Related papers: Nearest point problem in countably normed spaces

Best approximation (BA) is an interesting field in functional analysis that has attracted a lot of attention from many researchers for a very long period of time up-to-date. Of greatest consideration is the characterization of the Chebyshev…

We show that a normed linear space is isometrically isomorphic to an inner product space if and only if it is a strongly $n$-point homogeneous metric space for any (or every) $n \geqslant 3$. The counterpart for $n=2$ is the Banach-Mazur…

We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular…

Let $\mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $\mathfrak{M}$-universal if every $X\in\mathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find…

Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We…

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of…

We revisit the definition of effective local compactness, and propose an approach that works for arbitrary countably-based spaces extending the previous work on computable metric spaces. We use this to show that effective local compactness…

This paper concerns the self-similarity of topological spaces, in the sense defined in math.DS/0411344. I show how to recognize self-similar spaces, or more precisely, universal solutions of self-similarity systems. Examples include the…

The countable uniform power (or uniform box product) of a uniform space $X$ is a special topology on ${}^{\omega}X$ that lies between the Tychonoff topology and the box topology. We solve an open problem posed by P. Nyikos showing that if…

It is a classical theorem of Alexandroff that a locally compact Hausdorff space has a one-point Hausdorff compactification if and only if it is non-compact. The one-point Hausdorff compactification is indeed obtained by adding the so called…

It is shown that a Banach space with locally uniformly convex dual admits an equivalent norm which is itself locally uniformly convex. It follows that on any such space all continuous real-valued functions may be uniformly approximated by…

In algorithms for finite metric spaces, it is common to assume that the distance between two points can be computed in constant time, and complexity bounds are expressed only in terms of the number of points of the metric space. We…

The property of almost every point being a Lebesgue point has proven to be crucial for the consistency of several classification algorithms based on nearest neighbors. We characterize Lebesgue points in terms of a 1-Nearest Neighbor…

The main purpose of this paper is to find the fixed point in such cases where existing literature remain silent. In this paper we introduce partial completeness, a new type of contraction and many other definitions. Using this approach the…

For a separable locally compact but not compact metrizable space $X$, let $\alpha X = X \cup \{x_\infty\}$ be the one-point compactification with the point at infinity $x_\infty$. We denote by $EM(X)$ the space consisting of admissible…

This paper proposes the problem of point-and-count as a test case to break the what-and-where deadlock. Different from the traditional detection problem, the goal is to discover key salient points as a way to localize and count the number…

We prove that the existence of best coapproximation to any element of the normed linear space out of any one dimensional subspace and its coincidence with the best approximation to that element out of that subspace characterizes a real…

We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n log log n) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric…

We give a partial solution to a question by Alas, Junqueria and Wilson by proving that under PFA the one-point compactification of a locally compact, discretely generated and countably tight space is also discretely generated. After this,…

In this paper, we study some topological characteristics of the n-normed spaces. We observe convergence sequences, closed sets, and bounded sets in the n-normed spaces using norms of quotient spaces that will be constructed. These norms…