Let be an inner product space and . If and , then
Equality holds if and only if , , or . In fact, if (1) holds and is a normed linear space, then is an inner product space.
If is a normed linear space and and then
Equality holds if and only if , or . The constant is best possible. For example, let be the set of ordered pairs of real numbers, with norm of equal to Let and where is a small positive number. After a bit of routine calculation, it is easily seen that the best possible constant is .
The inequality (2) has been generalized in the case where is a normed linear space over the reals. In that case one can show:
where if and if . The case is the Dunkl and Williams inequality.
If is a normed linear space and then (3) holds with if and only if is an inner product space.
The inequality (2) can be improved slightly to get:
Equality holds in (4) if and only if and span an in the underlying real vector space with and (or ) as the vertices of the unit parallelogram.
- 1 C.F. Dunkl, K.S. Williams, A simple norm inequality. Amer. Math. Monthly, 71 (1) (1964) 53-54.
- 2 W.A. Kirk, M.F. Smiley, Another characterization of inner product spaces, Amer. Math. Monthly, 71, (1964), 890-891.
- 3 A.M. Alrashed, Norm inequalities and Characterizations of Inner Product Spaces, J. Math. Anal. Appl. 176, (2) (1993), 587-593.
- 4 J.L. Massera, J.J. Schäffer, Linear differential equations and functional analysis, Annals of Math. 67 (2)(1958), 517-573. (on page 538)
- 5 L.M. Kelly, The Massera-Schäeffer equality, Amer. Math. Monthly, 73, (1966) 1102-1103.