Neutrosophic Topological Vector Spaces and its Properties
E. Kungumaraj1, E. Lathanayagam2, Utpal Saikia3, M. Clement Joe Anand4, Sakshi Taaresh Khanna5, Nivetha Martin6, Mohit Tiwari7, Seyyed Ahmad Edalatpanah8
1Sakthi Institute of Information and Management Studies, Pollachi, Coimbatore, Tamil Nadu - 642001, India
2Akshaya College of Engineering and Technology, Kinathukadavu, Coimbatore, Tamil Nadu - 642109, India.
3Department of Mathematics, Silapathar College, Dhemaji, Assam – 787059, India.
4Department of Mathematics, Mount Carmel College (Autonomous), Affiliated to Bengaluru City University, Bengaluru - 560052, Karnataka, India.
5Department of Computer Science, Ram Lal Anand College, University of Delhi- 110021, Delhi, India.
6Department of Mathematics, Arul Anandar College (Autonomous), Karumathur-625514, Tamil Nadu, India.
7Department of Computer Science and Engineering, Bharati Vidyapeeth’s College of Engineering, Delhi -110063, Delhi, India.
8Department of Applied Mathematics, Ayandegan Institute of Higher Education, Tonekabon, Iran.
Emails: kungum99522@gmail.com; lathashrilishanth@gmail.com; utpalsaikiajorhat@gmail.com; arjoemi@gmail.com; sakshitaareshkhanna@gmail.com; nivetha.martin710@gmail.com; mohit.tiwari@bharatividyapeeth.edu; sayyed.edalatpanah48@gmail.com
Abstract
The algebraic structures Group, Ring, Field and Vector spaces are important innovations in Mathematics. Most of the theoretical concepts of Mathematics are based on the theorems related to these algebraic structures. Initially many mathematicians developed theorems related to all these algebraic structures. In 20th century most of the researchers introduced the theorems on the algebraic structures with Fuzzy and Intuitionistic fuzzy sets. Recently in 21st century the researchers concentrated on Neutrosophic sets and introduced the algebraic structures like Neutrosophic Group, Neutrosophic Ring, Neutrosophic Field, Neutrosophic Vector spaces and Neutrosophic Linear Transformation. In the current scenario of relating the spaces with the structures, we have introduced the concepts of Neutrosophic topological vector spaces. In this article, the study of Neutrosophic Topological vector spaces has been initiated. Some basic definitions and properties of classical vector spaces are generalized in Neutrosophic environment over a Neutrosophic field with continuous functions. Neutrosophic linear transformations and their properties are also included in Neutrosophic Topological Vector spaces. This article is an extension work of fuzzy and intuitionistic fuzzy vector spaces which were introduced in fuzzy and intuitionistic fuzzy environments. Even though it is an extension work, Neutrosophic Topological Vector space will play an important role in Neural Networks, Image Processing, Machine Learning and Artificial Intelligence Algorithms.
Keywords: Topological Vector Spaces; Fuzzy, Intuitionistic Fuzz and Neutrosophic Topological Vector space; Neutrosophic continuous function; Neutrosophic proper function; Neutrosophic Linear Transformation.