Abstract
In this work, two different structures are proposed which is fuzzy real normed space (FRNS) and fuzzy real Pre-Hilbert space (FRPHS). The basic concept of fuzzy norm on a real linear space is first presented to construct space, which is a FRNS with some modification of the definition introduced by G. Rano and T. Bag. The structure of fuzzy real Pre-Hilbert space (FRPHS) is then presented which is based on the structure of FRNS. Then, some of the properties and related concepts for the suggested space FRN such as -neighborhood, closure of the set named , the necessary condition for separable, fuzzy linear manifold (FLM) are discussed. The definition for a fuzzy seminorm on is also introduced with the prove that a fuzzy seminorm on is FRNS. The relationship between the -convergent sequence, -Cauchy sequence and -completeness is then investigated in this work. The structure of FRPHS with some important properties concerning on this space are introduced and proved. In addition, the property of orthogonality with some important properties for these spaces is included, for example the annihilator of the set . The relation between FRPHS and FRNS is investigated in the present work. Finally, after introducing the structure of FRPHS, it leads naturally to the definition of the most important class of FRPHS, namely the fuzzy real Hilbert space (FRHS).
Highlights
The concept of fuzzy set is a generalization of classical set theory which is first introduced by zadeh 1.Triangular conorms are an indispensable tool for the interpretation of the conjunction in fuzzy sets
A real linear space is considered and a structure of fuzzy normed space is introduced with a new characterization triangular conorm
Definition 2 Let Δc is a triangular conorm, V be a linear space over the field R .Let a fuzzy subset
Summary
The concept of fuzzy set is a generalization of classical set theory which is first introduced by zadeh 1.Triangular conorms are an indispensable tool for the interpretation of the conjunction in fuzzy sets. Later on many author viz. Mazumdar & Samanta 18, Hasankhani, Nazari & Saheli 19, Mukherjee & Bag 20 have introduced the structure of fuzzy Pre-Hilbert space from different points and studied a few properties and some of applications. Induced norms of these spaces may have very important applications in quantum particle physics for more details; see 21, 22.
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