Let E be a Banach space. One often wants to measure how far E is from being a Hilbert space. In this paper we define the skewness s(E) of a Banach space E, 0 ≤ s(E) ≤ 2, which describes the asymmetry of the norm. We show that s(E) = s(E*) for all Banach spaces E. Further, s(E) = 0 if and only if E is a (real) (Formula presnt) Hilbert space and s(E) = 2 if and only if E is quadrate, so 5(E) < 2 implies £ is reflexive. We discuss the computation of s(LP) and describe its asymptotic behavior near p = 1, 2 and ∞ Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.
ASJC Scopus subject areas
- Applied Mathematics