CONTINUITY OF ORTHOGONAL SETS

In this note, we introduce the operator approach for orthogonality in linear spaces. In particular, we represent the concept of orthogonal vectors using an operator associated with them, in normed spaces. Moreover, we investigate some of continuity properties of this kind of orthogonality. More precisely, we show that the set valued function F(x; y) = {µ : µ ∈ C, p(x − µy, y) = 1} is upper and lower semi continuous, where ∈ X} and where P p(x, y) = inf {p x,z 1 ,...,z p z 1 n−2 ,...,z ,y z n−2 1 ,...,z n−2 (x, y) : z (x, y) = kP x,z 1 1 , . . . , z ,...,z n−2 ,y n−2 k −1 denotes the projection parallel to y from X to the subspace generated by {x, z }. This can be considered as an alternative definition for numerical range in linear spaces. 1 , . . . , z n−2