Orthogonality in inner products is a binary relation that can be expressed in many ways without explicit mention to the inner product of the space.

Great part of such definitions have also sense in normed linear spaces. This simple observation is at the base of many concepts of orthogonality in these more general structures.

Various authors introduced such concepts over the last fifty years, although the origins of some of the most interesting results that can be obtained for these generalized concepts are, as usual in Mathematics, in previous of parallel works about convex sets, ellipsis or ellipsoids, duality, etc. (see, e.g., Gruber’s paper (1987) about the prior contributions of Carathéodory, Blaschke, or Radon).

1980 Mathematics Subject Classification (1985 revision): 46B20.