Related papers: Characterization of inner product spaces

Finite dimensional linear spaces (both complex and real) with indefinite scalar product [.,.] are considered. Upper and lower bounds are given for the size of an indecomposable matrix that is normal with respect to this scalar product in…

Let $\mathbb{F}$ be a field, and $n \geq p \geq r>0$ be integers. In a recent article, Rubei has determined, when $\mathbb{F}$ is the field of real numbers, the greatest possible dimension for an affine subspace of $n$--by--$p$ matrices…

We establish the optimal lower bound $\gtrsim N$ for counting the number of distinct inner products of pairs from any $N$ given vectors in $\R^2$. Essentially, we lift a related incidence structure defined by inner products in the plane to…

We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension $m\geq 3$ of a product $\Q_{k_1}^{n_1}\times…

A (bar-and-joint) framework is a set of points in a normed space with a set of fixed distance constraints between them. Determining whether a framework is locally rigid - i.e. whether every other suitably close framework with the same…

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and…

We introduce and study a strict monotonicity property of the norm in solid Banach lattices of real functions that prevents such spaces from having the local diameter two property. Then we show that any strictly convex 1-symmetric norm on…

In this paper, we consider the linear direct sum of a real normed linear space with an order unit space and with a base normed space to obtain respectively a new order unit space and a new base normed space. As a consequence, we find that…

Moser's Bernstein theorem \cite{moser61} says that an entire minimal graph of codimension 1 with bounded slope must be a hyperplane. An analogous result for arbitrary codimension is not true, by an example of Lawson-Osserman. Here, we show…

We prove several cases of Zimmer's conjecture for actions of higher-rank cocompact lattices on low dimensional manifolds. For example, if $\Gamma$ is a cocompact lattice in $\mathrm{Sl}(n, \mathbb R)$, $M$ is a compact manifold, and…

Gromov and Sormani conjectured that a sequence of three dimensional Riemannian manifolds with nonnegative scalar curvature and some additional uniform geometric bounds should have a subsequence which converges in some sense to a limit space…

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$, if for every $\epsilon > 0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less…

Let $R$ be a semilocal principal ideal domain. Two algebraic objects over $R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all…

We provide examples of nonseparable compact spaces with the property that any continuous image which is homeomorphic to a finite product of spaces has a maximal prescribed number of nonseparable factors.

A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…

Characterizations are given for 1-complemented hyperplanes of strictly monotone real Lorentz spaces and 1-complemented finite codimensional subspaces (which contain at least one basis element) of real Orlicz spaces equipped with either…

We characterize the finite dimensional asymmetric normed spaces which are right bounded and the relation of this property with the natural compactness properties of the unit ball, as compactness and strong compactness. In contrast with some…

The $\Lambda$-sequence spaces $\Lambda_p$ for $1< p\leq\infty$ and its generalization $\Lambda_{\hat{p}}$ for $1<\hat{p}<\infty$, $\hat{p}=(p_n)$ is introduced. The James constants and strong $n$-th James constants of $\Lambda_p$ for…

When performing classification tasks, raw high dimensional features often contain redundant information, and lead to increased computational complexity and overfitting. In this paper, we assume the data samples lie on a single underlying…

Let $A$ be a simple algebra over a field $F$. Under a mild cardinality assumption on $F$, we determine the greatest possible dimension for an $F$-affine subspace of $A$ that is included in the group of units $A^\times$, and we describe the…