Explore the vast range of titles available.

We prove that, for any countably compact subspace X of a Σ -product of real lines, the space C p ( X ) is uniformly ψ -separable, that is, has a uniformly dense subset of countable pseudocharacter. This result implies that C p ( K ) is uniformly ψ -separable whenever K is a Valdivia compact space. We show that the existence of a uniformly dense realcompact subset of C p ( X ) need not imply that C p ( X ) is realcompact even if the space X is compact. We also establish that C p ( X ) can fail to be ω -monolithic if it has a uniformly dense ω -monolithic subspace. Furthermore, an example is given of spaces X and Y such that both C p ( X ) and C p ( Y ) are Lindelöf but C p ( X × Y ) has no uniformly dense Lindelöf subspace.

We present a theory of backward stochastic differential equations in continuous time with an arbitrary filtered probability space. No assumptions are made regarding the left continuity of the filtration, of the predictable quadratic variations of martingales or of the measure integrating the driver. We present conditions for existence and uniqueness of square-integrable solutions, using Lipschitz continuity of the driver. These conditions unite the requirements for existence in continuous and discrete time and allow discrete processes to be embedded with continuous ones. We also present conditions for a comparison theorem and hence construct time consistent nonlinear expectations in these general spaces.

A space X is sequentially separable if there is a countable D ⊂ X such that every point of X is the limit of a sequence of points from D. Neither “sequential + separable” nor “sequentially separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.

We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Székely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hubert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hubert spaces.

We develop a version of Afriat's theorem that is applicable in a variety of choice environments beyond the setting of classical consumer theory. This allows us to devise tests for rationalizability in environments where the set of alternatives is not the positive orthant of a Euclidean space and where the rationalizing utility function is required to satisfy properties appropriate to that environment. We show that our results are applicable, amongst others, to choice data on lotteries, contingent consumption, and intertemporal consumption.

In this paper, we introduced Revised fuzzy cone Metric space with its topological properties. Likewise A necessary and sufficient condition for a Revised fuzzy cone metric space to be precompact is given. We additionally show that each distinct Revised fuzzy cone metric space is second countable and that a subspace of a separable Revised fuzzy cone metric space is separable.

A subset MM of a topological vector space XX is called lineable (respectively, spaceable) in XX if there exists an infinite dimensional linear space (respectively, an infinite dimensional closed linear space) Y⊂M∪{0}Y \\subset M\\cup \\{0\\}. In this article we prove that, for every infinite dimensional closed subspace XX of C[0,1]\\mathcal {C}[0,1], the set of functions in XX having infinitely many zeros in [0,1][0,1] is spaceable in XX. We discuss problems related to these concepts for certain subsets of some important classes of Banach spaces (such as C[0,1]\\mathcal {C}[0,1] or Müntz spaces). We also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of C[0,1]\\mathcal {C}[0,1], as well as oscillating and annulling properties of subspaces of C[0,1]\\mathcal {C}[0,1].

The spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data. In this paper, we investigate the concept of spatial distribution for data in infinite dimensional Banach spaces. Many technical difficulties are encountered in such spaces that are primarily due to the noncompactness of the closed unit ball. In this work, we prove some Glivenko-Cantelli and Donsker-type results for the empirical spatial distribution process in infinite dimensional spaces. The spatial quantiles in such spaces can be obtained by inverting the spatial distribution function. A Bahadur-type asymptotic linear representation and the associated weak convergence results for the sample spatial quantiles in infinite dimensional spaces are derived. A study of the asymptotic efficiency of the sample spatial median relative to the sample mean is carried out for some standard probability distributions in function spaces. The spatial distribution can be used to define the spatial depth in infinite dimensional Banach spaces, and we study the asymptotic properties of the empirical spatial depth in such spaces. We also demonstrate the spatial quantiles and the spatial depth using some real and simulated functional data.

We establish that a Čech-complete space X must be subcompact if it has a dense subspace representable as the countable union of closed subcompact subspaces of X . In particular, if a Čech-complete space contains a dense σ -compact subspace then it is subcompact. This result is new even for separable Čech-complete spaces. Furthermore, if X is a compact space of countable tightness then X \\ A is subcompact for any countable set A ⊂ X . We also show that any G δ -subset of a dyadic compact space is subcompact and give a comparatively simple proof of the fact that X \\ A is subcompact for any linearly ordered compact space X and any countable set A ⊂ X .

We characterize the metric spaces whose free spaces have the bounded approximation property through a Lipschitz analogue of the local reflexivity principle. We show that there exist compact metric spaces whose free spaces fail the approximation property.