Functionals generated dually by minimal projectors, and criteria for Banach spaces to be Hilbert
Let B be a real Banach space and \(0\neq x\in B\). Assume there is a unique hyperplane \(D_ x\) such that for each hyperplane D in B with \(x\not\in D\| P(D,x)\| >\| P(D_ x,x)\|\), where P(D,x) is the projection onto D along sp(x). For any \(z\in B\) set \(\phi_ x(z)=a\| x\|\), where \(z=ax+P(D_ x,x)z.\) Theorem 1. Let B be a reflexive Banach space such that for each hyperplane D in \(B^*\) there exists a unique minimal projection onto D. Then the functional \(\phi_ x\) is well-defined for each nonzero \(x\in B.\) Theorem 2. Let B be a uniformly smooth and strictly convex Banach space (Note that such space satisfies the hypothesis of theorem 1.) Then B is isometrically isomorphic to a Hilbert space iff \(\| x\| \phi_ x(y)=\| y\| \phi_ y(x)\) for all nonzero x,y\(\in B\).
uniformly smooth and strictly convex Banach space
scientific article; zbMATH DE number 4004906