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Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. (English) Zbl 0757.46034

In a Hilbert space \(H\) the norm satisfies the so-called polarization identity: \[ \| x+y\|^ 2=\| x\|^ 2+2 \text{Re}\langle x,y\rangle+\| y\|^ 2. \] A number of authors (e.g. Reich, Kay, Bynum and Drew, Ishikawa, Prus and Smarzewski) have derived inequalities which generalize (in one way or another) the polarization identity to \(L^ p\)-spaces, or, more generally, uniformly convex or uniformly smooth Banach spaces. In this paper other such inequalities are proved which not only improve on several of the above quoted results, but which actually characterize uniformly convex and uniformly smooth spaces.

MSC:

46B20 Geometry and structure of normed linear spaces
47J20 Variational and other types of inequalities involving nonlinear operators (general)
46C99 Inner product spaces and their generalizations, Hilbert spaces
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