Related papers: A characterization of inner product spaces
A generalisation of the Cassels and Greub-Reinboldt inequalities in complex or real inner product spaces and applications for isotonic linear functionals, integrals and sequences are provided.
Let $\Lambda^{\mathbb{R}}$ denote the linear space over $\mathbb{R}$ spanned by $z^{k}$, $k \! \in \! \mathbb{Z}$. Define the real inner product (with varying exponential weights) $\langle \boldsymbol{\cdot},\boldsymbol{\cdot}…
We show the following version of the Schur's product theorem. If $M=(M_{j,k})_{j,k=1}^n\in{\mathbb R}^{n\times n}$ is a positive semidefinite matrix with all entries on the diagonal equal to one, then the matrix $N=(N_{j,k})_{j,k=1}^n$ with…
Let (X,d,p) be a pointed metric space. A pretangent space to X at p is a metric space consisting of some equivalence classes of convergent to p sequences (x_n), x_n \in X, whose degree of convergence is comparable with a given scaling…
For a metrizable space $X$ of density $\kappa$, let $PM(X)$ be the space of continuous bounded pseudometrics on $X$ endowed with the uniform convergence topology. In this paper, its topology shall be classified as follows: (i) If $X$ is…
The Hilali conjecture claims that a simply connected rationally elliptic space $X$ satisfies the inequality $\operatorname{dim} (\pi_*(X)\otimes \mathbb Q ) \leqq \operatorname{dim} H_*(X;\mathbb Q )$. In this paper we show that for any…
Several characterizations are given for a square matrix that can be written as the product of two positive (semidefinite) projections. Based on one of these characterizations, and the theory of alternating projections, a Matlab program is…
An infinite real sequence $\{a_n\}$ is called an invariant sequence of the first (resp., second) kind if $a_n=\sum_{k=0}^n {n \choose k} (-1)^k a_k$ (resp., $a_n=\sum_{k=n}^{\infty} {k \choose n} (-1)^k a_k$). We review and investigate…
We give necessary and sufficient conditions for a family of inner products in a finite-dimensional vector space $V$ over an arbitrary field $\mathbb{K}$ to have an orthogonal basis relative to all the inner products. Some applications to…
We characterize the situations in which certain accumulation properties of topological spaces are preserved under taking products.
We consider the question of the boundedness of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of matrices, $A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, due to an appropriate choice of matrices $\{B_{i}\}$. It…
A topological space $X$ is called strongly $\sigma$-metrizable if $X=\bigcup_{n\in\omega}X_n$ for an increasing sequence $(X_n)_{n\in\omega}$ of closed metrizable subspaces such that every convergence sequence in $X$ is contained in some…
In the framework of nonstandard analysis, Bang-He Li and the author defined the product of any two distributions on $R^n$ via their harmonic representations. The product of $\delta (x_{1},..., x_{n})$ and $\delta (x_{1})$ was calculated by…
We present a preliminary study of Schur norms $\|M\|_{S}=\max\{ \|M\circ C\|: \|C\|=1\}$, where M is a matrix whose entries are $\pm1$, and $\circ$ denotes the entrywise (i.e., Schur or Hadamard) product of the matrices. We show that, if…
We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with…
We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if…
A metric measure space $(X,\mu)$ is 1-regular if \[0< \lim_{r\to 0} \frac{\mu(B(x,r))}{r}<\infty\] for $\mu$-a.e $x\in X$. We give a complete geometric characterisation of the rectifiable and purely unrectifiable part of a 1-regular measure…
Embedding methods for product spaces are powerful techniques for low-distortion and low-dimensional representation of complex data structures. Here, we address the new problem of linear classification in product space forms -- products of…
In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural…
It is shown that an ordered vector space $X$ is Archimedean if and only if $\inf\limits_{\tau\in\{\tau\}, y\in L}(x_\tau -y) \ = 0$ for any bounded decreasing net $x_\tau\downarrow$ in $X$, where $L$ is the collection of all lower bounds of…