Charles R. Diminnie

2-INNER PRODUCT SPACES II1)

This paper i s ail extension of [ l ] . We continue the i n vest igat ions begun there concerning 2-inner product spaces (2-pre-Hilbert spaces) and deal especial ly with orthogonal and orthonormal s e t s in such spaces. The reader should consult [1] for def in i t ions and notations. Let (L, (. , . I . )) be a 2-inner product space and II . , . II the associated 2-norm. L e m m a 1. For a , b , c e L ( a , b l a ) = 0 and o (a,b|yc) = f ( a , b l c ) , jr r e a l .

An extension of pythagorean and isosceles orthogonality and a characterization of inner-product spaces

Abstract A new orthogonality relation in normed linear spaces which generalizes pythagorean orthogonality and isosceles orthogonality is defined, and it is shown that the new orthogonality is homogeneous (additive) if and only if the space is a real inner-product space.

Pythagorean euclidean four point properties

Characterizations of real inner product spaces among a class of metric spaces have been obtained based on homogeneity of metric pythagorean orthogonality, a metrization of the concept of pythagorean orthogonality as defined in normed linear spaces. In the present paper a considerable weakening of this hypothesis is shown to characterize real inner product spaces among complete, convex, externally convex metric spaces, generalizing a result of Kapoor and Prasad [9], and providing a connection with the many characterizations of such spaces using euclidean four point properties.