A Hilbert space characterization using a pair of decompositions (Q2215823)

The author obtains the following interesting characterization of real Hilbert spaces: If \(X\) is a \(2n\)-dimensional Banach space, and if there are \(n\)-dimensinal subspaces \(X_1\), \(X_2\), \(X_3\) and \(X_4\) with pairwise zero intersection and satisfying that \(X=X_1\oplus_{\ell_2}X_2\) and \(X=X_3\oplus_{\ell_2}X_4\), then \(X\) is a Hilbert space. The author also obtains an isomorphic variant of the above result, and even an infinite-dimensional approach. The paper is well written and exhibits a new way to characterize real or complex Hilbert spaces from an isomorphic and isometric point of view.