×

A note about inner products on Banach spaces. (English) Zbl 0651.46023

Two vectors x and y in a Banach space X are said to be orthogonal if \(\| x\| \leq \| x+\lambda y\|\) for every real \(\lambda\). It is shown that if X is reflexive, smooth, strictly convex and of at least three dimensions, then as long as X is not Hilbert space, there exists x,y,z which are mutually orthogonal but \(x+y\) is not orthogonal to z.
Reviewer: N.Ghoussoub

MSC:

46B20 Geometry and structure of normed linear spaces
46C99 Inner product spaces and their generalizations, Hilbert spaces
PDFBibTeX XMLCite