aspg-33-1329-1792223363 20261017074923 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 2022 2 1 Mehmet Celik Department Of Mathematics, Gaziantep University, Gaziantep, Turkey Necati Olgun Department Of Mathematics, Gaziantep University, Gaziantep, Turkey Banach and Hillbert spaces are the main important concepts in the study of classical functional analysis. This paper generalizes these two kinds of functional spaces into neutrosophic systems, where the concept of neutrosophic Banach space and neutrosophic Hillbert space will be defined and discussed for the first time over partial ordered neutrosophic spaces. Also, many related concepts such as neutrosophic Cauchy sequence, neutrosophic Bessel's inequality, and neutrosophic Parseval's identity will be established and proved. 2022 08 13 1329 https://creativecommons.org/licenses/by/4.0/ 10.54216/GJMSA.020101 https://americaspg.com/journal/33/article/1329 https://americaspg.com/storage/7241.pdf