By Jan Mycielski, Pavel Pudlak, Alan S. Stern

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Proof. By the use of Theorem 18, we have: 1 1 [(y, x)s + (y, x)i ] = [(y, x)s − (−y, x)s ] 2 2 1 = lim Re J˜ (x + ty) , y + lim+ J˜ (x − ty) , y t→0 2 t→0+ J˜ (x + ty) + J˜ (x − ty) = lim+ Re ,y t→0 2 (y, x)g = for all x, y ∈ X, and the statement is proved. Another result of this type which can be proved with the help of Theorem 19 is the following. 1. DEFINITION AND THE MAIN PROPERTIES 49 Theorem 26. Let (X, · ) be a normed space. Then for every J˜ a section of the normalised dual mapping J, we have the representation: (y, x)g = lim+ Re t→0 J˜ (x + ty) − J˜ (x − ty) ,x 2t for all x, y ∈ X.

Passing at limit after t, t → 0+ , we have Re [y, x] (∨+ · ) (x) · y = x for all x, y ∈ X, x = 0. , X is smooth. Conversely, let us assume that the norm · is Gˆateaux differentiable on X {0}. 2) we can write: x + ty − x Re [y, x + ty] x + 2ty − x + ty ≤ ≤ t x + ty t 2. , 1 ( x + ty − x ) x + ty t 23 ≤ Re [y, x + ty] 1 ( x + 2ty − x + ty ) x + ty t for all t > 0 and x, y ∈ X. Taking t → 0+ , we obtain ≤ lim Re [y, x + ty] = x (∨+ · ) (x) · y t→0+ because a simple computation shows that: x + 2ty − x + ty = (∨+ · ) (x) · y.

S 27 for all x, y ∈ X, x = 0. 6) exists, it follows that (∨+ · ) (x) · y = (∨− · ) (x) · y for all x, y ∈ X, x = 0, which shows that X is smooth. p. is continuous. We will omit the details. Bibliography [1] G. LUMER, Semi-inner product spaces, Trans. Amer. Math. , 100 (1961), 29-43. R. GILES, Classes of semi-inner product spaces, Trans. Amer. Math. , 116 (1967), 436-446. L. PAPINI, Un’ asservatione sui prodotti semi-scalari negli spasi di Banach, Boll. Un. Mat. , 6 (1969), 684-689. M. MILICI norm´es, Mat.