We show that if $E$ is a separable reflexive space, and $L$ is a weak-star closed linear subspace of $L\left( E \right)$ such that $L\cap K\left( E \right)$ is weak-star dense in $L$ , then $L$ has a unique isometric predual. The proof relies on basic topological arguments.

The microdynamics of water and oil in macroporous media with SiO2 or CaCO3 surfaces has been probed at various temperatures by magnetic field-cycling measurements of the spin-lattice relaxation rates. These measurements and an original theory of surface diffusion allowed us to obtain surface dynamical parameters, such as a coefficient of surface affinity of the liquid molecules and the surface diffusion coefficient. The water surface diffusion coefficients are compared to the volume self-diffusion coefficients of water in pores, measured by PGSE method, the latter values being more than an order of magnitude higher than the surface ones. Complementary information on the nature of the solid-liquid interface was given by NMR chemical shift experiments at high magnetic field.

Let K be an arbitrary compact space and C(K) the space of continuous functions on K endowed with its natural supremum norm. We show that for any subset B of the unit sphere of C(K)* on which every function of C(K) attains its norm, a bounded subset A of C(K) is weakly compact if, and only if, it is compact for the topology tp(B) of pointwise convergence on B. It is also shown that this result can be extended to a large class of Banach spaces, which contains, for instance, all uniform algebras. Moreover we prove that the space (C(K), tp(B)) is an angelic space in the sense of D. H. Fremlin.

Norms with moduli of smoothness of power type are constructed on spaces with the Radon-Nikodym property that admit pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type. It is shown that no norms with pointwise moduli of rotundity of power type can exist on nonsuperreflexive spaces. A new smoothness characterization of spaces isomorphic to Hilbert spaces is given.

We show that if X is a separable Banach space such that X* fails the weak* convex point-of-continuity property (C*PCP), then there is a subspace Y of X such that both Y* and (X/Y)* fail C*PCP and both Y and X/Y have finite dimensional Schauder decompositions.

A norm |⋅| on a Banach space X is locally uniformly rotund (LUR) if lim |x n — x| = 0 for every x n , x ∈ X for which lim2|x|2 + 2 |x n |2-|xn+x n|2 = 0. It is shown that a Banach space X admits an equivalent LUR norm provided there is a bounded linear operator T of X into c0(Γ) such that T* c*0(Γ) is norm dense in X*. This is the case e.g. if X* is weakly compactly generated (WCG).