A characterization of inner product spaces

Abstract

In 1935, Jordan and von Neumann characterized inner product spaces as normed linear spaces satisfying the property that in any parallelogram, the sum of the squares of the lengths of the diagonals is equal to the sum of the squares of the lengths of the sides. If the parallelogram is a rectangle, this reduces to the Pythagorean theorem. In 1943, Ficken proved that a normed linear space is an inner product space if, and only if, a reflection about a line in any two-dimensional subspace is an isometric mapping. In 1947, Lorch gave a list of six implications which characterize inner product spaces among normed linear spaces. In this paper, we prove a geometric characterization of inner product spaces which encompasses those already mentioned in the sense that the necessity of each of the previously mentioned characterizations is easily derived from the latter one without employing the inner product. An application to fixed point theorems is given in the last section of the paper.