Examples, properties and applications of fuzzy inner product spaces

In this paper a new definition for the fuzzy inner product spaces of the type Kramosil-Michalek and the type George-Veeramani is presented. In the new setting a fuzzy inner product space can naturally become a fuzzy normed space, and a classical inner product space can be considered as a special case of fuzzy inner product spaces. Several examples are given to illustrate that, the new definition is a nontrivial generalization for classical inner product spaces, and so it has rich contents in fuzziness. By virtue of this definition, some elementary properties are described in terms of the families of semi-inner products and a fuzzy version of Pythagorean theorem is given. As applications, a fixed point theorem for nonlinear contractions is established and the existence of solution of global optimization problem is obtained.

Automated summary

This paper presents a new definition for fuzzy inner product spaces of the Kramosil-Michalek and George-Veeramani types. In the new setting, a fuzzy inner product space can naturally be a fuzzy normed space, and a classical inner product space can be considered as a special case of fuzzy inner product spaces. Several examples are given to illustrate that the new definition is a nontrivial generalization for classical inner product spaces, with rich contents in fuzziness. By using this definition, some elementary properties are described in terms of families of semi-inner products, and a fuzzy version of the Pythagorean theorem is provided. As applications, a fixed point theorem for nonlinear contractions is established, and the existence of a solution for a global optimization problem is obtained.