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A remarkable Rosenthal L∞-theorem is extended to operators T : L∞(Γ, E) → F , where Γ is an infinite set, E a locally bounded (for instance, normed or p-normed) space, and F any topological vector space.

A concept of limitedness of sets in one tvs relative to another tvs is discussed. Special attention is devoted to the limitedness relative to spaces without copies of $$\ell _{\infty }$$ ℓ ∞ . Applications to copies of $$\ell _{\infty }$$ ℓ ∞ in vector-valued $$c_0$$ c 0 -type spaces, and to the complementability of the latter in the containing them similar $$\ell _{\infty }$$ ℓ ∞ -type spaces are provided.

We investigate various Localization Properties of ideals in relative to sequence spaces supported by them, particularly the Localization Property for Series. The results obtained enable us to prove the existence of a class of sequence spaces X in which every zero-density convergent series is subseries convergent (i.e., X has the Zero Density Convergence Property), but some lacunary convergent series are not (i.e., X fails the Lacunary Convergence Property).

In this note, the authors discuss the concepts of a Pettis operator, by which they mean a weak $$^*$$ –weakly continuous linear operator F from a dual Banach space to an $$L_1$$ -space, and of its Pettis integral, understood simply as the dual operator $$F^*$$ of F. Applications to radial limits in weak Hardy spaces of vector-valued harmonic and holomorphic functions are provided.

A unified approach, that uses countable sets with convergences and basics of the theory of LF-spaces, is given to some old results of the Silverman–Toeplitz type obtained in the late 1930's by H.J. Hamilton for matrix transformations of multiple sequences.

Let X be an F-space and \({\boldsymbol x=(x_n)}\) be a sequence of vectors in X. Ideals \({\mathcal{C}(\boldsymbol x)}\) of subseries convergence are considered. In particular, we show that a characterization of the class of Banach spaces not containing c0 obtained by using the ideals \({\mathcal{C}(\boldsymbol x)}\) breaks down in every Frechet space not isomorphic to a Banach space. On the other hand, the result can be extended to some F-spaces via the definition of a new class of F-spaces satisfying a stronger version of the condition (O) of Orlicz. A theorem discriminating between the finite and infinite dimensional case is obtained about the family \({\mathcal{C}(X)}\) of all ideals associated with the F-space X.

We study the BDS-integral, using the original definition, but with respect to a convexly bounded measure μ with values in an arbitrary sequentially complete tvs X . Denote by L 0 ( μ ) the space of μ -measurable R -valued functions. Then all bounded measurable functions are μ -integrable (in an elementary sense), and the space L 1 ( μ ) of BDS-integrable functions is a vector lattice and a topological vector space in its natural topology. We next distinguish the space L ∘ 1 ( μ ) as the largest vector subspace of L 1 ( μ ) that is solid in L 0 ( μ ) . We prove general convergence theorems for both L 1 ( μ ) and L ∘ 1 ( μ ) . In particular, we show that L ∘ 1 ( μ ) with its natural topology is a Dedekind σ -complete topological vector lattice with the σ -Lebesgue property, and that the Dominated Convergence Theorem holds in L ∘ 1 ( μ ) . If X contains no isomorphic copy of c 0 , then L ∘ 1 ( μ ) has the σ -Levi property (that is, the Beppo Levi Theorem holds). We identify L ∘ 1 ( μ ) as the domain for the Thomas–Turpin integral, and thus show that this integral is simply the restriction of the BDS-integral to L ∘ 1 ( μ ) . The last two statements answer the questions posed, respectively, by Thomas and Turpin, and left open since the seventies.

A topological vector lattice E is called (σ-)nestedly complete if every downward directed net (resp., decreasing sequence) of order intervals in E whose ‘diameters’ tend to zero has a nonempty intersection. Some characterizations of the (σ-)nested completeness are given, and it is shown that if E is metrizable and nestedly complete, so is each of its quotients E/I, where I is a closed ideal in E. Conversely, if a closed ideal I in E is (sequentially) complete and E/I is (σ-)nestedly complete, so is E. However, the nested completeness is not a three-space property: an example is given where both I and E/I are nestedly complete while E is not. It is also shown that the nested completeness and the related notion of nested density come up quite naturally when extending some positive linear operators. Finally, the nested and other completeness type properties of vector lattices C(S) are investigated.

A submeasure μ defined on the subsets of \({\mathbb{N}}\) is nonatomic if for every l ≥ 1 there exists a partition of \({\mathbb{N}}\) into a finite number of parts on which μ is bounded from above by 1/l. In this paper we answer several natural questions concerning nonatomic submeasures dF that are determined (like the standard density) by a family F of finite subsets of \({\mathbb{N}}\). We first show that if the number of n-element sets in F grows at most exponentially with n, then dF is nonatomic; but if this growth condition fails, then dF need not be nonatomic in general. We next prove that, for a nonatomic submeasure dF, the minimal number of sets in a 1/l-small partition of \({\mathbb{N}}\) can grow arbitrarily fast with l. We also give a simple example of a nonatomic submeasure that is not equivalent to a submeasure of type dF.

A maxitive measure is a nonnegative function η on a σ-algebra Σ and such that η(U j A j ) = sup j η( A j ) for all countable disjoint families of sets ( A j ) in Σ. A representation theorem for such measures is established, and next applied to represent Kothe function M -spaces as L ∞ -spaces.

It is shown that if two submeasures on N that are lim sup's of sequences of measures have the same zero sets, and one is nonatomic, so is the other, and they are (e-δ)-equivalent. Moreover, if a submeasure η is the lim sup of a sequence of lower semi-continuous (lsc) submeasures and is 0-dominated by the so-called core γ • of an lsc submeasure γ, then η is also (e-δ)-dominated by γ • . And if a submeasure η of this type is 0-dominated by a nonatomic submeasure, then it is nonatomic as well.

We use stable Kneser hypergraphs to construct ideals in N which are not nonatomic yet have the Nikodym property.

Given an ordered metric space (in particular, a Banach lattice)E, the gen- eralized Helly space H(E) is the set of all increasing functions from the interval (0; 1) to E considered with the topology of pointwise convergence, and E is said to have property ( ) if each of these functions has only countably many points of discontinuity. The main ob- jective of the paper is to study those ordered metric spacesC(K;E), whereK is a compact space, that have property ( ). In doing so, the guiding idea comes from the fact that there is a natural one-to-one correspondence between increasing functions f : (0; 1) ! C(K;E) (with countably many discontinuities) and continuous maps F : K ! H(E) (with metriz- able ranges). It leads to the investigation of general continuous metrizing maps (those with metrizable ranges), and especially of the so called separately metrizing maps, and the results obtained are then used to derive some permanence properties of the class of spaces C(K;E) with property ( ). For instance, it is shown that if K is the product of compact spaces Kj (j 2 J) such that each of the spaces C(Kj;E) has property ( ), so does C(K;E); and, for any compact space K, if both C(K) and a Banach lattice E have property ( ), so does C(K;E).

The paper is concerned with order-topological characterizations of topological Riesz spaces, in particular spaces of measurable functions, not containing Riesz isomorphic or linearly homeomorphic copies of c 0 c_{0} or ℓ ∞ \ell _{\infty } .

Some weakenings of property (K) of Antosik for locally convex spaces are introduced: local property (K) and, for spaces with Schauder-type decompositions, two variants of property (K) defined in terms of block- and tail-sequences. It is shown that if a space enjoys any of these new properties, then it is Banach-Mackey. An application to the barrelledness of the spaces of Pettis integrable functions is given, and examples are provided to distinguish between the various K-type properties.

Let S(μ, E) be the space of (classes of μ-a.e. equal) simple functions defined on a (non-trivial) measure space with values in a locally convex space E. The following results hold: S(μ,E) is quasi-barrelled (resp. bornological) if and only if E is quasi-barrelled (resp. bornological) and E′(β(E′,E)) has the property (B) of Pietsch; S(μ, E) is barrelled if and only if S(μ,K) is barrelled and E is barrelled and nuclear; S(μ, E) is never ultrabornological; and S(μ, E) is a DF-space if and only if E is a DF-space.

It is well known that the normed space of Pettis integrable functions from a finite measure space to a Banach space is not complete in general. Here we prove that this space is always barrelled; this tells us that we may apply two important results to this space, namely, the Banach-Steinhaus uniform boundedness principle and the closed graph theorem. The proof is based on a theorem stating that a quasi-barrelled space having a convenient Boolean algebra of projections is barrelled. We also use this theorem to give similar results for the spaces of Bochner integrable functions.

For a sequence x=(x n ) in a Banach space, define C(x) to be the set of all elements (e n ) of the Cantor cube K={0,1}ℕ for which the series \( \sum\nolimits_{n = 1}^\infty {\varepsilon _n x_n } \) is subseries convergent. The main result of the paper is that a Banach space X contains no isomorphic copy of c 0 if and only if, for every sequence x in X, the set C(x) is F σ in K. A similar equivalence involving ‘ideals of Pettis integrability’ is also shown.

In a response to a question asked by Leonard and Whitfield (1983) we show that, under the Continuum Hypothesis, the Banach space l ∞ / C 0 {l_\infty }/{C_0} is primary.

Let X and Y be Banach spaces. Then Kw*(X*, Y) denotes the Banach space of compact and weak*-weakly continuous linear operators from X* into Y, endowed with the usual operator norm. Let us write E⊃l∞ to indicate that a Banach space E contains an isomorphic copy of l∞. The purpose of this note is to prove the followingTheorem. Kw*(X*, Y) ⊃ l∞if and only if either X ⊃ l∞or Y ⊃ l∞.

For a σ \sigma -algebra Σ \Sigma and a Banach space X , c a ( Σ , X ) X,ca\left ( {\Sigma ,X} \right ) is the Banach space of all vector measures from Σ \Sigma to X X . If Σ \Sigma admits a nonzero atomless finite positive measure, then c a ( Σ , X ) ⊃ l ∞ ca\left ( {\Sigma ,X} \right ) \supset {l_\infty } (or c 0 {c_0} ) if and only if there is a noncompact bounded linear operator from l 2 {l_2} to X X (Theorem 1). Otherwise, c a ( Σ , X ) ⊃ l ∞ ca\left ( {\Sigma ,X} \right ) \supset {l_\infty } (or c 0 {c_0} ) if and only if X ⊃ l ∞ X \supset {l_\infty } (or c 0 {c_0} ) (Theorem 2).

The paper is devoted to a study of some aspects of the theory of (topological) Riesz space valued measures. The main topics considered are the following. First, the problem of existence (and, particularly, the so-called proper existence) of the modulus of an order bounded measure, and its relation to a similar problem for the induced integral operator. Second, the question of how properties of such a measure like countable additivity, exhaustivity or so-called absolute exhaustivity, or the properties of the range space, influence the properties of the modulus of the measure. Third, the problem of exhibiting (or constructing) Banach lattices that are "good'' in many respects, and yet admit a countably additive measure whose modulus is not countably additive. A few applications to weakly compact operators from spaces of bounded measurable functions to Banach lattices are also presented.

The notion of a topological vector space of Bochner measurable functions is introduced and studied. Among the main results obtained are characterizations of completeness and of containment of copies of $c_0$ or $\ell_\infty$.

Inspired by some results of Laυric (1992), we investigate those Banach lattices E that have the following property ( λ ): Every increasing function f : [0, 1] → E has only countably many points of discontinuity. We show that the space of regular functions on [0, 1] contains no copy of l ∞ and yet it lacks property ( λ ), thus answering in the negative a question of Lavric. On the positive part, our main results show that if a Banach lattice E has property ( λ ), then so do the Banach lattices C ( K,E ) and L p ( μ,E ) (1 p ∞) for a wide class of compact spaces K and every measure μ .

A simpler proof is given for the recent result of I. Labuda and the author that a series in the space Lo (lambda) is subseries convergent if each of its lacunary subseries converges.

Let $\eta$ be a strictly positive submeasure on $\mathsf N$. It is shown that the space $\omega(\eta)$ of all real sequences, considered with the topology $\tau_{\eta}$ of convergence in submeasure $\eta$, is (pathwise) connected iff $\eta$ is core-nonatomic. Moreover, for an arbitrary submeasure $\eta$, the connected component of the origin in $\omega(\eta)$ is characterized and shown to be an ideal. Some results of similar nature are also established for general topological vector-lattice groups as well as for the topological vector groups of Banach space valued sequences with the topology $\tau_{\eta}$.

Let ( X , σ, μ) be a finite measure space, A a complete, solid lattice of scalar functions defined on X , and E a normed space. We study barrelledness properties of the space A(E) of strongly measurable functions f : X → E such that ‖ f (·)‖ ∈ A.

In a response to a question asked by Leonard and Whitfield (1983) we show that, under the Continuum Hypothesis, the Banach space lo/C0 is primary.

For a a-algebra 1 and a Banach space X, ca(X, X) is the Banach space of all vector measures from X to X . If E admits a nonzero atomless finite positive measure, then ca(X, X) D 1. (or c0) if and only if there is a noncompact bounded linear operator from 12 to X (Theorem 1). Otherwise, ca(?, X) D 1. (or c0) if and only if X D l00 (or c0) (Theorem 2). Let E be a a-algebra on a nonempty set S, and X be a Banach space. Then ca(X, X) denotes the Banach space of all countably additive vector measures ,u: E X under the sup-norm Jl,ull = supE Ilu(E)HI. (This norm is of course equivalent with the more commonly used semivariation norm.) If Z is a Banach space, we write Z D 1 (or c0) to indicate that Z contains an isomorphic copy of lo (or c0). The purpose of this paper is to give a complete answer, in terms of X and X, to the question stated in the title. In [5, Corollary 4] it has been recently shown that the subspace cca(X, X) of ca(X, X), consisting of measures with relatively norm compact ranges, contains an isomorph of l1 if and only if X D 100. Somewhat surprisingly, it is in general not true that ca(X, X) D 1. implies X D l0 (the converse is always valid trivially). Here are our main results. Theorem 1. Suppose the u-algebra X admits a nonzero atomless finite positive measure. Then the following are equivalent. (a) ca(l, X) :) 100' (b) There exists a noncompact bounded linear operator from 12 to X. (c) ca(X, X) D co . Theorem 2. Suppose every nonzero finite positive measure on E is purely atomic. Then: (A) ca(X, X) D c0 X X D 10. (B) ca(l, X) D co X* X D co . Received by the editors June 20, 1989; the results of this paper were presented at the Polish-GDR Seminar on Functional Analysis held in Georgenthal (GDR), March 31-April 5, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46G 10, 46E27, 46B20, 46B25.

Let L 0 ( Z ) {L_0}(Z) be the F F -space of all Bochner measurable functions from a probability space to a Banach space Z Z . We prove that every countably additive vector measure taking values in L 0 ( Z ) {L_0}(Z) has bounded range. This generalizes a recent result due to M. Talagrand and, independently, N. J. Kalton, N. T. Peck and J. W. Roberts, asserting the same for the case when Z Z is the space of scalars.

Every infinite dimensional Banach or Frechet space X has a dense Baire (hence barrelled) subspace E of uncountable codimension such that every closed subspaee M of X with M∩E={0} is finite dimensional. This result solves negatively a problem raised recently by M. De Wilde and B. Tsirulnikov.

The main result of the paper is the following: If an F-space X is covered by a family ( E α : α ∈ N N ) of sets such that E α ⊂ E β whenever α ⩽ β , and f is a linear map from X to a topological linear space Y which is continuous on each of the sets E α , then f is continuous. This provides a very strong negative answer to a problem posed recently by J. Kakol and M. Lopez Pellicer. A number of consequences of this result are given, some of which are quite curious. Also, inspired by a related question asked by J. Kakol, it is shown that if a linear map is continuous on each member of a sequence of compact sets, then it is also continuous on every compact convex set contained in the linear span of the sequence. The construction applied to prove this is then used to interpret a natural linear topology associated with the sequence as the inductive limit topology in the sense of Ph. Turpin, and thus derive its basic properties.