the α-duals of c 0 (Λ), cp(Λ), and c ∞(Λ) for 1<p <∞.


Introduction and well-known results.
We write ω for the set of all complex sequences x = (x k ) ∞ k=0 , φ, l ∞ , c and c 0 for the sets of all finite, bounded, convergent sequences, and sequences convergent to naught, respectively, further cs, bs, and l 1 for the sets of all convergent, bounded, and absolutely convergent series.
By e and e (n) (n ∈ N 0 ), we denote the sequences such that e k = 1 for k = 0, 1,..., and e (n) n = 1 and e (n) k = 0 for k = n.For any sequence x = (x k ) ∞ k=0 , let x [n] = n k=0 x k e (k) be its n-section.
Let X, Y ⊂ ω and z ∈ ω.Then we write for the multiplier space of X and Y .The sets M(X, l 1 ), M(X, cs), and M(X, bs) are called the α-, β-, and γ-duals of X.
A Fréchet subspace X of ω is called an FK space if it has continuous coordinates, that is, if convergence in X implies coordinatewise convergence.An FK space X ⊃ φ is said to have AK if, for every sequence x = (x k ) ∞ k=0 ∈ X, x [n] → x (n → ∞); and it is said to have AD if φ is dense in X.A BK space is an FK space which is a Banach space.
If X is a p-normed space, then we write X * for the set of all continuous linear functionals on X, the so-called continuous dual of X, with its norm • is given by Let X ⊃ φ be an FK space.Then the set Given any infinite matrix A = (a nk ) ∞ n,k=0 of complex numbers and any sequence x ∈ ω, let A n (x) = ∞ k=0 a nk x k (n = 0, 1,...), and let A(x) = (A n (x)) ∞ n=0 provided the series converge, and n=0 be a nondecreasing sequence of positive integers tending to infinity, throughout.We define the matrices ∆ and M by , and use the convention that any symbol with a negative subscript has the value zero.The sets were studied in [1], and their first duals were given there.If p = 1, then we omit the index p, i.e., we write w 0 (µ) = w 1 0 (µ), etc.Following the notation introduced in [3], we say that a nondecreasing sequence Λ = (λ n ) ∞ n=0 of positive reals tending to infinity is exponentially bounded if there are reals s and t with 0 < s ≤ t < 1 such that for some subsequence (λ n(ν) ) ∞ ν=0 of Λ, we have ν=0 is a strictly increasing sequence of nonnegative integers, then we write K ν for the set of all integers k with n(ν) ≤ k ≤ n(ν + 1) − 1, and ν and max ν for the sum and maximum taken over all k in K ν .
If X p (Λ) denotes any of the sets w (1.6)
Therefore the following result gives the second duals.