[Paper Review] A New View on Soft Normed Spaces

Tunay Bılgın, Sadi Bayramov+2 more|arXiv (Cornell University)|Mar 20, 2014

Fuzzy and Soft Set Theory3 citations

TL;DR

This paper introduces a novel framework for soft normed linear spaces using soft points, redefining soft vector spaces and soft norms over real parameter sets. It establishes foundational properties of soft norms and proves key results on soft continuous linear operators, including operator norm equivalence and submultiplicativity, providing a rigorous algebraic-structural foundation for soft functional analysis.

ABSTRACT

In this paper, we work on the structure of soft linear spaces over a field K and investigate some of its properties. Here, we use the concept of the soft point which was introduced in [2,6]. We then introduce the soft norm in soft linear spaces. Finally, we examine the properties of this soft normed space and present some investigations about soft continuous operators in the space.

Motivation & Objective

• To redefine soft vector spaces using soft points for improved structural clarity.
• To introduce a new definition of soft norm in soft linear spaces based on soft points.
• To investigate properties of soft normed spaces and soft continuous linear operators.
• To establish operator norm equivalence and submultiplicativity in soft operator theory.

Proposed method

• Uses soft points (denoted $\tilde{x}_e$) as fundamental elements in soft vector spaces, where each point is defined by a unique parameter $e \in \mathbb{R}$ and value $x \in X$.
• Defines soft vector space operations: $\tilde{x}_e + \tilde{y}_{e'} = \widetilde{(x+y)}_{(e+e')}$ and $\tilde{r} \cdot \tilde{x}_e = \widetilde{(rx)}_{(re)}$ for soft real numbers $\tilde{r}$.
• Introduces soft norm $\|\tilde{x}_e\|$ as a mapping from $SV(\tilde{X})$ to non-negative soft real numbers, satisfying standard norm axioms.
• Defines soft operator norm as $\|T\| = \sup_{\|\tilde{x}_e\| \leq \tilde{1}} \|T(\tilde{x}_e)\|$, proving equivalence to $\sup_{\tilde{x}_e \neq \tilde{\theta}_0} \frac{\|T(\tilde{x}_e)\|}{\|\tilde{x}_e\|}$.
• Proves that the soft operator norm satisfies $\|T\| \geq \tilde{0}$, $\|\tilde{r}T\| = |\tilde{r}|\|T\|$, and $\|T + S\| \leq \|T\| + \|S\|$.
• Establishes submultiplicativity: $\|S \circ T\| \leq \|S\|\|T\|$ and $\|T^n\| \leq \|T\|^n$ for soft operators on the same space.

Experimental results

Research questions

• RQ1 How can soft vector spaces be redefined using soft points to ensure algebraic consistency and vector space structure?
• RQ2 What are the necessary and sufficient conditions for a soft norm to satisfy standard norm axioms in soft linear spaces?
• RQ3 How can soft continuous linear operators be characterized via a well-defined soft operator norm?
• RQ4 What are the key algebraic properties of the soft operator norm, such as homogeneity and triangle inequality?
• RQ5 Does the soft operator norm satisfy submultiplicativity under composition?

Key findings

• The soft vector space $SV(\tilde{X})$ forms a proper vector space under the defined operations of soft vector addition and scalar multiplication.
• The soft norm $\|\tilde{x}_e\|$ satisfies all standard norm axioms: non-negativity, homogeneity, and triangle inequality.
• The soft operator norm is equivalent to the supremum of $\|T(\tilde{x}_e)\| / \|\tilde{x}_e\|$ over all non-zero soft vectors.
• The soft operator norm satisfies $\|T\| = \sup_{\|\tilde{x}_e\| \leq \tilde{1}} \|T(\tilde{x}_e)\|$, proving a standard characterization in functional analysis.
• The soft operator norm satisfies $\|S \circ T\| \leq \|S\|\|T\|$, establishing submultiplicativity.
• For any soft operator $T$, $\|T^n\| \leq \|T\|^n$, confirming consistency with classical operator norm behavior.