For a Banach space X, Susumu Okada raised the question of whether the unit hall of the dual space X* is weak* separable if X* is weak* separable. The problem occurred in the theory of manifolds modelled on locally convex spaces. We answer the question in the negative but show that it is true for particular types of spaces.

Some sufficient conditions are presented for the lower radical construction in a variety of algebras to terminate at the step corresponding to the first infinite ordinal. An example is also presented, in a variety satisfying some non-trivial identities, of a lower radical construction terminating in four steps.

This article generalizes the well known theorem that the geometric morphisms from the category of sets to a category of set-valued sheaves on a topological space correspond to the irreducible components of the topological space. As irreducible components are not available in any topos more general than a spatial one, they are characterized in terms of filters of open sets - which are available in any topos. It is then seen that the theorem phrased in these terms generalizes to sheaves on any lattice with reasonable distributivity conditions.

In this note, the Morse-Smale characteristic of a differentiable manifold is defined and certain of its properties are studied.

Let E be a Banach space, M a closed subspace of E with the 3-ball property. It is known that M is proximinal in E, and that its metric projection admits a continuous selection. This means that there is a continuous (generally non-linear) map π: E → M satisfying ‖x−π(x)‖ = d(x, M) for all x in E. Here it is shown that the same conclusion holds under a much weaker hypothesis on M, which we call the 1½-ball property. We also establish that if M has the 1½-ball property in E, then there is a continuous Hahn-Banach extension map from M* to E*.

For convex optimal control problems without explicit pure state constraints, the structure of dual problems is now well known. However, when these constraints are present and active, the theory of duality is not highly developed. The major difficulty is that the dual variables are not absolutely continuous functions as a result of singularities when the state trajectory hits a state constraint. In this paper we recognize this difficulty by formulating the dual probram in the space of measurable functions. A strong duality theorem is derived. This pairs a primal, state constrained convex optimal control problem with a dual convex control problem that is unconstrained with respect to state constraints. In this sense, the dual problem is computationally more attractive than the primal.