A new brief proof of the fact stated in the title is presented.

In previous papers we have proved that if G is a ω*-closed subspace of the conjugate space B* of a normed linear space B, then every b ∈ B can be extended within B, from G to the whole B*, with an arbitrarily small increase of the norm. Here we give some extensions of this result to the case when B* is replaced by a normed linear space E and B by any linear subspace V of E*, and some applications to separation and Helly type theorems.

The notion of asymptotic centre of a bounded sequence of points in a uniformly convex Banach space was introduced by Edelstein in order to prove, in a quasi-constructive way, fixed point theorems for nonexpansive and similar maps.

Similar theorems have also been proved by, for example, adding a compactness hypothesis to the restrictions on the domain of the maps. In such proofs, which are generally less constructive, it may be possible to weaken the uniform convexity hypothesis.

In this paper Edelstein's technique is extended by defining a notion of asymptotic centre for an arbitrary set of nonempty bounded subsets of a metric space. It is shown that when the metric space is uniformly rotund and complete, and when the set of bounded subsets is a filter base, this filter base has a unique asymptotic centre. This fact is used to derive, in a uniform way, several fixed point theorems for nonexpansive and similar maps, both single-valued and many-valued.

Though related to known results, each of the fixed point theorems proved is either stronger than the corresponding known result, or has a compactness hypothesis replaced by the assumption of uniform convexity.

The usual statements of the classical adjoint-functor theorems contain the hypothesis that the codomain category should admit arbitrary intersections of families of monomorphisms with a common codomain. The aim of this article is to formulate an adjoint-functor theorem which refers, in a similar manner, to arbitrary internal intersections of “families of monomorphisms” in the case where the categories under consideration are suitably defined relative to a fixed elementary base topos (in the usual sense of Lawvere and Tierney).

Let π be a convex lattice polygon with b boundary points and c (≥ 1) interior points. We show that for any given c, the number b satisfies b ≤ 2c + 7, and identify the polygons for which equality holds.

An elementary proof is given of a sufficient optimality condition recently proven by B.D. Craven. This proof avoids the use of a transposition theorem and this allows for a strengthening of Craven's result.

Suppose that a finite soluble group G is the product AB of subgroups A and B. Our question is the following: what conclusions can be made about G if A and B are suitably restricted? First we shall prove that the p–length of G is restricted by the derived lengths of the Sylow p–subgroups of A and B, if A and B are p–closed and p′-closed. Moreover, if in such a group the Sylow p–subgroups of A and B are modular, the p–length of G is at most 1. Next we obtain a general estimate for the derived length of the group G = AB of odd order in terms of the derived lengths of A and B. Furthermore it will be possible to bound the nilpotent length of G and also the p–length of G in terms of other invariants of special subgroups of G.

Let E be a separated, locally convex topological vector space and F a commutative family of nonexpansive mappings defined on a quasi-complete convex (not necessarily bounded) subset X of E. In this paper, it is proved that if one of the mappings in F is condensing with a bounded range then the family F has a common fixed point in X. This result improves several well-known results and supplements a recent result of E. Tarafdar (Bull. Austral. Math. Soc. 13 (1975), 241–254) for such mappings.

Sufficient conditions are obtained for all solutions of a class of second order nonlinear functional differential equations to be nonoscillatory.

The Fibonacci group F(2, 7) has been known to be cyclic of order 29 for about five years. This was first established by computer coset enumerations which exhibit only the result, without supporting proofs. The working in a coset enumeration actually contains proofs of many relations that hold in the group. A hand proof that F(2, 7) is cyclic of order 29, based on the working in computer coset enumerations, is presented here.

The indecomposable representations in characteristic two of the groups PSL(2, q) where q is congruent to 3 or 5 modulo 8 are classified. For q = 3 or 5 the classification is obtained by explicit construction of modules, using the Green correspondence to prove completeness. For larger q, the classification is obtained using equivalences between appropriate categories of modules.

Let Si, i ∈ I, be a set of semigroups such that Si ∩ Sj = U, if i ≠ j, and such that U is a unitary subsemi-group of Si for each i in I. The semigroup amalgam [{Si | i ∈ I}; U] determined by this system is the partial groupoid G = USi in which a product of two elements is defined if and only if they both belong to the same Si and their product is then taken as their product in Si. In 1962, J.M. Howie showed that the amalgam G is embeddable in the free product of the Si, amalgamating U. To prove this result it suffices to find any semigroup in which G can be embedded. In this paper, by taking convenient representations of the Si, adapting a method recently (1975) used by T.E. Hall for inverse semigroups, we provide a short method of constructing such a semigroup.

A short elementary proof is given of the theorem of Marggraff which states that a primitive permutation group which contains a cycle fixing k points is (k+1)-fold transitive. It is then shown that the method of proof actually yields a generalization of Marggraff's theorem.

Initial value problems of the form x′ + A(t, x)x = f(t, x), x(0) = a, t ≥ 0, are considered in a real, separable, reflexive Banach space. Results concerning the existence of solutions on (0, ∞) are given by considering linear systems of the form x′ + A(t, u(t))x = f(t, u(t)). Here u(t) belongs to a suitable function space.

It is shown that a separable, quasi-reflexive Banach space of deficiency one admits an equivalent norm such that its second conjugate space is smooth; this answers a question raised by Ivan Singer [Bull. Austral. Math. Soc. 12 (1975), 407–416.

In this note we show that BVk, [a, b], the linear space of functions of bounded kth variation on [a, b], is a Banach space under the norm ‖ • ‖k, where

In this note we establish theorems on compactness of lattice packings.