Jordan-von Neumann theorem for Saworotnow’s generalized Hilbert space. (English) Zbl 0888.46038
The Jordan-von Neumann theorem proved in 1935 says that a normed space \(X\) for which the parallelogram identity holds
\[
|x+y|^2+|x-y|^2= 2|x|^2+2|y|^2, \qquad \forall x,y\in X
\]
can be made into an inner product space \((X,\langle ,\rangle)\) with \(\langle x,x\rangle=|x|^2\). In this paper the author works with proper \(H^*\)-algebras \(A\). For a left module \(X\) over \(A\) he defines a generalized normed space \((X,N)\), where \(N:X\to A\) is a norm-like mapping. The main theorem says that if \((X,N)\) is a normed \(A\)-module over \(A\), then \(N\) satisfies the parallelogram identity if and only if \(X\) is a Saworotnow’s pre-Hilbert module with respect to the generalized inner product \([x,y]\) such that \(N(x)^2= [x,x]\), \(\forall x\in X\).
Reviewer: Kh.N.Boyadzhiev (Ada)
MSC:
| 46K15 | Hilbert algebras |
Keywords:
Jordan-von Neumann theorem; parallelogram identity; inner product space; proper \(H^*\)-algebras; left module; Saworotnow’s pre-Hilbert module; generalized inner productReferences:
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