By Sever S. Dragomir, Susan Boriotti, Donna Dennis

Semi-inner items, that may be certainly outlined ordinarily Banach areas over the genuine or advanced quantity box, play a massive function in describing the geometric houses of those areas. This new publication dedicates 17 chapters to the examine of semi-inner items and its functions. The bibliography on the finish of every bankruptcy incorporates a checklist of the papers stated within the bankruptcy. The reader might locate additional information at the topic by way of consulting the checklist of papers supplied on the finish of the paintings. The e-book is meant to be used by way of either researchers and postgraduate scholars drawn to practical research. It additionally presents important instruments to mathematicians utilizing useful research in different domain names similar to: linear and non-linear operator concept, optimization thought, video game concept or different comparable fields.

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**Semi-Inner Products and Applications **

Semi-inner items, that may be certainly outlined often Banach areas over the true or complicated quantity box, play an incredible function in describing the geometric homes of those areas. This new booklet dedicates 17 chapters to the research of semi-inner items and its functions. The bibliography on the finish of every bankruptcy includes a record of the papers mentioned within the bankruptcy.

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**Extra info for Semi-Inner Products and Applications **

**Example text**

Let J˜ be a section of the normalised dual mapping J. 7) x ( x + ty − x ) ˜ y , ≥ Re Jx, t for all x, y ∈ X and t > 0. 2 x 2. THE CONNECTION BETWEEN (·, ·)s(i) AND THE DUALITY MAPPING 39 On the other hand, for t = 0 and x + ty = 0, we get x + ty − x t x + ty = J˜ (x + ty) , x + ty − x − x x + ty = t x + ty t x + ty 2 Re J˜ (x + ty) , x + t Re J˜ (x + ty) , y − x = x + ty x + ty t x + ty Re J˜ (x + ty) , y ≤ x + ty because Re J˜ (x + ty) , x ≤ x x + ty . 8) x + ty ( x + ty − x ) ≤ Re J˜ (x + ty) , y t for all t > 0 and x, y ∈ X.

3) t→0 for all x, y ∈ X, x = 0. 2), we also have: ( x + sy − x ) ( x + ty − x ) ≤ Re [y, x] ≤ x s t − where s < 0 and t < 0. 4) Re [y, x] = x (∨ · ) (x) · y for all x, y ∈ X, x = 0. 4), we obtain the continuity of [·, ·] in the sense of Definition 7. 24 2. SEMI-INNER PRODUCTS IN THE SENSE OF LUMER-GILES Further on, we will state a result due to Nath [15] containing a characterisation of strictly convex spaces in terms of semi-inner product in Lumer-Giles’ sense. Theorem 12. p. which generates its norm.

M. MILICI 39 (1987), 325-334. M. MILICI mationes dans des espaces norm´es, Pub. L’Inst. Mat. (Belgrade), 48(62) (1990), 110-118. M. MILICI norm´e et son utilisation, Pub. L’Inst. Mat. (Belgrade), 42 (1987), 63-70. ˇ C, ´ Sur les espaces semi-lisses, Mat. Vesnik, 36 (1984), 222-226. M. MILICI [6] G. GODINI, Geometrical properties of a class of Banach spaces including the spaces c0 and C p (1 ≤ p < ∞) , Math. , 243(3) (1979), 197-212. 55 CHAPTER 5 (Q) and (SQ)-Inner Product Spaces 1. (Q) – Inner Product Spaces In the paper [1] (see also [2] and [3]) the author introduced the following generalisation of inner products in a real linear space that extends this concept in a different manner than the extensions due to Lumer-Giles, Tapia or Miliˇci´c.