[Paper Review] Cone Normed Linear Spaces

T. K. Samanta, Sanjay Roy+1 more|arXiv (Cornell University)|Sep 11, 2010

Approximation Theory and Sequence Spaces3 citations

TL;DR

This paper introduces cone normed linear spaces by defining a cone norm on a real vector space using a cone in a Banach space, establishes cone convergence and Cauchy sequences, and proves that in finite-dimensional cone normed linear spaces with a normal cone, a subset is compact if and only if it is closed and bounded—extending classical topological properties to cone-based norms.

ABSTRACT

In this paper, we introduce cone normed linear space, study the cone convergence with respect to cone norm. Finally, we prove the completeness of a finite dimensional cone normed linear space.

Motivation & Objective

• To formalize the concept of a cone normed linear space by generalizing the notion of norm using a cone in a Banach space.
• To define and study cone convergence and cone Cauchy sequences in this new framework.
• To investigate the topological properties of cone normed linear spaces, particularly boundedness, closedness, and compactness.
• To establish the equivalence between compactness, closedness, and boundedness in finite-dimensional cone normed linear spaces with a normal cone.
• To extend classical functional analytic results—such as completeness and sequential compactness—to the cone normed setting.

Proposed method

• Define a cone norm $\|\cdot\|_c: V \to E$ on a real vector space $V$, where $E$ is a Banach space and $C \subseteq E$ is a normal cone.
• Introduce cone convergence: $x_n \to x$ if for every $\epsilon \gg \theta$, there exists $n_0$ such that $\|x_n - x\|_c \ll \epsilon$ for all $n \geq n_0$.
• Define cone Cauchy sequences analogously: $\|x_n - x_m\|_c \ll \epsilon$ for all $m,n \geq n_0$.
• Use the normal constant $K$ of the cone to relate cone norm convergence to convergence in the underlying Banach space norm via $\|\|x_n - x\|_c\| \leq K\|\epsilon\|$.
• Prove sequential compactness in finite-dimensional spaces by expressing vectors in a basis, bounding coefficients via the cone norm, and applying the Bolzano-Weierstrass theorem.
• Establish that in finite-dimensional spaces, compactness is equivalent to being closed and bounded, using the cone norm and properties of the interior of the cone.

Experimental results

Research questions

• RQ1 How can the concept of a norm be generalized using a cone in a Banach space to define a cone normed linear space?
• RQ2 What conditions ensure that cone convergence implies convergence in the underlying Banach space norm, and vice versa?
• RQ3 Under what conditions is a subset of a finite-dimensional cone normed linear space compact?
• RQ4 Is the property of being closed and bounded sufficient for compactness in finite-dimensional cone normed linear spaces?
• RQ5 How do algebraic and topological properties (e.g., completeness, continuity) behave under cone norms compared to standard norms?

Key findings

• Cone convergence in a cone normed linear space is equivalent to the convergence of the cone norm values to the zero element $\theta$ in the underlying Banach space.
• In a finite-dimensional cone normed linear space with a normal cone, a subset is compact if and only if it is closed and bounded.
• Every Cauchy sequence in a cone normed linear space is bounded, generalizing a standard result from normed spaces.
• The cone norm satisfies the triangle inequality and homogeneity, and the reverse triangle inequality holds: $|\|x\|_c - \|y\|_c| \leq \|x - y\|_c$ in the order induced by the cone.
• The sum and scalar multiplication of convergent or Cauchy sequences in a cone normed space remain convergent or Cauchy, respectively, preserving sequential structure.
• The topology induced by the cone norm is compatible with the order structure of the cone, and the interior of the cone plays a key role in defining open balls and convergence.