extracta mathematicae Vol. 15, N´ um. 2, 391 – 420 (2000) Sobczyk’s Theorems from A to B F´ elix Cabello S´ anchez ∗ , Jes´ us M.F. Castillo ∗ , David Yost Dpto. de Matem´aticas, Univ. de Extremadura, 06071-Badajoz, Spain e-mail: fcabello@unex.es, castillo@unex.es, dyost@unex.es AMS Subject Class. (1991): 46B03, 46B20, 46M10 1. Sobczyk’s theorem and how to prove it Sobczyk’s theorem is usually stated as: Every copy of c 0 inside a separable Banach space is complemented by a projection with norm at most 2. Never- theless, our understanding is not complete until we also recall: and c 0 is not complemented in ℓ ∞ . Now the limits of the phenomenon are set: although c 0 is complemented in separable superspaces, it is not necessarily complemented in a nonseparable superspace, such as ℓ ∞ . The history of complemented and uncomplemented subspaces of Banach spaces is traced back in another article of this volume [48]. It is probably worth mentioning that it starts with two propositions: Every closed subspace of a Hilbert space is complemented by a norm one projection and ℓ 1 contains uncomplemented subspaces. The first result easily follows by proving that the metric projection onto a closed subspace acts linearly; the second result holds since the kernels of quotient maps ℓ 1 → X are necessarily uncomplemented when X has not been previously chosen a subspace of ℓ 1 (and recalling that all separable Banach spaces are quotients of ℓ 1 ). A more interesting question for us is: why should one suspect that c 0 is complemented inside separable superspaces? A previous result in this dir- ection had been proved by Phillips [47]: Every copy of ℓ ∞ inside a Banach space is complemented by a norm one projection. In other words, the spaces ℓ ∞ (Γ) are injective. Since it was well known (and can be easily proved) that every Banach space is isometric to a subspace of some ℓ ∞ (Γ) it is clear that the injective spaces are precisely the ℓ ∞ (Γ) spaces and their complemented subspaces. In order to determine all the injective spaces the story starts with * Supported in part by DGCYT project PB 97-0377. 391