From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal κ is fully indestructible by <κ-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal. Lave...

The Banach space E has the weakly compact approximation property (W.A.P.) if there is C < ∞ so that the identity map IE can be uniformly approximated on any weakly compact subset D ⊂ E by weakly compact operators V on E satisfying ‖V ‖ ≤ C. We show that the spaces N(`, `) of nuclear operators ` → ` have the W.A.P. for 1 < q ≤ p < ∞, but that the Hardy space H does not have the W.A.P.

It is well known that in applied and computational mathematics, cardinal B-splines play an important role in geometric modeling (in computeraided geometric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth. More recently, in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of ...

Via two short proofs and three constructions, we show how to increase the model-theoretic precision of a widely used method for building ultrafilters. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck” in the inductive construction of a regul...

In this paper, we study trees of uncountable regular heights containing ascending paths of small width. This combinatorial property of trees generalizes the concept of a cofinal branch and it causes trees to be non-special not only in V, but also in every cofinality-preserving outer model of V. Moreover, under certain cardinal arithmetic assumptions, the non-existence of such paths through a tr...

We prove four theorems concerning the number of normal measures a non-(κ+ 2)-strong strongly compact cardinal κ can carry.

Introduction. The two sections of this note are independent, but they are related by the fact that both use the results of [5 ] to obtain information on the properties of weakly compact sets in Banach spaces. In the first section we prove some results on a class of compact sets which is believed to include all weakly compact subsets of Banach spaces. We are interested in the properties of the n...

Let E be an ideal of L0 over a σ-finite measure space (Ω,Σ, μ), and let E∼ stand for the order dual of E. For a real Banach space (X, ‖ · ‖X ) let E(X) be a subspace of the space L0(X) of μ-equivalence classes of strongly Σ-measurable functions f : Ω −→ X and consisting of all those f ∈ L0(X) for which the scalar function ‖f(·)‖X belongs to E. For a real Banach space (Y, ‖ · ‖Y ) a linear opera...

Using Koszmider’s strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ ≤ λ, κ = κ and 2 = κ, and η is an ordinal with κ ≤ η < κ and cf(η) = κ. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra B such that ht(B) = η + 1, wdα(B) = κ for every α < η and wdη(B) = λ (i.e. there is a local...