Properties of Isosceles, Pythagorean and Isosceles-Pythagorean Vectors in the Characterization of Hilbert Spaces

Abstract

All Hilbert spaces are Banach spaces but the converse is not necessarily true. Characterization of Banach spaces as Hilbert spaces has had different approaches for various Banach spaces. It has been shown that a separable Banach space which is almost transitive with vector orthogonalities for dimension greater than three is a Hilbert space. It worthy to note that micro transitivity together with Isosceles (I), Pythagorean (P) and Isosceles Pythagorean (IP) orthogonalities in the unit sphere have some essential properties that can be considered in characterization of Hilbert spaces. In this study, separable micro transitive Banach spaces are examined and their characterization as Hilbert spaces is achieved by applying the I-vector property in affine sets along with the P and IP-vector properties. In particular, by letting a separable Banach space 𝑋 of 𝑑𝑖𝑚𝑋 ≥ 2 possessing micro transitivity property with I, P, and IP vectors, then 𝑋 is a Hilbert space. The results of this research are expected to be useful in algebra and differential operators, particularly for calculating wave functions and formulation of theory