Download PDF by Sever S. Dragomir: Advances in Inequalities of the Schwarz, Triangle and

By Sever S. Dragomir

ISBN-10: 1594549036

ISBN-13: 9781594549038

The aim of this e-book is to provide a complete advent to numerous inequalities in internal Product areas that experience vital purposes in a number of themes of up to date arithmetic equivalent to: Linear Operators idea, Partial Differential Equations, Non-linear research, Approximation concept, Optimisation thought, Numerical research, chance conception, information and different fields.

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Additional info for Advances in Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner Product Spaces

Example text

P, ·, x, y) is superadditive as an index set mapping on Pf (N) . , σ (p, ·, x, y) is monotonic nondecreasing as an index set mapping on S+ (R) . 75) pi xi 2 pi yi i∈I 2 ≥ i∈I p i x i , yi i∈I for I ∈ Pf (N) \ {∅} , p ∈ S+ (R) and x, y ∈ S (H) . 75) if and only if there exists a scalar λ ∈ K such that xi = λyi , i ∈ I. 75). (1) Let αi ∈ R, xi , yi ∈ H, i ∈ {1, . . , n} . 76) xi n 2 i=1 2 yi x i , yi i=1 n ≥ − i=1 1 2 n xi 2 2 sin αi i=1 yi 2 2 n i=1 i=1 n + 1 2 n xi i=1 xi , yi sin2 αi − sin αi 2 cos2 αi yi 2 cos2 αi i=1 n xi , yi cos2 αi ≥ 0.

76) 2 x 2 , a, x ∈ H; in which the equality holds if and only if there exists a scalar λ ∈ K (R, C) so that x = λa. As noted by T. Precupanu in [29], independently of Buzano, U. 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.

We follow the proof in [15]. 10) x 2 − | x, e |2 y 2 − | y, e |2 ≥ | x, y − x, e e, y |2 . 12) y − | x, e e, y |)2 ( x ≥ x 2 − | x, e |2 y 2 − | y, e |2 for any x, y, e ∈ H with e = 1. 10). 10) is obvious. Corollary 4 (Dragomir, 1985). 14) x y ≥ 2 | x, e e, y | . Remark 11. Assume that A : H → H is a bounded linear operator on H. 15) Ay ≥ | x, Ay − x, e e, Ay | + | x, e e, Ay | ≥ | x, Ay | for any y ∈ H. 16) Ay = sup {| x, Ay − x, e e, Ay | + | x, e e, Ay |} x =1 for any y ∈ H. 17) A = sup {| x, Ay − x, e e, Ay | + | x, e e, Ay |} y =1, x =1 for any e ∈ H, e = 1, a representation that has been obtained in [15, Eq.

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Advances in Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner Product Spaces by Sever S. Dragomir


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