Certain corrections are needed in this paper concerning the definition and properties of the weak topology on 12. Instead of characterising it in terms of converging sequences as on p. 7, one should define the weak topology as the coarsest topology for which the evaluations f --+ f o h(t)dr(t), f E ]2, are continuous for all continuous h(t) of compact support. With this topology, 12 is a Tihono...

where 1 sp < co. This definition is far from being intrinsic. For an intrinsic definition of WrTp(Mm, N”) see [ 11. In this space, beside the standard topology induced by the norm (1. ]ll,p, we also have weak topology and weak convergence. Let fk, f E W’~p(Mm), where 1 < p < co. We say that fk converges to f in weak topology iff fk + f in Lp and the set (lIVfk[lp)k is bounded. Weak convergence ...

We prove that in a certain class E of nonseparable Banach spaces the norm topology of the dual ball is definable in terms of its weak∗ topology. Thus, if X, Y ∈ E and f : BX∗ −→ BY ∗ is a weak ∗-to-weak∗ homeomorphism, then f is automatically norm-to-norm continuous.

We investigate possible preduals of the measure algebra M(G) of a locally compact group and the Fourier algebra A(G) of a separable compact group. Both of these algebras are canonically dual spaces and the canonical preduals make the multiplication separately weak∗-continuous so that these algebras are dual Banach algebras. In this paper we find additional conditions under which the preduals C0...

We prove that a continuous linear operator T on a topological vector space X with weak topology is mixing if and only if the dual operator T ′ has no finite dimensional invariant subspaces. This result implies the characterization of hypercyclic operators on the space ω due to Herzog and Lemmert and implies the result of Bayart and Matheron, who proved that for any hypercyclic operator T on ω, ...

The topology of weak lensing fields is studied using the 2-dimensional genus statistic. Simulated fields of the weak lensing convergence are used to focus on the effect of nonlinear gravitional evolution and to model the statistical errors expected in observational surveys. For large smoothing angles, the topology is in agreement with the predictions from linear theory. On smoothing angles smal...

We introduce a new topological semantics for belief logics in which the belief modality is interpreted as the interior of the closure of the interior operator. We show that the system wKD45, a weakened version of KD45, is sound and complete w.r.t. the class of all topological spaces. Moreover, we point out a problem regarding updates on extremally disconnected spaces that appears in the setting...