Angles in Normed Linear Spaces and a Characterization of Real Inner Product Spaces

Abstract

A number of writers have defined a concept of angle in a normed linear space or metric space by means of the law of cosines, and have studied the properties of these angles obtaining, in some cases, characterizations of real inner product spaces.(For a summary of earlier results see MARTIN and VALENTINE [5] or VALENTINE and WAYMENT [7]). In the present paper a concept of angle is defined in a similar way, referring to unit vectors rather than arbitrary vectors, and it is shown that if the notion of angle satisfies certain natural geometric properties the space must be a real inner product space. Finally the related notion of orthogonality is considered, and a counterexample is provided to show that, in contrast to the results of JAMES [3] and the present authors [2], among others, the assumption that this orthogonality is homogeneous does not characterize real inner product spaces.

An orthogonality in normed linear spaces based on angular distance inequality

“…The concept of angle between two vectors in a normed linear space has been introduced in different ways [8,12], so that they coincide with the standard definition of angle in inner product spaces. Now inspired by the new approach to the Singer orthogonality we define acute and obtuse angles in normed linear spaces as follows:…”

“…Dimmine et al [8] showed that this notion of orthogonality is not homogeneous in general. He also mentioned the problem of whether the additivity of the orthogonality characterizes inner product spaces or not.…”

“…We show that in the two dimensional case there exists a strictly convex normed linear space which is not an inner product space, but the orthogonality ⊥ w is additive. In fact, by considering the example which was mentioned by Diminnie et al [8], we have a strictly convex normed linear space which is not an inner product space, but the orthogonality ⊥ w is homogeneous (see [8, p. 203]). Now we can see that the orthogonality ⊥ w is also additive in this space.…”

A formula for the g-angle between two subspaces of a normed space

“…In particular, the angle θ = θ(x, y) between two nonzero vectors x and y in X is defined by cos θ := x,y x y where x := x, x 1 2 denotes the induced norm on X. One may observe that the angle θ in X satisfies the following basic properties (see [3]). In a normed space, the concept of angles between two vectors has been studied intensively (see, for instance, [1,2,5,8,11,12,13]).…”

“…One may observe that the angle θ in X satisfies the following basic properties (see [3]). In a normed space, the concept of angles between two vectors has been studied intensively (see, for instance, [1,2,5,8,11,12,13]). Here we shall be interested in the notion of angles between two subspaces of a normed space using a semi-inner product.…”