A characterization of inner product spaces based on a property of height's transformation (Q1842030)

The main result offered by the authors is the following. Let \(E\) be a real normed space of dimension greater than 1 and (as ``candidate for inner product'') \[ \sigma (x,y) : = \lim_{t \to 0 + } \bigl( \| x + ty \|^ 2 - \| x \|^ 2 \bigr) / (2t). \] Then \(f : E \to E\) with \(f(x) = 0\) iff \(x = 0\) satisfies \[ f \bigl( y - \| x \|^{-2} \sigma (x,y)x \bigr) = f(y) - \bigl \| f(x) \bigr \|^{-2} \biggl( \bigl \| f(y) \bigr \|^ 2 - \sigma \bigl[ f(y) - f(x), f(y) \bigr] \biggl) f(x) \] if, and only if, \(E\) is an inner product space and \(\| f(x) \| = \gamma \| x \|\) \((\gamma > 0\) constant).