The following changes should be made in the statement of Theorem 3.1 and its proof.

A set function is said to be λ-convex if whenever A and B are in Σ with A ∩ B = φ, then

At the 29th Meeting of the British Mathematical Colloquium held at Edinburgh in March 1977, D. H. Fremlin announced the following result.

Let Φ denote the proposition that there exists a K-analytic Hausdorff space, having metrizable compacta, which is not Souslin. (We refer the reader to Fremlin's paper (1) for the relevant definitions)

Theorem 1.1. (Fremlin) (i) Assume . (ii) Assume Martin's Axiom together with . Then Φ is false.

Let X be any real or complex Banach space. If T is a bounded linear operator on X then denote by N(T) the null space of T and by R(T) the range space of T.

Now if X is finite dimensional and N(T) = N(T2) then also R(T) = R(T2). Therefore X admits a direct sum decomposition

.

Indeed it is easy to see that N(T) = N(T2) implies that and, using dimension theory of finite dimensional spaces, that N(T) and R(T) span the whole space (see, for example, (2, pp. 271–73))

Let G be a permutation group of degree m. Let x be an irreducible complex character of G. If A = (aij) is an m-square matrix, the generalised matrix function of A based on G and x is defined by

For example if G = Sm, the full symmetric group, and x is the alternating character, then d = determinant. If G = Sm and x is identically 1, then d = permanent.

Keller (6) considered a generalisation of a problem of Minkowski (7) concerning the filling of Rn by congruent cubes. Hajós (4) reduced Minkowski's conjecture to a problem concerning the factorization of finite abelian groups and then solved this problem. In a similar manner Hajós (5) reduced Keller's conjecture to a problem in the factorization of finite abelian groups, but this problem remains unsolved, in general. It occurs also as Problem 80 in Fuchs (3). Seitz (10) has obtained a solution for cyclic groups of prime power order. In this paper we present a solution for cyclic groups whose order is the product of two prime powers.

R will denote a Dedekind domain and Pone of its prime ideals. A P-primary module will be an R-module all of whose non-zero elements have annihilators that are powers of the prime P. In all that follows E is such a module.

The height of 0 ≠ x ∈ E will be max{n : x ∈ PnE}. It is denoted by h(x). If this maximum does not exist we will say h(x)∞.

Clearly the condition is equivalent to E having no non-zero elements of infinite height. Adopting the terminology of (2, Ch. XI) where such modules over the ring of integers are studied, we will call these modules separable and reduced.

The permanent of an m x n matrix A = (aij), m ≤ n, is defined by

where the summation is over all one-to-one functions σ from {1, … , m} to { 1, …, n}. In other words, the permanent of A is the sum of all the diagonal products of A, that is, all the products of m entries of A no two of which lie in the same row or in the same column. Thus the permanent of A may be evaluated by first multiplying all the row sums of A and then subtracting from the product all terms that contain as factors two or more entries from the same column of A. This is the idea behind the formulas of Binet and of Ryser.

Following (1) we say that a subgroup H of a group G is almost subnormal in G if H is of finite index in some subnormal subgroup of G, or, equivalently, if |Hn : H| is finite for some n, where Hn is the n-th term of the normal closure series of H in G. The aim of this article is to prove, in answer to a question of R. Baer, the following analogue of the well known result of Roseblade and Stonehewer (3) that in any group the join of a pair of finitely generated subnormal subgroups is always subnormal:

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X. The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper (1); see, for example, (14), (15), (3), (4), (12), (2), and (6). Most of these investigations have been concerned with generalising the space X and taking E to be the scalar field or a locally convex space.

Let H1(U2) be the Hardy space of the bidisc as described in (3). Each function f ∈ H1(U2) has a Taylor expansion of the form . For 0<p<∞, a doubly-indexed sequence is said to be a multiplier of H1(U2) into lp if

This paper is concerned with the cases p = 2 and p = 1. Theorem 1 characterises the multipliers of into l2 and is an analogue in two variables of an old result of Hardy and Littlewood. Theorem 2 characterises the sequences (an)n≥0 such that (an+m)n,m≥0 is a multiplier of H1(U) into l1

In recent years there has been considerable interest in Banach spaces with the Radon-Nikodym Property; see (1) for a summary of the main known results on this class of spaces.We may define this property as follows: a Banach space X has the Radon-Nikodym Property if whenever T ∈ ℒ (L1, X)(where L1 = L1(0, 1)) then T is differentiable i.e.

where g:(0, 1)→X is an essentially bounded strongly measurable function. In this paper we examine analogues of the Radon-Nikodym Property for quasi-Banach spaces. If 0>p > 1, there are several possible ways of defining “differentiable” operators on Lp, but they inevitably lead to the conclusion that the only differentiable operator is zero.