Elliptic distances in Hilbert spaces (Q1972869)

Let \(X_0\) denote the set of all nonzero vectors in a pre Hilbert space of dimension at least 2. The author determines all mappings \(f: X_0 \to X_0\) satisfying the functional equation \[ {\bigl|f(x)f(y) \bigr |\over\bigl \|f(x)\bigr \|\cdot \bigl\|f(y)\bigr \|}= {|xy|\over\|x\|\cdot\|y\|}\quad (x,y\in X_0) \] and also all \(f:X_0 \to X_0\) with \[ {f(x)f(y) \over\bigl\|f(x) \bigr\|\cdot \bigl\|f(y) \bigr \|}= {xy\over\|x\|\cdot\|y\|}\quad (x,y\in X_0). \] He also deals with elliptic and spherical geometry on the basis of the distance notions \[ \cos\varepsilon(x,y)={|xy|\over\|x\|\cdot\|y \|}\quad\text{and}\quad\cos\sigma(x,y)={xy\over\|x\|\cdot\|y \|}, \] respectively. Finally, he describes all the corresponding isometries of \(X_0\).