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Dense and subspace dense subsets in finite-dimensional spaces

Salah Herzi, Habib Marzougui

Abstract

This note is motivated by the article of Bamerni, Kadets and Kiliçman [J. Math. Anal. Appl. 435 (2), 1812--1815 (2016)]. We consider the remaining problem which claims that if $A$ is a dense subset of a finite dimensional space $X$, then there is a nontrivial subspace $M$ of $X$ such that $A\cap M$ is dense in $M$. We show that the above problem has a negative answer when $X=\mathbb{K}^{n}$ ($\mathbb{K}= \mathbb{R}$ or $\mathbb{C}$) for every $n\geq 2$.

Dense and subspace dense subsets in finite-dimensional spaces

Abstract

This note is motivated by the article of Bamerni, Kadets and Kiliçman [J. Math. Anal. Appl. 435 (2), 1812--1815 (2016)]. We consider the remaining problem which claims that if is a dense subset of a finite dimensional space , then there is a nontrivial subspace of such that is dense in . We show that the above problem has a negative answer when ( or ) for every .

Paper Structure

This paper contains 2 sections, 7 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Subspace dense subset of $\mathbb{K}^{n}$

Key Result

Theorem 2.1

Let $n\geq 2$ be an integer and let $\alpha=(\alpha_{1},\dots, \alpha_{n})$ be an $n$-tuple of negative real numbers such that $1, \alpha_{1},\dots, \alpha_{n}$ are linearly independent over $\mathbb{Q}$. Set $A_{\alpha}=\mathbb{N}^{n}+\mathbb{N}[\alpha_{1},\dots, \alpha_{n}]^{T}$. Then $A_{\alpha}$

Theorems & Definitions (11)

  • Theorem 2.1: The case $X=\mathbb{R}^{n}$
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4: The case $X=\mathbb{C}^{n}$
  • Lemma 2.5: Kronecker's theorem: complex version
  • proof
  • proof : Proof of Theorem \ref{['t24']}
  • Proposition 2.6: Counterexample in $\mathbb{C}^{2}$
  • proof
  • Proposition 2.7: Counterexample in $\mathbb{R}^{2}$
  • ...and 1 more