Conditions for uniqueness of a projector with unit norm

Suppose that in a normed linear space B there exists a projector with unit norm onto a subspace D. A sufficient condition for this projector to be unique is the existence of a setM D ^* which is total on D, each functional in which attains its norm on the unit sphere in D and has a unique extension onto B with preservation of norm. As corollaries to this fact, we obtain a series of sufficient conditions for uniqueness (some of which were previously known) as well as a necessary and sufficient condition for uniqueness.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 45–49, July, 1977.The author thanks M. I. Kadets for useful discussions.