J.R. Giles

Characterisation of normed linear spaces with Mazur's intersection property

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak* denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties.

On numerical ranges of operators on locally convex spaces

Numerical range theory for linear operators on normed linear spaces and for elements of normed algebras is now firmly established and the main results of this study are conveniently presented by Bonsall and Duncan in (1971) and (1973). An extension of the spatial numerical range for a class of operators on locally convex spaces was outlined by Moore in (1969) and (1969a), and an extension of the algebra numerical range for elements of locally m-convex algebras was presented by Giles and Koehler (1973). It is our aim in this paper to contribute further to Moores work by extending the concept of spatial numerical range to a wider class of operators on locally convex spaces.

A distance function property implying differentiability

In a real normed linear space X , properties of a non-empty closed set K are closely related to those of the distance function d which it generates. If X has a uniformly Gâteaux (uniformly Frechet) differentiable norm, then d is Gâteaux (Frechet) differentiable at x ∈ X / K if there exists an such that and is Geteaux (Frechet) differentiable on X / K if there exists a set P + ( K ) dense in X / K where such a limit is approached uniformly for all x ∈ P + ( K ). When X is complete this last property implies that K is convex.

SEPARABLE DETERMINATION OF FRECHET DIFFERENTIABILITY OF CONVEX FUNCTIONS

For a continuous convex function on an open convex subset of any Banach space a separability condition on its image under the subdifferential mapping is sufficient to guarantee the generic Frechet differentiability of the function. This gives a direct insight into the characterisation of Asplund spaces.

Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

CONTINUITY CHARACTERISATIONS OF DIFFERENTIABILITY OF LOCALLY LIPSCHITZ FUNCTIONS

Recently David Preiss contributed a remarkable theorem about the differentiability of locally Lipschitz functions on Banach spaces which have an equivalent norm differentiable away from the origin. Using his result in conjunction with Frank Clarkes non-smooth analysis for locally Lipschitz functions, continuity characterisations of differentiability can be obtained which generalise those for convex functions on Banach spaces. This result gives added information about differentiability properties of distance functions.

The numerical ranges of unbounded operators

On a Banach space the numerical range of an unbounded operator has a certain density property in the scalar field. Consequently all hermitian and dissipative operators are bounded. For a smooth or separable or reflexive Banach space the numerical range of an unbounded operator is dense in the scalar field.

Comparable differentiability characterisations of two classes of Banach spaces

We characterise Banach spaces not containing l 1 by a differentiability property of each equivalent norm and show that a slightly stronger differentiability property characterises Asplund spaces.

On the characterisation of Asplund spaces

A Banach space is an Asplund space if every continuous convex function on an open convex subset is Frechet differentiable on a dense G 8 subset of its domain. The recent research on the Radon-Nikodým property in Banach spaces has revealed that a Banach space is an Asplund space if and only if every separable subspace has separable dual. It would appear that there is a case for providing a more direct proof of this characterisation.

Comparable characterisations of two classes of Banach spaces by subdifferentials

We characterise Banach spaces not containing l 1 and Banach spaces which are Asplund spaces by continuity properties of the subdifferential mappings of their equivalent norms.