Inner product spaces and Stewart’s relation. (English) Zbl 0879.46005
A characterization of inner product spaces is given using Stewart’s equality:
For every three elements \(x\), \(y\), \(z\) in \(X\) and every \(u\in[y,z]\) one has \(xy^2\cdot uz+ xz^2\cdot uy- xu^2\cdot yz= uy\cdot uz\cdot yz\).
Here \([x,y]= \{(1- t)x+ ty\mid t\in [0,1]\}\) and \(xy= |x-y|\).
For every three elements \(x\), \(y\), \(z\) in \(X\) and every \(u\in[y,z]\) one has \(xy^2\cdot uz+ xz^2\cdot uy- xu^2\cdot yz= uy\cdot uz\cdot yz\).
Here \([x,y]= \{(1- t)x+ ty\mid t\in [0,1]\}\) and \(xy= |x-y|\).
MSC:
| 46C15 | Characterizations of Hilbert spaces |
