A characterization of real Hilbert spaces using the Bochnak complexification norm
Let \(E\) be a real Banach space. The algebraic complexification \(\tilde E\) of \(E\) is the tensor product \(E\otimes {\mathbb R}^2\), which identifies with \(E\oplus iE\). There are infinitely many ways to extend the norm on \(E\) to a complex complete norm on \(\tilde E\). A norm on \(\tilde E\) is a natural complexification norm for \(\| .\| _E\) if for every \(x,y\in E\), \(\| x\| _{\tilde E}=\| x\| _E\) and \(\| x+iy\| _{\tilde E}=\| x-iy\| _{\tilde E}\). The largest natural complexification norm on \(\tilde E\) is the projective tensor norm on \(E\otimes\ell_2^2\) which is called the Bochnak norm, denoted by \(\| .\| _B\). It was proved by \textit{G. A. Muñoz, Y. Sarantopoulos} and \textit{A. Tonge} [Stud. Math. 134, 1--33 (1999; Zbl 0945.46010)] that \[ \| x+iy\| _{B}\leq\inf_{t\in[0,2\pi]}\| x\cos t-y\sin t\| _E+\| x\sin t+y\cos t\| _E=B(x+iy). \] In the paper under review, the authors show that \(B\) coincides with the Bochnak norm (i.e., \(B\) is a natural complexification norm on \(\tilde E\)) if and only if \(E\) is a real Hilbert space.
scientific article; zbMATH DE number 1966462