Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A natural correspondence between quasiconcave functions and fuzzy norms

Javier Cabello Sánchez, Daniel Morales González

Abstract

In this note we show that the usual notion of fuzzy norm defined on a linear space is equivalent to that of quasiconcave function, in the sense that every fuzzy norm $N:X\times\mathbb{R}[0,1]$ defined on a (real or complex) linear space X is uniquely determined by a quasiconcave function $f:X\to[0, 1]$. We explore the minimum requirements that we need to impose to some quasiconcave function $f:X\to[0, 1]$ in order to define a fuzzy norm $N:X\times\mathbb{R}[0,1]$. Later we use this equivalence to prove some properties of fuzzy norms, like a generalisation of the celebrated Decomposition Theorem.

A natural correspondence between quasiconcave functions and fuzzy norms

Abstract

In this note we show that the usual notion of fuzzy norm defined on a linear space is equivalent to that of quasiconcave function, in the sense that every fuzzy norm defined on a (real or complex) linear space X is uniquely determined by a quasiconcave function . We explore the minimum requirements that we need to impose to some quasiconcave function in order to define a fuzzy norm . Later we use this equivalence to prove some properties of fuzzy norms, like a generalisation of the celebrated Decomposition Theorem.
Paper Structure (6 sections, 8 theorems, 16 equations)

This paper contains 6 sections, 8 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Applications
  4. Finite-dimensional spaces
  5. Decomposition Theorem
  6. Concluding comments and remarks

Key Result

Theorem 2.1

The map is bijective.

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 1.2: bag2003finite
  • Remark 1.3
  • Example 1.4: Font2017
  • Definition 1.5
  • Theorem 2.1: Characterisation of fuzzy norms
  • proof
  • Proposition 3.1: Crou
  • Theorem 3.2: Crou
  • Definition 3.3: bag2005bounded, p. 536
  • ...and 13 more