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The objective of this work is to define for the first time the concept of inner product in a neutrosophic vector space and a refined neutrosophic vector space. Also, this paper introduces many interesting properties of neutrosophic real inner products and establishes the theoretical foundations of neutrosophic functional analysis such as neutrosophic normed spaces, refined neutrosophic normed spaces, neutrosophic real inner product spaces, refined neutrosophic inner product spaces, orthogonal basis, neutrosophic and refined neutrosophic Cauchy- Schwartz inequality.