[Paper Review] On soft linear spaces and soft normed linear spaces

Sujoy Das, Pinaki Majumdar+1 more|arXiv (Cornell University)|Aug 2, 2013

Fuzzy and Soft Set Theory16 references31 citations

TL;DR

This paper introduces soft linear spaces and soft normed linear spaces by extending soft set theory to functional analysis, defining soft vectors, soft norms, and studying completeness, equivalent norms, and convex soft sets. The key contribution is establishing foundational properties of soft normed linear spaces, including that every soft normed linear space is a soft metric space and that soft spheres and closures of convex soft sets are convex.

ABSTRACT

In this paper an idea of soft linear spaces and soft norm on soft linear spaces are given and some of their properties are studied. Soft vectors in soft linear spaces are introduced and their properties are studied. Completeness of soft normed linear space is also studied and equivalent soft norms and convex soft sets are studied in soft normed linear space settings.

Motivation & Objective

• To extend soft set theory to functional analysis by introducing soft linear spaces.
• To define soft vectors and study their properties within soft linear spaces.
• To introduce soft norms and investigate their properties, including completeness and equivalence.
• To explore convex soft sets in the context of soft normed linear spaces.
• To establish connections between soft normed linear spaces and soft metric spaces.

Proposed method

• Defining soft linear spaces as parametrized families of subsets over a universe, with operations like soft addition and scalar multiplication.
• Introducing soft vectors as functions from a parameter set to the universe, with component-wise operations.
• Defining a soft norm as a mapping from a soft linear space to soft real numbers satisfying norm axioms.
• Proving that every soft normed linear space induces a soft metric via the standard metric formula.
• Studying completeness using Cauchy soft sequences and their convergence in soft normed spaces.
• Analyzing equivalent soft norms and convex soft sets using soft real number parameters and segment definitions.

Experimental results

Research questions

• RQ1 How can soft set theory be extended to define linear algebraic structures such as soft linear spaces?
• RQ2 What are the defining properties of soft vectors and how do they behave under soft operations?
• RQ3 How can a soft norm be defined on a soft linear space, and what are its implications for metric structure?
• RQ4 Under what conditions is a soft normed linear space complete, and how does this relate to soft Cauchy sequences?
• RQ5 Are soft spheres and closures of convex soft sets themselves convex in soft normed linear spaces?

Key findings

• Every soft normed linear space is a soft metric space, as the soft norm induces a soft metric via the standard distance formula.
• Soft spheres in soft normed linear spaces are convex soft sets, as shown by the triangle inequality applied to soft convex combinations.
• The closure of any convex soft set in a soft normed linear space is also convex, preserving convexity under topological closure.
• The intersection of any family of convex soft sets in a soft normed linear space is convex, demonstrating stability under intersection.
• Soft norms can be equivalent, and such equivalence preserves topological and metric properties in soft normed linear spaces.
• Soft vectors in soft linear spaces satisfy component-wise linearity and compatibility with soft scalar multiplication, forming a parametrized vector-like structure.