In the standard theorems on weak convergence of stochastic processes, it is invariably assumed that the parameter set is a bounded interval. The object of this paper is to indicate that analogues of these theorems for unbounded intervals are also valid. We shall confine our attention to the results of Skorohod [l], and in particular to those results concerning his Ji topology. Let £ be a comple...

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets. Several works have been devoted to deriving limit theorems for random sets. For i.i.d. random compact sets in R, the law of large numbers was initially proved by Artstein and Vitale [1] and the central limit theorem by Cressie [3...

0. The aim of this paper is to announce several results concerning smoothing Hubert manifolds and some of their consequences for homeomorphisms. All manifolds are assumed to be Hausdorff, paracompact, and separable with local model the oo-dimensional separable Hubert space H. (All separable infinite dimensional Fréchet spaces are homeomorphic to H.) As usual, we denote by DTM the unit disc and ...

To determine what spaces are the images of “nice” spaces under “nice” mappings is one of the central questions of general topology in 1 . In the past, many noteworthy results on images of metric spaces have been obtained. For a survey in this field, see 2 , for example. A characterization for a quotient compact image of a locally separable metric space is obtained in 3 . Also, such a quotient i...

We begin by recalling that if Γ is a group, andH a subgroup of Γ, then Γ is calledH-separable if for every g ∈ Γ\H, there is a subgroup K of finite index in Γ such that H ⊂ K but g / ∈ K. The group Γ is called subgroup separable (or LERF) if Γ is H-separable for all finitely generated subgroups H. As is well-known LERF is a powerful property in the setting of low-dimensional topology which has ...

Inspired by the recent theory of Entropy-Transport problems and $\mathbf{D}$-distance Sturm on normalised metric measure spaces, we define a new class complete separable distances between spaces possibly different total mass. We provide several explicit examples such distances, where prominent role is played geodesic based Hellinger-Kantorovich distance. Moreover, discuss some limiting cases th...

In connection with those paragraphs of my paper Applications of the theory of Boolean rings to general topology (Trans. Amer. Math. Soc. vol. 41 (1937) pp. 375-481) dealing with regular spaces, I have long been curious to know whether certain results proved there could be used to obtain the well known theorem that a separable (Hausdorff) space is metrizable if (and only if) it is regular. Since...

Let M⊂B(H) be a von Neumann algebra acting on the (separable) Hilbert space H. We first prove that M is finite if and only if, for every x∈M all vectors ξ,η∈H, coefficient function u↦⟨uxu∗ξ|η⟩ weakly almost periodic topological group UM of unitaries in (equipped with weak operator topology). The main device unique invariant mean C∗-algebra WAP(UM) functions UM. Next, we direct sum diffuse, abel...

This paper treats a holomorphic self-mapping f : Ω → Ω of a bounded domain Ω in a separable Hilbert space H with a fixed point p. In case the domain is convex, we prove an infinitedimensional version of the Cartan-Carathéodory-Kaup-Wu Theorem. This is basically a rigidity result in the vein of the uniqueness part of the classical Schwarz lemma. The main technique, inspired by an old idea of H. ...

For two not necessarily commutative topological groups G and T , let H(G, T ) denote the space of all continuous homomorphisms from G to T with the compact-open topology. We prove that if G is metrizable and T is compact then H(G, T ) is a k-space. As a consequence we obtain that if G1 is a dense subgroup of G then H(G1, T ) is homeomorphic to H(G,T ), and if G is separable h-complete, then the...