Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 10.54216/GJSMA https://www.americaspg.com/journals/show/1329 2022 2022 Department Of Mathematics, Gaziantep University, Gaziantep, Turkey Mehmet .. Department Of Mathematics, Gaziantep University, Gaziantep, Turkey Necati Olgun 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 2022 08 13 10.54216/GJMSA.020101 https://www.americaspg.com/articleinfo/33/show/1329