## Advances in Inequalities of the Schwarz, Grüss, and Bessel Type in Inner Product SpacesThe 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

1 | |

2 | |

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 |

245 | |

247 | |

### Common terms and phrases

77ie constant applications assumptions of Theorem Bessel's inequality Boas-Bellman Cauchy-Bunyakovsky-Schwarz inequality choose complex inner product complex number field Convex Functions deduce the desired desired inequality Er=i exp 2wimk F a finite fc=i fc=l following corollary following Griiss type following result holds following reverse follows by Theorem Griiss inequality Griiss type inequality Gruss Hilbert space Holder inequality i=l i=l inequality for real inequality in inner inner product space Integral Inequalities Jn Jn Lebesgue integrable Lemma Let H Math measure space Mellin transform n-tuples obtained omit the details orthornormal family orthornormal vectors PiXi proof follows prove the sharpness real inner product real numbers real or complex Remark reverse of Bessel's Reverses of Schwarz's RGMIA Res S.S. DRAGOMIR Schwarz's inequality second inequality sequences of vectors smaller constant space H Stieltjes integral sufficient condition t=i t=i triangle inequality vectors in H x,y G H