A Characterization of Inner Product Spaces

Abstract

The object of this note is to show that if a lIall is a Euclidean norm, then it is the norm arising from an inner product on A. (The converse is obvious.) It follows that a Euclidean norm does satisfy the triangle inequality, although this was not assumed in the definition. (We remark that in Euclid, the triangle inequality is deduced, not assumed; see [7, Book I, Proposition 20].) Note that E2 may be regarded as asserting that the side-angle-side rule of congruence for triangles holds in the very special case where the two corresponding angles are not merely congruent but actually coincide. See Figure 1. The restriction t > 1 is included in E2 only to make this geometric interpretation as clear as possible. It follows trivially from El and E2 that if hlall = llbil, then IIua + vbII = IlIva + ubiI for each u, v E St. Ficken [5] showed that if the latter condition holds in a normed linear space, then the norm arises from an inner product. Our result improves on his in that we do not assume the triangle inequality. In addition, the proof we present is shorter than his. We wish to thank the referees for suggesting improvements in the presentation.