Abstract. The main aim of this monograph is to survey some 
recent results obtained by the author related to reverses of the 
Schwarz, triangle and Bessel inequalities. Some Grüss’ type in- 
equalities for orthonormal families of vectors in real or complex 
inner product spaces are presented as well. Generalizations of the 
Boas-Bellman, Bombieri, Selberg, Heilbronn and Peˇ cari´ c inequali- 
ties for finite sequences of vectors that are not necessarily orthogo- 
nal are also provided. Two extensions of the celebrated Ostrowski’s 
inequalities for sequences or real numbers and the generalization 
of Wagner’s inequality in inner product spaces are pointed out. Fi- 
nally, some Grüss type inequalities for n-tuples of vectors in inner 
product spaces and their natural applications for the approxima- 
tion of the discrete Fourier and Mellin transforms are given as well.

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

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