Advances in Inequalities of the Schwarz, Grüss, and Bessel Type in Inner Product Spaces

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Nova Publishers, 2005 - Mathematics - 249 pages
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The theory of Hilbert spaces plays a central role in contemporary mathematics with numerous applications for Linear Operators, Partial Differential Equations, in Nonlinear Analysis, Approximation Theory, Optimisation Theory, Numerical Analysis, Probability Theory, Statistics and other fields. The Schwarz, triangle, Bessel, Gram and most recently, Gr ss type inequalities have been frequently used as powerful tools in obtaining bounds or estimating the errors for various approximation formulae occurring in the domains mentioned above. Therefore, any new advancement related to these fundamental facts will have a flow of important consequences in the mathematical fields where these inequalities have been used before.
 

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Contents

Reverses for the Schwarz Inequality
1
2 An Additive Reverse of the Schwarz Inequality
2
3 A Generalisation of the Cassels Inequality
9
4 Quadratic Reverses of Schwarzs Inequality
18
5 More Reverses of Schwarzs Inequality
27
Bibliography
35
CHAPTER 2
37
2 Griiss Inequality in Inner Product Spaces
41
2 Bombieri Type Inequalities
134
3 Pecaric Type Inequalities
148
Bibliography
159
Some Griiss Type Inequalities for nTuples of Vectors
161
2 The Version for Norms
163
3 The Version for InnerProducts
177
4 More Griiss Type Inequalities
184
5 Some Inequalities for Forward Difference
193

3 Companions of Griiss Inequality
52
4 Other Griiss Type Inequalities
61
Bibliography
69
CHAPTER 3 Reverses of Bessels Inequality
71
2 Reverses of Bessels Inequality
72
3 Another Reverse for Bessels Inequality
84
4 More Reverses of Bessels Inequality
97
5 General Reverses of Bessels Inequality
109
Bibliography
121
Other Inequalities in Inner Product Spaces
123
Generalisations of Bessels Inequality
125
6 Bounds for a Pair of nTuples of Vectors
201
Bibliography
215
CHAPTER 6 Other Inequalities in Inner Product Spaces
217
2 Another Ostrowski Type Inequality
221
3 The Wagner Inequality in Inner Product Spaces
224
5 Other Bombieri Type Inequalities
230
6 Some PreGriiss Inequalities
235
Bibliography
245
Index
247
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