×

On a characterisation of inner product spaces. (English) Zbl 1016.46024

Summary: It is well known that for a Hilbert space \(H\) the minimum value of the functional \(F_\mu(f) = \int_H\|f-g\|^2 d\mu(g)\), \(f\in H\), is achieved at the mean of \(\mu\), for any probability measure \(\mu\) with strong second moment on \(H\). We show that the validity of this property for measures on a normed space having support at three points with norm 1 and arbitrarily fixed positive weights implies the existence of an inner product that generates the norm.

MSC:

46C15 Characterizations of Hilbert spaces
PDFBibTeX XMLCite
Full Text: EuDML