Uniform Boundedness in Vector - Valued Sequence Spaces

Let µ be a normal scalar sequence space which is a K-space under the family of semi-norms M and let X be a locally convex space whose topology is generated by the family of semi-norms X. The space µ{X} is the space of all X valued sequences x = {x k } such that {q(x k)} ∈ µ{X} for all q ∈ X. The space µ{X} is given the locally convex topol-ogy generated by the semi-norms π pq (x) = p({q(x k)}), p ∈ X, q ∈ M. We show that if µ satisfies a certain multiplier type of gliding hump property, then pointwise bounded subsets of the β-dual of µ{X} with respect to a locally convex space are uniformly bounded on bounded subsets of µ{X}.


Introduction
In [Sw3], in order to establish uniform boundedness results for sequence spaces, we introduced a gliding hump property in which the "humps" in the sequence space are multiplied by elements of another sequence space in order to facilitate the convergence of a series whose elements are those composed of the "humps".In this note we show that this gliding hump property can be employed to establish uniform boundedness results for the operator valued β-dual of certain vector-valued sequence spaces.
We describe the vector-valued sequence spaces which will be considered.Let µ be a normal scalar-valued sequence space containing c 00 , the space of all sequences which are eventually 0, and which is a Hausdorff locally convex K-space whose topology is generated by the family of semi-norms M. Let X (Y ) be a Hausdorff locally convex space whose topology is generated by the family of semi-norms X (Y) and let L(X, Y ) be the space of all continuous linear operators from X into Y .Let µ{X} be the space of all X-valued sequences such that {q(x k )} ∈ µ for every q ∈ X .Since µ is normal, µ{X} is a vector space and we supply µ{X} with the locally convex topology generated by the family of semi-norms π pq (x) = p({q(x k )}) for x = {x k } ∈ µ{X}, p ∈ M, q ∈ X .For perfect sequence spaces these spaces were introduced by Pietsch ([P]) and considered in [Ro],[F], [FP] and [Sw3]; they include such spaces as the space of absolutely pth power summable series l p {X}.One of the basic problems in this area is to determine which properties such as barrelledness of µ{X} are inherited from µ and X.For example, sufficient conditions for quasi-barrelledness and barrelledness are given in [FP], [F].The β-dual of µ{X} with respect to Y , µ{X} βY , is defined to be all sequences T = {T k } ⊂ L(X, Y ) such that We consider sufficient conditions for a family B ⊂ µ{X} βY which is pointwise bounded on µ{X} to be uniformly bounded on bounded subsets of µ{X},i.e., we seek a uniform boundedness or Banach-Mackey principle for the pair µ{X}, µ{X} βY .

Main Result
We first establish a lemma which will be used in the proof of our main result.The pair (X, Y ) is said to satisfy the property UB (Uniform Boundedness) if for every family F ⊂ L(X, Y ) which is pointwise bounded on X is uniformly bounded on bounded subsets of X ([Sw2]12.5).For example , if X is barrelled , then any pair (X, Y ) has UB by the Uniform Boundedness Principle for barrelled spaces ( If I is an interval, the characteristic function of I is denoted by C I and if x = {x k } is an X-valued sequence, then C I x will denote the coordinatewise product of C I and x. Lemma 1. Assume that a k > 0 and a k+1 ≥ a k for every k and that (X, Y ) satisfies UB.If B ⊂ µ{X} βY is pointwise bounded on µ{X}, A ⊂ µ{X} is coordinatewise bounded and Proof : If the conclusion fails, there is a continuous semi-norm r on Y , . Now for every j , {x k j : k} is bounded by hypothesis and since c 00 {X} ⊂ µ{X} , {T k j : k} is pointwise bounded on X.By the UB property, {T k j x k j : k} is bounded in Y .There exists k 2 > k 1 such that . Now just continue this construction and relabel.
A scalar version of this lemma is given in [Sw3].Our main result involves the gliding hump properties introduced in [Sw 3].Let λ be a scalar-valued sequence space which contains c 00 .For the space µ{X} these are given in the following definitions.Definition 2. µ{X} has the strong λ gliding hump property (strong λ-GHP) if whenever {I k } is an increasing sequence of intervals and {x k } is a bounded sequence in µ{X}, then for every t = {t k } ∈ λ the coordinate sum of the series P t k C I k x k belongs to µ{X}.Definition 3. µ{X} has the weak λ gliding hump property (weak λ-GHP) if whenever {I k } is an increasing sequence of intervals and {x k } is a bounded sequence in µ{X}, there is a subsequence {n k } such that the coordinate sum Examples of spaces satisfying both strong and weak λ-GHP are given in [Sw3].For example, l p has strong l p -GHP for 0 < p ≤ ∞ and l ∞ and c 0 have strong c 0 -GHP.
The proof of our main result employs a matrix theorem due to Antosik and Mikusinski which we now state for the convenience of the reader.Theorem 4. Let M = [y ij ] be an infinite matrix with entries from Y .Suppose (1) lim i y ij = 0 for every j, (2) for every increasing sequence of positive integers {m j } there is a subsequence {n j } of {m j } such that For a proof (of a more general result), see [Sw 2],2.2.2.A matrix satisfying the conditions of Theorem 4 is called a K-matrix and Theorem 4 is called the Antosik-Mikusinski Diagonal Theorem.
Theorem 5. Assume that λ contains a vector {b k } with b k > 0 for every k.Assume that µ has strong λ-GHP and that (X, Y ) has UB.If A ⊂ µ{X} is bounded and B ⊂ µ{X} βY is pointwise bounded on µ{X}, then B • A is bounded.
Proof : Suppose the conclusion fails.Since µ is a K-space, A is coordinatewise bounded so we may apply Lemma 1 with a k = k/b k .Let the notation be as in Lemma 1 and define the matrix We claim that M is a K-matrix in the sense of Antosik and Mikusinski .First, the columns of M converge to 0 by the pointwise boundedness of B. Let z = {z i } be the coordinatewise sum of the series z = P ∞ j=1 b j C I j x j .We claim that z ∈ µ{X}.Let q ∈ X .By the strong λ-GHP the series , where e k is the sequence with a 1 in the kth coordinate and 0 in the other coordinates, converges in µ to the element {q(z i )} ∈ µ.That is, z ∈ µ{X}.Therefore, z → 0 by the pointwise boundedness of B. Since the same argument can be applied to any subsequence, M is a K-matrix.By the Antosik-Mikusinski Diagonal Theorem the diagonal of M converges to 0. But this contradicts the conclusion of Lemma 1 since r(m ii ) > 1.
Remark 6.The proof above also applies if µ has the weak λ-GHP and X is normed since in this case it is only necessary to check a single element q ∈ X to establish that z ∈ µ{X}.

Remark 7.
The assumption that the pair has the UB property is necessary in Theorem 5.For assume that F ⊂ L(X, Y ) is a subset which is pointwise bounded on X and let A ⊂ X be bounded.If x ∈ X, T ∈ L(X, Y ), put x 1 = (x, 0, 0, ...), T 1 = (T, 0, 0, ...)and A pointwise bounded on µ{X}.Thus, if the conclusion of Theorem 5 holds, then F A is bounded so the pair (X, Y ) would have UB.
Remark 8. Without some assumption such as the GHP, the conclusion of Theorem 5 cannot hold.For, let X be a non-barrelled normed space and let {x k : k} ⊂ X be bounded in X and let B = {y k : k} ⊂ X 0 be weak* bounded in X 0 with sup{|hy k , x k i| : k} = ∞.Set

A 1 =
{x k e k : k} and B 1 = {y 1 k = (y k , y k , ...) : k}.Then A 1 ⊂ c 00 {X} and B 1 ⊂ c 00 {X} βR , B 1 is pointwise bounded ,c 00 {X} does not have l 1 -GHP and B 1 • A 1 = {hy k , x k i : k} is unbounded so the conclusion of Theorem 5 fails to hold.A scalar version of Theorem 5 is given in [Sw3 ], Corollary 7; however the proof of this result in [Sw3] relies on duality methods and cannot be employed to obtain the operator-valued version given in Theorem 5.The Antosik-Mikusinski diagonal Theorem is used in place of the duality methods.The assumptions on µ are also somewhat stronger in [Sw3].Another scalar Banach-Mackey type result is given in [F]3.7.