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.