  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 Gruss' type inequalities 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 Pecaric inequalities for
finite sequences of vectors that are not necessarily orthogonal 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. Finally, some Gruss type inequalities for
n-tuples of vectors in inner product spaces and their natural applications for
the approximation of the discrete Fourier and Mellin transforms are given as
well.
