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.
1. Introduction.
Let ( X , 11-11) be a real normed linear space. We define the angle A ( x , y) between vectors x, y c X as follows.
Definition 1.1. If x, y c X , x, y+O, define
The first characterization theorem is based on certain natural geometric pro- Definition 1.2. Letdenote any one of the relations s, = , or 2 .
i) A ( . , .) has the straight angle property if and only if perties of the angle A (-,-).
A(%, y) + A ( -x, y) -%5 for all x, EX, x, y.t.0, ii) A ( . ; ) is additive if and only if A (x, ax+by)+A ( a x f b y , y)-A(x, y) for all independent x, y in X, and all a, b > 0,
iii) A ( -; ) has the angle sum property if and only if A ( z , Y)+A (y, y-x)+A (x, x -y ) -n