A MULTIPLIER GLIDING HUMP PROPERTY FOR SEQUENCE SPACES CHARLES SWARTZ New Mexico State University -

We consider the Banach-Mackey property for pairs of vector spaces E and E′ which are in duality. Let A be an algebra of sets and assume that P is an additive map from A into the projection operators on E. We define a continuous gliding hump property for the map P and show that pairs with this gliding hump property and another measure theoretic property are Banach-Mackey pairs,i.e., weakly bounded subsets of E are strongly bounded. Examples of vector valued function spaces, such as the space of Pettis integrable functions, which satisfy these conditions are given.


INTRODUCTION
H. Lebesgue introduced the gliding hump technique of proof to establish several uniform boundedness results for concrete function spaces such as L[0,1] ( [L]).Subsequently, Schur and Hellinger/Toeplitz also used the gliding hump method to establish similar uniform boundedness principles for concrete function spaces ([Sc], [HT]).The early proofs of abstact uniform boundedness principles by Banach, Hahn and Hilldebrandt all employed gliding techniques ( [B], [Ha], [Hi]).Absract gliding hump assumptions have been used to treat a number of topics in sequence spaces;for example, Noll used a "strong gliding hump" property to establish the weak sequential completeness of the beta dual of a sequence space ( [N] ; see [BF] for a list of various gliding hump properties for sequence spaces).In this paper we introduce a gliding hump assumption involving multipliers from a scalar sequence space which is particularly useful in establishing uniform boundedness results for a vector-valued sequence space and its beta dual; in particular, our results establish Banach-Mackey properties for sequence spaces.

DEFINITIONS AND EXAMPLES
We begin with the notations and assumptions which will be used.Let X be a Hausdorff locally convex space and let E be a vector space of X-valued sequences containing c 00 (X), the space of all Xvalued sequences which are eventually 0. We assume that E has a Hausdorff locally convex topology under which E is a K-space, i.e., the coordinate maps Let λ be a vector space of scalar valued sequences which contains c 00 the space of sequences which are eventually 0. The β-dual of λ, λ β , is defined to be {t = {t k } : t k s k converges f or every s = {s k } ∈ λ}.If s ∈ λ and t ∈ λ β , we set t • s = t k s k ; λ and λ β are in duality with respect to the bilinear pairing (s, t) → s • t.
Definition 1. E 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 E, then for every t = {t k } ∈ λ the coordinate sum of the series t k χ I k x k belongs to E. Definition 2. E 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 E, there is a subsequence {n k } such that the coordinate sum t k χ In k x k belongs to E for every t ∈ λ.
We refer to the elements of λ in Definitions 1 and 2 as multipliers since their coordinates multiply the blocks {χ I k } determined by {I k } and {x k }.The weak λ − GHP is like the strong gliding humps property introduced by Noll ( [N]) where the multipliers consist only of the constant sequence {1}.After giving examples of spaces with λ-GHP we will make remarks comparing λ-GHP with other gliding hump properties.
We proceed to give an extensive list of examples of spaces with λ-GHP.The reader may want to skip ahead to section 3 where the main results are established and then refer back to the examples.For our first example we need a definition.Definition 3. E satisfies the boundedness property (B) if for every increasing sequence of intervals {I k } and every bounded set A ⊂ E, the set {χ For example, if I is the family of all intervals in N and the maps χ I : E → E, x → χ I x, I ∈ I are equicontinuous, then (B) holds.This is the case if p(χ I x) ≤ p(x) holds for every I ∈ I, § ∈ E and continuous seminorm p on E. Proposition 4. If E is a locally complete space with property (B), then E has strong l 1 − GHP .
Proof: Let {I k } be an increasing sequence of intervals and {x k } ⊂ E be bounded.By (B) absolutely convergent in E and, therefore, converges to an element x ∈ E by local completeness.Since X is a K − space, x is also the coordinate sum of the series.
Proposition 4 gives a large supply of spaces with l 1 − GHP .We also have Example 5. l ∞ and c 0 have strong c 0 − GHP ; l p has strong l p − GHP for 0 < p ≤ ∞.
We now give examples of non-complete scalar sequence spaces with weak l p − GHP .
Example 6.Let 1 ≤ p < ∞.Let P be the power set of N and let µ : P → [0, ∞) be a finitely additive set function with µ({j}) > 0 for every j.Put l p (µ) = L p (µ), the space of all pth power µ-integrable functions with the norm [RR] for details on the integration with repect to finitely additive set functions; the assumption µ({j}) > 0 for every j makes l p (µ) a K −space].We show that l p (µ) has weak l p −GHP .Let {I k } be an increasing sequence and . We claim that f ∈ l p (µ) and the series converges to f in l p (µ) by using Theorem 4.6.10 of [RR].Put The claim is thus justified, and it follows that l p (µ) has weak l p − GHP .
Problem.Does l p (µ) have strong l p ?
We next give examples of vector-valued sequence spaces with λ − GHP .Let X be a family of semi-norms which generate the topology of X .Let µ be a normal (scalar) K-space whose topology is generated by the family of semi-norms M.
We make the following assumptions on µ: These assumptions are satisfied by many of the classical sequence spaces.
We define µ{X} to be the space of all X − valued sequences x = {x k } such that {p(x k )} ∈ µ for every p ∈ X .Since µ is normal, µ{X} is a vector space.We assume that µ{X} has the locally convex topology generated by the semi-norms Spaces of this type were considered in [FP] and [F].
The spaces l p {X} and c 0 {X} are the usual spaces of pth power convergent and null sequences, respectively.As in Example 5 it is easily seen that l ∞ {X} and c 0 {X} have strong c 0 − GHP and l p {X} has strong l p − GHP .More generally.we have Proposition 7. If µ has strong λ − GHP , then µ{X} has strong λ − GHP .
Proof: Let {I k } be an increasing sequence of intervals and {x k } ⊂ µ{X} be bounded.Let t ∈ λ and put Proposition 8.If µ has weak λ − GHP and X is normed, then µ{X} has weak λ − GHP .
Proof: Continue the notation from Proposition 7 and let be the norm on X.For every k { x k j } ∞ j=1 ∈ µ and {{ x k j } j : k} is bounded in µ so by weak λ − GHP there is a subsequence Propositions 7 and 8 give a large supply of spaces with λ − GHP many of which are not sequentially complete [ e.g., l p {X}or c 0 {X}].
We now give other examples of (non-monotone) vector-valued sequence spaces.
Example 9. Let CS(X) be the space of all X-valued sequences {x k } such that the series x k is Cauchy in X.If X is the scalar field, CS(X) is the space cs of convergent series.We define a topology on CS(X) induced by the semi-norms p ({x k }) = sup{p( j∈I x j ) : I ∈ I}, p ∈ X .
We claim that CS(X) has strong l 1 − GHP .Suppose {I k } is increasing and Example 10.Let BS(X) be all X-valued sequences {x k } such that the partial sums { n k=1 x k } are bounded.If X is the scalar field, BS(X) is the space of bounded series bs.As above define a topology on BS(X) by the semi-norms p ({x k }) = sup{p( j∈I x j : ) and topologize BV (X) by the semi-norms {p : p ∈ X }.
We show that BV (X) has strong As noted earlier the weak λ − GHP resembles the strong gliding hump property introduced by Noll where the mutipliers consist of the single constant sequence {1} ( [N]).A weaker gliding hump property is the zero−GHP ; E has zero−GHP if x k → 0 in E and {I k } increasing implies there exists a subsequence Sw3] 12.5).We give an example of a space with l 1 − GHP but without zero − GHP .
Example 12. Let E be l 2 with the weak topology.Since E is sequentially complete, E has strong l 1 − GHP by Proposition 4.However, E fails to have zero − GHP [consider {k} and {e k }].

MAIN RESULTS
We now prove several uniform boundedness results for spaces with weak λ − GHP.The (scalar x k ; E and E β are then in duality with respect to the bilinear pairing (x, y) → y • x.
If Z and Z are two vector spaces in duality, we denote the weak (strong) topology of Z with resect to this duality by σ(Z, Z )(β(Z, Z )).Recall that the pair Z, Z is a Banach-Mackey pair if σ(Z, Z ) bounded sets in Z are β(Z, Z ) bounded, and X is a Banach-Mackey space if X, X is a Banach-Mackey pair ( [Wi] 10.4).
We begin with a basic lemma.
For every j {x k j : k} is bounded in X by hypothesis and {y k j : k} is σ(X , X) bounded since B has σ(X , X) bounded coordinates.Since X is Banach-Mackey, { y k j , x k j : k} is bounded for every j so lim k . Now just continue this construction and relabel.
We now establish our first uniform boundedness result for E and its β-dual.In what follows e k is the canonical vector with a 1 in the kth coordinate and 0 in the other coordinates.
Theorem 2. Let X be a Banach-Mackey space and suppose that E has weak λ − GHP .Assume (2) Proof: If the conclusion fails, Lemma 1 applies.Let the notation be as in Lemma 1 and let {n j } be the subsequence in the definition of the weak λ − GHP .Define a linear operator ) and since this series converges for every t ∈ λ, {y • χ I n j x n j } belongs to λ β and y ([Wi] 11.2.6, [Sw1] 26.15).Thus, by hypothesis, {T e k } = {χ But this contradicts the conclusion of Lemma 1.
A similar uniform boundedness result for spaces with zero − GHP is given in [Sw3] 12.5.7.
Corollary 3.Under the hypothesis of Theorem 2 if E ⊂ E β , then E is a Banach-Mackey space.
We have a general criterion for the hypothesis in Corollary 3 to hold.If z ∈ X, we define e k ⊗ z to be the sequence with z in the kth coordinate and 0 in the other coordinates.We say that E is an AK-space if the series ∞ k=1 e k ⊗ x k converges to x = {x k } ∈ E in the topology of E for all x.
Example 5. CS(X) is an AK-space so it follows from Proposition 4, Corollary 3 and Example 2.9 that CS(X) is a Banach-Mackey space when X is a Banach-Mackey space.
For the vector-valued sequence spaces µ{X}, we have Example 6.It is easily checked that µ{X} is an AK-space when µ is an AK-space.If (3) X is a Banach-Mackey space and either µhas strong λ − GHP or µ has weak λ − GHP and X is normed, (2) holds and µ is anAK-space, then µ{X} is a Banach-Mackey space [Proposition 4, Corollary 3 and Propositions 2.7 or 2.8].
In particular, c 0 {X} is a Banach-Mackey space when X is a Banach-Mackey space; this was established by Mendoza ([M]).It also follows that l p {X} is a Banach-Mackey space for 1 ≤ p < ∞; Fourie has given a general criterion for spaces of the type µ{X} to be Banach-Mackey spaces ([F] 3.7) but his result does not cover l 1 {X}.
We also have a general uniform boundedness result for the spaces µ{X} and their β-duals.
We consider conditions which guarantee that E, E β form a Banach-Mackey pair and then consider specific examples.From Theorem 2, we obtain Corollary 8. Assume that X is a Banach-Mackey space, E has weak λ − GHP and (2) holds.If E is such that σ(E, E β ) bounded sets are bounded in the topology of E, then E, E β is a Banach-Mackey pair.
When E is a monotone space [or more generally when E has the signed weak GHP] and X is weak * sequentially complete, then (E β , σ(E β , E)) is sequentially complete so E, E β form a Banach-Mackey pair ([Sw3] 12.4.1,[Wi] 10.4).This result applies to l ∞ {X} and c 0 {X} when X is weak * sequentially complete; however, our assumption on X being a Banach-Mackey space is weaker.