Related papers: On Vector Spaces with Formal Infinite Sums

Let D be a division ring with centre F. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In [1], authors have conjectured that every division ring is generated as a vector space over its centre by all of…

Let $V$ be an infinite-dimensional vector space over a field. In a previous article, we have shown that every endomorphism of $V$ splits into the sum of four square-zero ones but also into the sum of four idempotent ones. Here, we study…

This paper deals with some basic constructions of linear and multilinear algebra on finite-dimensional diffeological vector spaces. We consider the diffeological dual formally checking that the assignment to each space of its dual defines a…

We define formal orbifolds over an algebraically closed field of arbitrary characteristic as curves together with some branch data. Their \'etale coverings and their fundamental groups are also defined. These fundamental group approximates…

We consider a generalized angle in complex normed vector spaces. Its definition corresponds to the definition of the well known Euclidean angle in real inner product spaces. Not surprisingly it yields complex values as `angles'. This…

We give a complete classification in canonical forms on finite-dimensional vector spaces over the real numbers.

We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual…

We prove that every endomorphism of an infinite-dimensional vector space splits as the sum of four idempotents and as the sum of four square-zero endomorphisms, a result that is optimal in general.

We present bases for certain spaces of meromorphic vector-valued rational-weight mock modular forms constructed using Rademacher sums.

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes…

Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the…

We show that the definition of an algebraic basis for a vector space allows the construction of an isomorphism with the one here called Algebraic Vector Space. Although the concept does not bring anything new, we mention some of the…

A vector space partition $\mathcal{P}$ in $\mathbb{F}_q^v$ is a set of subspaces such that every $1$-dimensional subspace of $\mathbb{F}_q^v$ is contained in exactly one element of $\mathcal{P}$. Replacing "every point" by "every…

Suppose $(\mathcal{C},\mathbb{E},\mathfrak{s})$ is an $n$-exangulated category. We show that the idempotent completion and the weak idempotent completion of $\mathcal{C}$ are again $n$-exangulated categories. Furthermore, we also show that…

We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching…

We consider the problem of constructing matrices of linear forms of constant rank by focusing on the associated vector bundles on projective spaces. Important examples are given by the classical Steiner bundles, as well as some special…

We provide a new characterization of enriched accessible categories by introducing the two new notions of virtual reflectivity and virtual orthogonality as a generalization of the usual reflectivity and orthogonality conditions for locally…

We study ample stable vector bundles on minimal rational surfaces. We give a complete classification of those moduli spaces for which the general stable bundle is both ample and globally generated. We also prove that if $V$ is any stable…

Formal semantics and distributional semantics are distinct approaches to linguistic meaning: the former models meaning as reference via model-theoretic structures; the latter represents meaning as vectors in high-dimensional spaces shaped…

We develop a homotopy theory for additive categories endowed with endofunctors, analogous to the concept of a model structure. We use it to construct the homotopy theory of a Hovey triple (which consists of two compatible complete cotorsion…