Let E be a smooth, strictly convex and reflexive Banach space, let Y * be a closed linear subspace of the dual space E * of E and let Π Y * be the generalized projection of E * onto Y * . Then, the mapping is called the generalized conditional expectation with respect to Y * , where J is the normalized duality mapping from E into E * . In this paper, we prove two results which are related to norm one linear projections and generalized conditional expectations in Banach spaces. The multivalued

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... erator J : E → E * is called the normalized duality mapping of E. From the Hahn-Banach theorem, Jx = ∅ for each x ∈ E. We know that E is smooth if and only if reflexive, strictly convex and smooth, then J is single-valued, one-to-one and onto. In this case, the normalized duality mapping J * from E * into E is the inverse of J, that is, J * = J −1 ; see [43] for more details. Let E be a smooth Banach space and let J be the normalized duality mapping of E. We define the function φ : for all x, y ∈ E. We also define the function φ * : for all x * , y * ∈ E * . It is easy to see that ( x − y ) 2 ≤ φ(x, y) for all x, y ∈ E. Thus, in particular, φ(x, y) ≥ 0 for all x, y ∈ E. We also know the following: additionally assumed to be strictly convex, then φ(x, y) = 0 ⇔ x = y. (2.3) Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. For an arbitrary point x of E, the set {z ∈ C : φ(z, x) = min y∈C φ(y, x)} is always nonempty and a singleton. Let us define the mapping Π C of E onto C by z = Π C x for every x ∈ E, i.e., φ(Π C x, x) = min y∈C φ(y, x) for every x ∈ E. Such Π C is called the generalized projection of E onto C; see Alber [1]. The following lemma is due to Alber [1] and Kamimura and Takahashi [31]. Lemma 2.1 ([1, 31]). Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E and let (x, z) ∈ E × C. Then, the following hold: From this lemma, we can prove the following lemma. Lemma 2.2. Let M be a closed linear subspace of a smooth, strictly convex and reflexive Banach space E and let (x, z) ∈ E × M Then, z = Π M x if and only if J(x) − J(z), m = 0 for any m ∈ M.