By Sever S. Dragomir

The aim of this publication is to provide a accomplished creation to a number of inequalities in internal Product areas that experience vital purposes in a variety of subject matters of up to date arithmetic corresponding to: Linear Operators concept, Partial Differential Equations, Non-linear research, Approximation thought, Optimisation idea, Numerical research, chance concept, information and different fields.

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**Example text**

21). Remark 13. 9). 2]. √ Corollary 5. Let x, y ∈ H such that x , y ≤ 2. 25) x y ≥ | x, y |2 2 − x 2 1 2 2− y 2 + | x, y | 1 − x 2 1 2 − y 2 + | x, y |2 . Proof. Follows by Theorem 12 on choosing a = x, y y, b = y, x x. We omit the details. 2. 3. A Refinement for Orthonormal Families. 9) [15, Theorem 3] (see also [8, Theorem] or [18, Theorem 3]): Theorem 13 (Dragomir, 1985). Let (H; ·, · ) be an inner product space over the real or complex number field K and {ei }i∈I an orthonormal family in I.

93) 2 i=1 pi yi − p i x i , yi i=1 n ≥ 2n 2 i=1 n p2k x2k k=1 2 2 n p2k y2k 2 − k=1 p2k x2k , y2k 12 k=1 n + n p2k−1 x2k−1 k=1 2 p2k−1 y2k−1 2 k=1 2 n − p2k−1 x2k−1 , y2k−1 12 (≥ 0) . k=1 Remark 8. 91) have been obtained by Dragomir and Mond in [2] for the case of scalar sequences x and y. S. DRAGOMIR and B. MOND, On the superadditivity and monotonicity of Schwarz’s inequality in inner product spaces, Contributions, Macedonian Acad. of Sci and Arts, 15(2) (1994), 5-22. S. DRAGOMIR and B.

77) 2 [ a, b − a b ] ≤ a, x x, b ≤ 1 x 2 2 [ a, b + a b ]. The main aim of the present section is to obtain similar results for families of orthonormal vectors in (H; ·, · ) , real or complex space, that are naturally connected with the celebrated Bessel inequality and improve the results of Busano, Richard and Kurepa. 2. A Generalisation for Orthonormal Families. We say that the finite family {ei }i∈I (I is finite) of vectors is orthonormal if ei , ej = 0 if i, j ∈ I with i = j and ei = 1 for each i ∈ I.